Question

Find all solutions of the equation in the interval [0, 2). (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION.)

Answer 1

First, we will put all the expressions on the same side:

$sinx=cosx$

We know:

$sinx=cos(\frac{\pi}{2}-x)$

And:

$cosx=cosy$

so x can be:

$x=y+2k\pi$

or,

$x=-y+2\pi k$

We have:

$cosx=cos(\frac{\pi}{2}-x)$$x=\frac{\pi}{2}-x+2k\pi$

and it can be:

$x+x=\frac{\pi}{2}+2k\pi$

The second equation:

$x=-(\frac{\pi}{2}-x)+2k\pi$

Then,

$x-x=-\frac{\pi}{2}+2k\pi$

This is the second equation, but x - x = 0, so we use the first equation obtained:

$2x=\frac{\pi}{2}+2k\pi$$x=\frac{\pi}{4}+k\pi$

Where k is a number belonging to Z.