Question

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

Answer 1

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

`x = pi/4 + n pi` any `n in ZZ`

Explanation:

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

So `x = pi/4` is a solution.

Also `sin(pi/4 + pi) = cos(pi/4 + pi) = -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 = pi/4 + n pi` captures all the solutions.