UBC Wiki - User contributions [en]

Course:MATH110/Archive/2010-2011/003/Teams/Neuchatel/Homework 12

2011-01-28T09:41:38Z

AviHarry:

{| class="wikitable" width="800px"
!style="background: #777777; color: #D0D0D0; text-align: left; padding:3px;" width="100%"|Team Homework 12
|-
|style="background: #FFFFFF; padding:12px;"|
Starting with the function
:<math>P(t) = \frac{1}{1+e^{-t}}</math>
Your goal is to modify the function so that we can use it to model a real-life problem. We want to be able to control the following things:
* Change the height of the horizontal asymptote on the right, we'll denote it by <math>K</math>.
* Change the <math>y</math>-intercept to any number between <math>0</math> and <math>K</math>
BONUS (just the point below, not what comes after)
* Change the slope of the curved part. Find a way so that the slope can go from very close to zero to almost vertical. (If you graph it, it should be quite clear).

Once you've played with the function enough, try to find an application of the graph to model something. It can be anything which starts at a value and then goes to another one (think for a population, it goes from 0 to it's carrying capacity). Explain what you are modelling and how you decide to attribute a numerical value to each of the 2 or 3 parameters that you researched just above. Then use the model to make a prediction. For example, if your model is suppose to describe a population for which you have its initial population and carrying capacity (potentially its rate of increase if you solved the bonus part), then use that data to make a prediction for the population in 20 years, or use the model to predict when will the population reach 95% of its carrying capacity).

When doing this last part, explain well where you're taking your data from (real data or imagined data), what it is that you're modelling and how you are doing the math to answer a predictive question.
|}

In order to change the height of the horizontal asymptote on the right, and use K. 
 We put K at 1/("1"+e^-t), so it become 1/(K+e^-t). 
 In order to makes it heiger, K should be less than 1, also can't smaller than 0.
 So, 0<K<1. 
 If want it make right side of horizontal asymptote lower, than K should be bigger than 1.
 So, its 1<K<∞. 

To change the y-intercept between any number 0 and k we changed the value of x from 1 to 0.1 where:
:<math>P(t) = \frac{1}{x+e^{-t}}</math>
The smaller the value of x the larger the y intercept so we figured that having x= 0.1 would give us a good starting population for our bacteria growth.

To see our new y intercept simply look at the comparisons of the original equation and ours.

[[File:Group1graph.gif]]

So so far we have the function: <math>P(t) = \frac{1}{0.1+e^{-t}}</math>

Finally, now that we have come up with a model to describe the population of bacteria as a function of time, we simply substitute a time value in which we are interested into the equation. At this juncture, we are interested to know what the bacteria population after <math> {\frac{1}{2}}</math> year will be. Hence,
 
:<math>P({\frac{1}{2}})= \frac{1}{{\frac{?}{?}}+?^{-({\frac{1}{2}})}} \approx ?</math>
 
Therefore, the population of bacteria, according to the modified model, is approximately ?(unit) after 1/2 year.

To change the slope of the curve so that the slope can go from close to zero to almost vertical we must simply replace the value "e" to a larger value such as "50". This will distinctively change the slope of the curve to a near straight line approaching zero.

[[File:B532d798ec926e169adf3c1a55f0cdf0.png]]

[[File:MSP1195719e4c9eb213d60ea000034ge6d8e17465a0c.gif]]
‎‎[[File:MSP1195919e4c9eb213d60ea000014f69567e43gbac8.gif]]
[[User:AviHarry|AviHarry]]

Course:MATH110/Archive/2010-2011/003/Teams/Neuchatel/Homework 12

2011-01-28T09:17:50Z

AviHarry:

Course:MATH110/Archive/2010-2011/003/Teams/Neuchatel/Homework 12

2011-01-28T09:02:34Z

AviHarry:

File:B532d798ec926e169adf3c1a55f0cdf0.png

2011-01-28T07:27:56Z

AviHarry: uploaded a new version of "File:B532d798ec926e169adf3c1a55f0cdf0.png"

File:MSP1195719e4c9eb213d60ea000034ge6d8e17465a0c.gif

2011-01-28T07:18:18Z

AviHarry:

File:MSP1195919e4c9eb213d60ea000014f69567e43gbac8.gif

2011-01-28T07:17:43Z

AviHarry:

