# For what values of x is sin x = cos x ?

Jul 12, 2015

From geometric definition of $\sin$ and $\cos$ and properties of right angled triangles, find:

$x = \frac{\pi}{4} + n \pi$ any $n \in \mathbb{Z}$

#### Explanation:

In Q1 this is basically the $\frac{\sqrt{2}}{2}$, $\frac{\sqrt{2}}{2}$, $1$ right angled isosceles triangle with angles $\frac{\pi}{4}$, $\frac{\pi}{4}$ and $\frac{\pi}{2}$. It needs to be isosceles in order that $\sin \left(x\right) = \cos \left(x\right)$

So $x = \frac{\pi}{4}$ is a solution.

Also $\sin \left(\frac{\pi}{4} + \pi\right) = \cos \left(\frac{\pi}{4} + \pi\right) = - \frac{\sqrt{2}}{2}$ in Q3

There are no solutions in Q2 or Q4 since $\sin$ and $\cos$ have opposite signs in those quadrants.

So $x = \frac{\pi}{4} + n \pi$ captures all the solutions.