Solution assignment 09 Trigonometric equations

Solve the following equation for $0\leq{x}\leq{2\pi}$ :

$\cos(x+0.5)=\sin(x)$

We write the right-hand side as:

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

and thus we have to solve:

$\cos(x+0.5)=\cos(\displaystyle\frac{\pi}{2}-x)$

$x+0.5=\displaystyle\frac{\pi}{2}-x+2k\pi, k=0, \pm1, \pm2, ...$

$2x=\displaystyle\frac{\pi}{2}-0.5+2k\pi, k=0, \pm1, \pm2, ...$

$x=0.53$

This is the only solution in the given domain.