Course:MATH110/Archive/2010-2011/003/Teams/Neuchatel/Homework 12

2011-01-28T03:06:14Z

AviHarry:

{| class="wikitable" width="800px"
!style="background: #777777; color: #D0D0D0; text-align: left; padding:3px;" width="100%"|Team Homework 12
|-
|style="background: #FFFFFF; padding:12px;"|
Starting with the function
:<math>P(t) = \frac{1}{1+e^{-t}}</math>
Your goal is to modify the function so that we can use it to model a real-life problem. We want to be able to control the following things:
* Change the height of the horizontal asymptote on the right, we'll denote it by <math>K</math>.
* Change the <math>y</math>-intercept to any number between <math>0</math> and <math>K</math>
BONUS (just the point below, not what comes after)
* Change the slope of the curved part. Find a way so that the slope can go from very close to zero to almost vertical. (If you graph it, it should be quite clear).

Once you've played with the function enough, try to find an application of the graph to model something. It can be anything which starts at a value and then goes to another one (think for a population, it goes from 0 to it's carrying capacity). Explain what you are modelling and how you decide to attribute a numerical value to each of the 2 or 3 parameters that you researched just above. Then use the model to make a prediction. For example, if your model is suppose to describe a population for which you have its initial population and carrying capacity (potentially its rate of increase if you solved the bonus part), then use that data to make a prediction for the population in 20 years, or use the model to predict when will the population reach 95% of its carrying capacity).

When doing this last part, explain well where you're taking your data from (real data or imagined data), what it is that you're modelling and how you are doing the math to answer a predictive question.
|}

1.Change the height of the horizontal asymptote on the right, we'll denote it by K. 
 So in order to change the height of the horizontal asymptote on the right, and use K. 
 We put K at 1/("1"+e^-t), so it become 1/(K+e^-t). 
 In order to makes it heiger, K should be less than 1, also can't smaller than 0.
 So, 0<K<1. 
 If want it make right side of horizontal asymptote lower, than K should be bigger than 1.
 So, its 1<K<∞. 

2.To change the y-intercept between any number 0 and k we changed the value of x from 1 to 0.1 where:
:<math>P(t) = \frac{1}{x+e^{-t}}</math>
The smaller the value of x the larger the y intercept so we figured that having x= 0.1 would give us a good starting population for our bacteria growth.

To see our new y intercept simply look at the comparisons of the original equation and ours.

[[File:Group1graph.gif]]

So so far we have the function: <math>P(t) = \frac{1}{0.1+e^{-t}}</math>

Finally, now that we have come up with a model to describe the population of bacteria as a function of time, we simply substitute a time value in which we are interested into the equation. At this juncture, we are interested to know what the bacteria population after <math> {\frac{1}{2}}</math> year will be. Hence,
 
:<math>P({\frac{1}{2}})= \frac{1}{{\frac{?}{?}}+?^{-({\frac{1}{2}})}} \approx ?</math>
 
Therefore, the population of bacteria, according to the modified model, is approximately ?(unit) after 1/2 year.

3. Change the slope of the curved part. Find a way so that the slope can go from very close to zero to almost vertical. (If you graph it, it should be quite clear).

Thread:User talk:AviHarry/Group stuff/reply

2011-01-28T02:56:44Z

AviHarry: Reply to Group stuff

No Worries, I'll see what I can do. ; )

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T10:08:03Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[User:AviHarry|AviHarry]]

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle. The Cosine length will in this case, always be the adjacent side of the triangle and the Sine length will always be the opposite side of the triangle.

Let's hypothetically make the cosine length sqrt3/2. We now have enough information to able to convert the Pythagoras Theorem to get the value for the opposite sine length.

(c^2 = a^2 + b^2)

(1^2 = a^2 + sqrt3/2^2)

Since we don't have one of the legs of the theorem we must arrange the formula so that we can isolate the piece to the puzzle we don't know.

(c^2 - b^2 = a^2)

(1^2 - sqrt3/2^2 = a^2)

(1 - 3/4 = a^2)

(0.25 = a^2)

(0.5 = a)

This method is quite simple and can also be used to solve the Cosine length, as long as we have one of the legs in the triangle.

