The
real sine function induces a
bijective,
strictly increasing
function
-
and the
real cosine function
induces a bijective, strictly decreasing function
-
Proof
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
The
real tangent function
induces a
bijective,
strictly increasing
function
-
and the
real cotangent function
induces a bijective strictly decreasing function
-
Proof
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)
Due to the bijectivity of sine, cosine, tangent and cotangent on suitable interval, there exist the following inverse functions.
The
inverse function
of the real
sine function
is
-
and is called
arcsine.
The
inverse function
of the real
cosine function
is
-
and is called
arccosine.
Arkustangens
The
inverse function
of the real
tangent function
is
-
and is called
arctangent.
The
inverse function
of the real
cotangent function
is
-
and is called
arccotangent.
For example, for the arctangent, we have, due to
fact,
![{\displaystyle {}{\begin{aligned}(\arctan x)^{\prime }&={\frac {1}{\frac {1}{\cos ^{2}(\arctan x)}}}\\&={\frac {1}{\frac {\cos ^{2}(\arctan x)+\sin ^{2}(\arctan x)}{\cos ^{2}(\arctan x)}}}\\&={\frac {1}{1+\tan ^{2}(\arctan x)}}\\&={\frac {1}{1+x^{2}}}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ea3dbec9ef0cb321053d854dcb3411a664e73c69)
![{\displaystyle \Box }](https://wikimedia.org/api/rest_v1/media/math/render/svg/029b77f09ebeaf7528fc831fe57848be51f2240b)