Science:Math Exam Resources/Courses/MATH110/December 2012/Question 07 (a)

MATH110 December 2012

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Question 07 (a)
Let $f(x)=\sin x-\cos x$. Find all values of $x$ satisfying the equation $f(x)=0$ in the interval $[0,\pi )$. Justify your answer.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
In order to solve the equation $f(x)=0$ it is helpful to know the values of $\sin x$ and $\cos x$ at certain angles. Try using the unit circle or drawing out the graphs of the two functions.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The equation $f(x)=0$ can be written as $\sin x-\cos x=0$ or $\sin x=\cos x$ Now it remains to find where $\sin x$ is equal to $\cos x$ in the interval $[0,\pi )$. There are multiple ways to do this. Unit Circle The coordinates of points on the unit circle are given by $\sin \theta$ and $\cos \theta$, so finding where $\sin \theta =\cos \theta$ is the same as finding where the x and y coordinates of a point on the unit circle are the same. This occurs when the angle is equal to $\pi /4$ radians. Drawing the graph If you draw the graphs of $\sin x$ and $\cos x$ very carefully, you can make an educated guess about where the two functions are equal. However, this method will never be as accurate as knowing key values of $\sin x$ and $\cos x$ from the unit circle. Trig ratios One last method is writing out $\sin x$ and $\cos x$ as trig ratios, where, given a right triangle with angle x, $\sin x=o/h$ and $\cos x=a/h$. Setting this equal gives: $o/h=a/h$, or $o=a$. The opposite and adjacent sides of a right triangle are equal when the right triangle is isoceles, meaning each angle is 45 degrees or $x=\pi /4$.