[[Image:Pi Circle Yo.png]]

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T10:07:13Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle. The Cosine length will in this case, always be the adjacent side of the triangle and the Sine length will always be the opposite side of the triangle.

Let's hypothetically make the cosine length sqrt3/2. We now have enough information to able to convert the Pythagoras Theorem to get the value for the opposite sine length.

(c^2 = a^2 + b^2)

(1^2 = a^2 + sqrt3/2^2)

Since we don't have one of the legs of the theorem we must arrange the formula so that we can isolate the piece to the puzzle we don't know.

(c^2 - b^2 = a^2)

(1^2 - sqrt3/2^2 = a^2)

(1 - 3/4 = a^2)

(0.25 = a^2)

(0.5 = a)

This method is quite simple and can also be used to solve the Cosine length, as long as we have one of the legs in the triangle.

[[Image:Pi Circle Yo.png]]

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T10:04:36Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle. The Cosine length will in this case, always be the adjacent side of the triangle and the Sine length will always be the opposite side of the triangle.

Let's hypothetically make the cosine length sqrt3/2. We now have enough information to able to convert the Pythagoras Theorem to get the value for the opposite sine length.

(c^2 = a^2 + b^2)

(1^2 = a^2 + sqrt3/2^2)

(c^2 - b^2 = a^2)

(1^2 - sqrt3/2^2 = a^2)

(1 - 3/4 = a^2)

(0.25 = a^2)

(0.5 = a)

[[Image:Pi Circle Yo.png]]

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T10:04:07Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle. The Cosine length will in this case, always be the adjacent side of the triangle and the Sine length will always be the opposite side of the triangle.

Let's hypothetically make the cosine length sqrt3/2. We now have enough information to able to convert the Pythagoras Theorem to get the value for the opposite sine length.

(c^2 = a^2 + b^2)

(1^2 = a^2 + sqrt3/2^2)

(c^2 - b^2 = a^2)

(1^2 - sqrt3/2^2 = a^2)

(1 - 3/4 = a^2)

(0.25 = a^2)

(0.5 = a)

[[Image:Pi Circle Yo.png.jpg]]

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T10:02:16Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle. The Cosine length will in this case, always be the adjacent side of the triangle and the Sine length will always be the opposite side of the triangle.

Let's hypothetically make the cosine length sqrt3/2. We now have enough information to able to convert the Pythagoras Theorem to get the value for the opposite sine length.

(c^2 = a^2 + b^2)

(1^2 = a^2 + sqrt3/2^2)

(c^2 - b^2 = a^2)

(1^2 - sqrt3/2^2 = a^2)

(1 - 3/4 = a^2)

(0.25 = a^2)

(0.5 = a)

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T10:01:01Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle. The Cosine length will in this case, always be the adjacent side of the triangle and the Sine length will always be the opposite side of the triangle.

Let's hypothetically make the cosine length sqrt3/2. We now have enough information to able to convert the Pythagoras Theorem to get the value for the opposite sine length.

(c^2 = a^2 + b^2) w

(1^2 = a^2 + sqrt3/2^2)

(c^2 - b^2 = a^2)

(1^2 - sqrt3/2^2 = a^2)

(1 - 3/4 = a^2)

(0.25 = a^2)

(0.5 = a)

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T10:00:33Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle. The Cosine length will in this case, always be the adjacent side of the triangle and the Sine length will always be the opposite side of the triangle.

Let's hypothetically make the cosine length sqrt3/2. We now have enough information to able to convert the Pythagoras Theorem to get the value for the opposite sine length.

(c^2 = a^2 + b^2)

(1^2 = a^2 + sqrt3/2^2)

(c^2 - b^2 = a^2)

(1^2 - sqrt3/2^2 = a^2)

(1 - 3/4 = a^2)

(0.25 = a^2)

(0.5 = a)

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T09:57:20Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle. The Cosine length will in this case, always be the adjacent side of the triangle and the Sine length will always be the opposite side of the triangle.

Let's hypothetically make the cosine length sqrt3/2. We now have enough information to able to convert the Pythagoras Theorem to get the value for the opposite sine length.

(c^2 = a^2 + b^2)

(1^2 = a^2 + sqrt3/2^2)

(c^2 - b^2 = a^2)

(1^2 - sqrt3/2^2 = a^2)

(1 - 3/4 = a^2)

(0.25 = a^2)

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T09:41:51Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle. The Cosine length will in this case, always be the adjacent side of the triangle and the Sine length will always be the opposite side of the triangle.

