Science:Math Exam Resources/Courses/MATH110/December 2012/Question 07 (a)/Solution 1

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 $x=\pi /4$.