# Sine and cosine/Real/Properties/Fact

The functions

and

have the following properties for .

- We have and .
- We have and .
- The addition theorems
and

hold.

- We have

The functions

- $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \cos x,$

and

- $\mathbb {R} \longrightarrow \mathbb {R} ,x\longmapsto \sin x,$

have the following properties for ${}x,y\in \mathbb {R}$.

- We have ${}\cos 0=1$ and ${}\sin 0=0$.
- We have ${}\cos {\left(-x\right)}=\cos x$ and ${}\sin {\left(-x\right)}=-\sin x$.
- The addition theorems
- ${}\cos(x+y)=\cos x\cdot \cos y-\sin x\cdot \sin y\,$

and

- ${}\sin(x+y)=\sin x\cdot \cos y+\cos x\cdot \sin y\,$

hold.

- We have
- ${}(\cos x)^{2}+(\sin x)^{2}=1\,.$