Let's hypothetically make the cosine length sqrt3/2. We now have enough information to able to convert the Pythagoras Theorem to get the value for the opposite sine length.

(c^2 = a^2 + b^2)

(1^2 = a^2 + sqrt3/2^2)

(c^2 - b^2 = a^2)

(1^2 - sqrt3/2^2 = a^2)

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T09:41:33Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle. The Cosine length will in this case, always be the adjacent side of the triangle and the Sine length will always be the opposite side of the triangle.

Let's hypothetically make the cosine length sqrt3/2. We now have enough information to able to convert the Pythagoras Theorem to get the value for the opposite sine length.

(c^2 = a^2 + b^2)
(1^2 = a^2 + sqrt3/2^2)

(c^2 - b^2 = a^2)
(1^2 - sqrt3/2^2 = a^2)

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T09:41:09Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle. The Cosine length will in this case, always be the adjacent side of the triangle and the Sine length will always be the opposite side of the triangle.

Let's hypothetically make the cosine length sqrt3/2. We now have enough information to able to convert the Pythagoras Theorem to get the value for the opposite sine length.

(c^2 = a^2 + b^2)
(1^2 = a^2 + sqrt3/2^2)

c^2 - b^2 = a^2
1^2 - sqrt3/2^2 = a^2

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T09:39:25Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle. The Cosine length will in this case, always be the adjacent side of the triangle and the Sine length will always be the opposite side of the triangle.

Let's hypothetically make the cosine length sqrt3/2. We now have enough information to able to convert the Pythagoras Theorem to get the value for the opposite sine length.

c^2 = a^2 + b^2
1^2 = a^2 + sqrt3/2^2

c^2 - b^2 = a^2
1^2 - sqrt3/2^2 = a^2

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T09:38:00Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle. The Cosine length will in this case, always be the adjacent side of the triangle and the Sine length will always be the opposite side of the triangle.

Let's hypothetically make the cosine length sqrt3/2. We now have enough information to able to convert the Pythagoras Theorem to get the value for the opposite sine length.

c^2 = a^2 + b^2 1^2 = a^2 + sqrt3/2^2

c^2 - b^2 = a^2 1^2 - sqrt3/2^2 = a^2

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T09:36:19Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle. The Cosine length will in this case, always be the adjacent side of the triangle and the Sine length will always be the opposite side of the triangle.

Let's hypothetically make the cosine length sqrt3/2. We now have enough information to able to convert the Pythagoras Theorem to get the value for the opposite sine length.

c^2 = a^2 + b^2 = c^2 - b^2 = a^2
1^2 = a^2 + sqrt3/2^2 = 1^2 - sqrt3/2^2 = a^2

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T09:21:36Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

Here we have a trigonometric circle with a right angle triangle labeled accordingly. The hypotenuse, also being the radius is a constant 1 around the circle. To calculate the length given the cosine length, or the reverse we simply apply the Pythagorean theorem within the triangle in the circle.

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T09:17:39Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem.

[[Image:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg]]

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T09:16:48Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem. This needs to be fleshed out with examples.

[[Image:]]

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

File:Defining-Trigonometric-functions-from-a-right-angle-triangle-inside-a-circle.jpg

2010-12-03T09:15:39Z

AviHarry:

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T09:13:06Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem. This needs to be fleshed out with examples.

[[Image:Pi Circle Yo.png]]

sacdsaf

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T09:12:55Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem. This needs to be fleshed out with examples.

[[Image:Pi Circle Yo.png]]

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T09:12:00Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem. This needs to be fleshed out with examples.

