Real sine and cosine function/Derivative/Fact

sine function

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

is differentiable, with

${\displaystyle {}\sin \!'(x)=\cos x\,,}$

and the cosine function

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

is differentiable, with

${\displaystyle {}\cos \!'(x)=-\sin x\,.}$