Sine and cosine/R/Properties of periodicity/Fact

The sine function and the cosine function fulfill in ${}\mathbb {R}$ the following periodicity properties.

We have ${}\cos {\left(x+2\pi \right)}=\cos x$ and ${}\sin {\left(x+2\pi \right)}=\sin x$ for all ${}x\in \mathbb {R}$ .
We have ${}\cos {\left(x+\pi \right)}=-\cos x$ and ${}\sin {\left(x+\pi \right)}=-\sin x$ for all ${}x\in \mathbb {R}$ .
We have ${}\cos {\left(x+\pi /2\right)}=-\sin x$ and ${}\sin {\left(x+\pi /2\right)}=\cos x$ for all ${}x\in \mathbb {R}$ .
We have ${}\cos 0=1$ , ${}\cos \pi /2=0$ , ${}\cos \pi =-1$ , ${}\cos 3\pi /2=0$ , and ${}\cos 2\pi =1$ .
We have ${}\sin 0=0$ , ${}\sin \pi /2=1$ , ${}\sin \pi =0$ , ${}\sin 3\pi /2=-1$ , and ${}\sin 2\pi =0$ .