[[Image:Untitled.png]]

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Basic Skills Project

2010-12-03T09:11:01Z

AviHarry: /* What Does This Have To Do With the Trigonometric Circle? */

==The Pythagorean Theorem==

=== What Is It? ===
'''History:'''

Developed by a man named Pythagoras was born in the late 6th century B.C. on the island of Samos Greece. He proved that for any right hand triangle, the two shorter sides squared and added together exactly equal the squared amount of the longest side. This looks like:
a^2 + b^2 = c^2

'''Pythagorean Basics:'''

1) Only works for triangles

2) Only works for triangles with a right hand angle (90 degrees)

[[File:500px-Pythagorean.svg.png]]

This diagram demonstrates the use of pythagorean theorem to calculate the area of the square on the hypothenus- c. This is possible by summing the squared areas of the smaller sides - a and b

Area of a triangle -> (1/2)(base)(height)

Christa Bicego

=== How Do We Know It's True? ===
[[User:BenJeffery|BenJeffery]]

There are many proofs of the theorem. One of the most straightforward is the following:

Take four right-angled triangles and connect them into a square as shown, so that they hypotenuses form an inner square as follows:

[[File:Proof_2.png]]

We can see that the length of each side of the outer square is <math>a+b</math>, and that the area must be <math>(a+b)^2</math>.

The pieces of the square are the inner square and our four triangles.

The area of each triangle can be given as <math>(1/2)ab</math>.

Meanwhile, the inner square has side length of <math>c</math>, and area of <math>c^2</math>.

From this, we can see that:

<math>(a+b)^2 = 2ab + c^2</math>

Expanding, we see that:

<math>a^2+2ab+b^2 = 2ab +c^2</math>

Balance the equation by subtracting <math>2ab</math> to get:

<math>a^2+b^2 = c^2</math>

This is, of course, the formula for the theorem!

If you prefer a more straight-forward visual proof, watch the following video. It clearly shows that the water in the two smaller squares (formed by squaring the length of the smaller sides) fits perfectly into the square formed from the hypotenuse.

{{#ev:youtube | CAkMUdeB06o | 400}}

=== How Do You Use It? ===

Explain how to apply the theorem to simple, as well as more complex examples. Verbal as well as mathematical examples are good.

Real-life application

There are many different real-life situations which involve the Pythagorean theorem.

For example, let’s assume two friends Fred and Matt want to meet at a specific point (ex: shopping mall). Fred is 8 Km away from the mall and Matt is 7 km away (assume they are at a right angle from each other in comparison to the mall), how do we find the distance between the two of them? By using the Pythagorean theorem we are able to determine the distance between them.

<math>5^2 + 7^2=74 </math>

<math>\sqrt {74} = 8.6 km </math>

8.6 is the distance that Fred and Matt are from each other

Another example of a real-life application of the Pythagorean theorem would be on how to figure out the necessary height of an object. Lets suppose you have a 15 meter high wall. You want to find out how long a ladder has to be if it is 5 meters from the base of the wall. Implementing the Pythagorean theorem one is able to solve this issue with little to no issues. By simply substituting 5 and 15 for a and b (in the formula) we can obtain the result which will give us the necessary height of the ladder.
Therefore

<math> 5^2+15^2=250</math>

<math> \sqrt {250}= 15.8m </math>

15.8 is the exact height the ladder must be to fit the given parameters of the wall.

A third way to apply the pythagorean theorem is in computer games. As awkward as this game may be, it shows the integration of mathematics into the gaming world.
{{#ev:youtube | jQ2QDnpImcg | 400}}

[[User:MarcoGasparian|MarcoGasparian]]

=== What Can We Do With It?===

The Pythagoras Theorem is defined by Google definitions as the formula used to find an unknown length of a right angled triangle, the two sides that meet to form the right angle equal to the long side connecting them. One may wonder how the Pythagoras Theorem came to be, In 500 BC a Scholar by the name of Pythagoras studying the ratios between the different lengths of the sides of a triangle. When he began to take a closer look at his calculations he came to the conclusion that the two shorter sides of the triangles squared and added together gave you the length of the longest side. From this he created the equation a2+b2=c2, c of course being the hypotenuse, a and b being the two shorter sides.
The Pythagorean theorem is useful in many situations where you need to find one or more lengths of a right angle triangle. Pythagorean Theorem can be used to find the length of the corner of a ceiling or floor while laying tiles. It does come to use in the real world as well as being used in mathematical equations.
Once you know the fundamentals of Pythagorean Theorem you can broaden the rage of questions that you do and you can solve much more difficult questions with multiple steps involved. A couple examples of other things you could use the thermo for would be Euclidean distance in various coordinate systems, Pythagorean trigonometric identities, along with complex arithmetic and various other questions.

