Solution assignment 07 Unit circle and simple formulas

Give the period of:

$\sin(x)$ $\sin(2x)$ $\sin(\displaystyle\frac{1}{3}x)$ $\cos(x)$ $\cos(4x)$ $\tan(\displaystyle\frac{1}{2}x)$

The period of $\sin(x)$ equals $2\pi$ (after one turn we start again). In other words: $0\leq{x}\leq{2\pi}$

For the function $\sin(2x)$ we have:

$0\leq{2x}\leq{2\pi}$

or:

$0\leq{x}\leq{\pi}$

and thus the period of this function is equal to $\pi$ .
For similar reasons the other functions have, subsequently, the following periods:

$2\pi$ $\pi$ $6\pi$ $2\pi$ $\displaystyle\frac{1}{2}\pi$ en $4\pi$