# Sine and cosine/R/Properties of periodicity/Fact

The sine function and the cosine function fulfill in ${\displaystyle {}\mathbb {R} }$ the following periodicity properties.
1. We have ${\displaystyle {}\cos {\left(x+2\pi \right)}=\cos x}$ and ${\displaystyle {}\sin {\left(x+2\pi \right)}=\sin x}$ for all ${\displaystyle {}x\in \mathbb {R} }$.
2. We have ${\displaystyle {}\cos {\left(x+\pi \right)}=-\cos x}$ and ${\displaystyle {}\sin {\left(x+\pi \right)}=-\sin x}$ for all ${\displaystyle {}x\in \mathbb {R} }$.
3. We have ${\displaystyle {}\cos {\left(x+\pi /2\right)}=-\sin x}$ and ${\displaystyle {}\sin {\left(x+\pi /2\right)}=\cos x}$ for all ${\displaystyle {}x\in \mathbb {R} }$.
4. We have ${\displaystyle {}\cos 0=1}$, ${\displaystyle {}\cos \pi /2=0}$, ${\displaystyle {}\cos \pi =-1}$, ${\displaystyle {}\cos 3\pi /2=0}$, and ${\displaystyle {}\cos 2\pi =1}$.
5. We have ${\displaystyle {}\sin 0=0}$, ${\displaystyle {}\sin \pi /2=1}$, ${\displaystyle {}\sin \pi =0}$, ${\displaystyle {}\sin 3\pi /2=-1}$, and ${\displaystyle {}\sin 2\pi =0}$.