Example:

c2= a2+b2 c2= 62 + 82 c2 = 36 + 64 c2 = 100 c = 10

http://www.youtube.com/watch?v=ku4rEwRxZOc

You can use the Pythagorean thermo in many different ways, one being that you can simply solve for one of the side (usually the hypothenuse) of a right angled triangle. This is fairly basic, if you watch the youtube video attached it is demonstrated.

The formual used is :

a^2+b^2 = c^2

Once you know this and have practiced it on triangles you can now bring this information to use in real like scenarios.
One common one would be a question about the length of a shadow from a street lamp or building. In order to complete one of these problems you must define what is a, b and c and then continue to follow the therum.

http://www.youtube.com/watch?v=kBw_i6tlQfU

If you then watch the video above you can see how the questions may progressively become harder but remain simple if you label the different variables and decide which you need to solve for. If you proceed to the pdf attachment there you can see many examples of easy as well as difficult pythagorean questions.

http://bodmas.org/bnd/docs/y08_maths_worksheet_pythagoras.pdf

=== What Does This Have To Do With the Trigonometric Circle?===

Given that the radius of a trigonometric circle is one, we can easily find out the sine length given the cosine length, or the reverse, using the Pythagorean theorem. This needs to be fleshed out with examples.

[[Image:]]

==Proposition for Basic Skills Project==

We propose to put up information about the Pythagorean Theorem. We could put up some proofs, various applications of the theorem, how to use it, and other such things. This could certainly tie in to distance and lines, as well, as getting the distance between two points on a graph is essentially the same thing.

[[User:BenJeffery|BenJeffery]]

----

Using information from the group members who responded (Christa, Marco, Ben):

==Things we all can do well==

10. Distance and Lines

12. Construction of Graphs

13a. Pythagorean Theorem

14. Areas and Volumes

15. Mathematical Writing

==Things some of us can do well==

3. Equations

4. Inequalities

5. Composition of Functions

8. Intersections of Functions

9. Reading Graphs of Functions

11. Operations on Graphs of Functions

13b. Trigonometry

== Things none of us can do well==

1. Basic Functions

2. Properties of Functions

6. Polynomial Long Division

7. Graphs of Functions

File:Untitled.png

2010-12-03T09:03:41Z

AviHarry:

File:Pi Circle Yo.png

2010-12-03T08:55:20Z

AviHarry:

User:AviHarry

2010-11-23T22:17:27Z

AviHarry: Created page with 'Well.... Guess you can call me a Ghost? Despite me being brown ; O HHAHAhahaha.. ha. ha h a h'

Well.... Guess you can call me a Ghost? Despite me being brown ; O HHAHAhahaha.. ha. ha h a h

Course talk:MATH110/003/Groups/Group 17/Basic Skills Project

2010-11-23T22:16:07Z

AviHarry:

Guys - Any ideas for our basic skills project offer?
Seeing as we can apparently all do Distance and Lines well, how about we offer to do an overview with examples of that part?

It's pretty straight-forward, so there shouldn't be too many challenges with getting it together. If no one answers this by tonight, I'll just put it up as our offer.

Cheers,
[[User:BenJeffery|BenJeffery]]

looks good to me! i would be between that or the pythagorean theorem (just because it seems more interesting) but for me, either or work. [[User:MarcoGasparian|MarcoGasparian]]

I'm comfortable with doing either- leaning more to pythagorean theorem

[[User:ChristaBicego|ChristaBicego]]

Was wondering what part of the guidelines Ben set out you guys would want me to do, and also how to compute them. I am a bit confused as to the guidelines of this project.

[[User:AviHarry|AviHarry]]

Course talk:MATH110/003/Groups/Group 17/Basic Skills Project

2010-11-23T22:15:20Z

AviHarry:

Sandbox:DavidKohler/Schedule

2010-11-01T20:25:56Z

AviHarry:

Simply write your name or student number next to an empty time slot that suits you. You're not allowed to move others of course. -- [[User:DavidKohler|DavidKohler]]

Tuesday November 2
* 3:00 - 3:20-Ghita Youssefi
* 3:20 - 3:40- Avi Harry
* 3:40 - 4:00-Albert Koenig
* 4:00 - 4:20
* 4:20 - 4:40
* 4:40 - 5:00

Wednesday November 3
* 9:20 - 9:40
* 9:40 - 10:00
* 10:00 - 10:20
* 10:20 - 10:40
* 10:40 - 11:00

Thursday November 4
* 2:40 - 3:00- Charly Huxford
* 3:00 - 3:20
* 3:20 - 3:40
* 3:40 - 4:00
* 4:00 - 4:20
* 4:20 - 4:40
* 4:40 - 5:00

Friday November 5
* 9:20 - 9:40- Sabrina Pannu
* 9:40 - 10:00
* 10:00 - 10:20
* 10:20 - 10:40
* 10:40 - 11:00

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Problem Solving 2

2010-10-20T06:48:25Z

AviHarry:

== Problem Set 2==

===Question 1===
Question 1: Five persons named their pets after each other. From the following clues, can you decide which pet belongs to Suzan's mother? Tosh owns a cat, Bianca owns a frog that she loves, Jaela owns a parrot which keeps calling her "darling, darling", Jun owns a snake, don't mess with him, Suzan is the name of the frog, The cat is named Jun, The name by which they call the turtle is the name of the woman whose pet is Tosh, Finally, Suzan's mother's pet is Bianca.
The question is asking, which pet belongs to Suzan's mother?
- we can figure this out by connecting the names of the different pets to their owners as the pets are all named after these individuals
First: the info from the question:
Tosh has a cat named Jun Bianca has a frog named Susan Jaela has a parot named - ? Jun has a snake named ? Suzans owns a turtle named ?
Suzans mother cannot be susan(susans mother cannot be her daughter...), or Bianca (Susan's mother's pet name is Bianca)
Tosh has a cat named Jun Bianca has a frog named Susan Jaela has a parot named - (Bianca or Tosh) Jun has a snake named (Jaela, Bianca or Susan) Suzans owns a turtle named (Tosh or Jaela)
The only way the question satisfies all points is if :
Tosh has a cat named Jun Bianca has a frog named Susan Jaela has a parot named - Tosh Jun has a snake named Bianca Suzans owns a turtle named Jaela
Therefore: Susans mother is Jun and she owns a snake

===Question 2===
''Bohao, Stewart, Dylan, Tim and Chan are the five players of a basketball team. Two are left handed and three right handed, Two are over 2m tall and three are under 2m, Bohao and Dylan are of the same handedness, whereas Tim and Chan use different hands. Stewart and Chan are of the same height range, while Dylan and Tim are in different height ranges. If you know that the one playing centre is over 2m tall and is left handed, can you guess his name?''

So, we need to find out who on the team is over 2m tall and left-handed.

Looking at the information about the players, we see that there are 3 right-handers and two left-handers. We know that Bohao and Dylan are the same handedness, while Tim and Chan are different. This tells us that either Tim or Chan have the same handedness as both Bohao and Dylan. This means that Bohao and Dylan must be right-handed, since they are in a group of 3 people that share the same hand.

Looking at height, we can use the same deduction. There are three people under 2m tall and two over 2m tall. If Stewart and Chan are of the same size, and Dylan and Tim are different, then we know that Stewart and Chan are in a group of three people, and therefore must be under 2m tall.

Finally, we need someone who is over 2m and left-handed. Bohao and Dylan are out, since they are right-handed. Chan and Stewart are out, since they are under 2m tall. From the list of 5 players, the only remaining player is Tim. Therefore, Tim is our centerman.

[[User:BenJeffery|BenJeffery]]

===Question 3===
Adam, Bobo, Charles, Ed, Hassan, Jason, Mathieu, Pascal and Sung have formed a baseball team. The following facts are true:

Adam does not like the catcher,
Ed's sister is engaged to the second baseman,
The centre fielder is taller than the right fielder,
Hassan and the third baseman live in the same building,
Pascal and Charles each won $20 from the pitcher at a poker game,
Ed and the outfielders play cards during their free time,
The pitcher's wife is the third baseman's sister,
All the battery and infield except Charles, Hassan and Adam are shorter than Sung,
Pascal, Adam and the shortstop lost $100 each at the race track,
The second baseman beat Pascal, Hassan, Bobo and the catcher at billiards,
Sung is in the process of getting a divorce,
The catcher and the third baseman each have two legitimate children,
Ed, Pascal Jason, the right fielder and the centre fielder are bachelors, the others are all married
The shortstop, the third baseman and Bobo all attended the fight,
Mathieu is the shortest player of the team,
Determine the positions of each player on the baseball team.

Note: On a baseball team there are three outfielders (right, centre and left), four infielders (first baseman, second baseman, third baseman and shortstop) and the battery (pitcher and catcher).

{| border="1"
|
|Adam
|Bobo
|Charles
|Ed
|Hassan
|Jason
|Mathieu
|Pascal
|Sung
|-
|Right Outfielder
|x
|x
|x
|x
|x
|x
|o
|x
|x
|-
|Center Outfielder
|x
|o
|x
|x
|x
|x
|x
|x
|x
|-
|Left Outfielder
|x
|x
|x
|x
|x
|x
|x
|x
|o
|-
|1st Baseman
|x
|x
|x
|x
|x
|x
|x
|o
|x
|-
|2nd Baseman
|x
|x
|x
|x
|x
|o
|x
|x
|x
|-
|3rd Baseman
|o
|x
|x
|x
|x
|x
|x
|x
|x
|-
|Shortstop
|x
|x
|x
|o
|x
|x
|x
|x
|x
|-
|Pitcher
|x
|x
|x
|x
|o
|x
|x
|x
|x
|-
|Catcher
|x
|x
|o
|x
|x
|x
|x
|x
|x
|}

The graph above depicts with "x's" and "o's" the correct placement of each player on the baseball field. "X" being were they aren't and "o" being where they are...obviously.

Here are some of the facts that I derived from the statements the question gave us to come up with an answer.

Hassan is not the third baseman because he lives with the third baseman.

Ed is not the 2nd baseman because his sister is engaged to the second baseman.

Adam is not the catcher because he does not like the catcher.

Mathieu is not the center fielder because the center fielder is taller than the right fielder.
Mathieu is the shortest of them all.

Pitcher has wife, therefore it can’ t be Ed, Pascal Jason, the right fielder and the centre
fielder

Pascal and Adam aren’ t shortstop because both of them and the shortstop lost 100$ at the
race track

Pascal, Hassan, Bobo cant be the second baseman or catcher because the second baseman
beat Pascal, Hassan, Bobo, and the catcher at billiards

All the battery and infield except Charles, Hassan and Adam are shorter than Sung,
Therefore, Charles, Hassan, and Adam are either battery or infield

Pascal and Charles can’ t be the pitcher because they won money from him at a poker
game

Ed can’ t be an outfielder because he plays cards with them

This list would go on and on, but you get the idea.
[[User:MarcoGasparian|MarcoGasparian]]
===Question 4===

===Question 5===
Homer finally had a week off from his job at the nuclear power plant and intended to spend all nine days of his vacation (Saturday through the following Sunday) sleeping late. But his plans were foiled by some of the people who work in his neighbourhood.On Saturday, his first morning off, Homer was wakened by the doorbell; it was a salesman of magazine subscriptions.On Sunday, the barking of the neighbour's dog abruptly ended Homer's sleep.On Monday, he was again wakened by the persistent salesman but was able to fall asleep again, only to be disturbed by the construction workers next door.In fact, the salesman, the neighbour's dog and the construction workers combined to wake Homer at least once each day of his vacation, with only one exception.The salesman woke him again on Wednesday; the construction workers on the second Saturday; the dog on Wednesday and on the final Sunday.No one of the three noisemakers was quiet for three consecutive days; but yet, no pair of them made noise on more than one day during Homer's vacation. On which day of his holiday was Homer actually able to sleep late?

During the 9 days of vacation, Homer will be able to sleep on Tuesday. In coming to this conclusion I analyzed the problem creating a chart of the 9 days Homer had vacation and simply listed the days he had been disturbed in his sleep. According to the problem Homer was awaken on the first Saturday,Sunday,Monday,Wednesday and the following Saturday and Sunday. This leaves us with Tuesday,Thursday, and Friday. A quick detail in the problem states, "no one of the three noisemakers was quiet for three consecutive days". Considering that the construction worker and the salesman do not have a consecutive three in the days of Thursday and Friday those days were ruled out leaving Tuesday to be the only day Homer got too sleep late.
[[User:AviHarry|AviHarry]]

Course:MATH110/Archive/2010-2011/003/Groups/Group 17/Problem Solving 2

2010-10-20T06:46:28Z

AviHarry: