Trigonometric functions/R/Inverse functions/Analytic properties/Section
The real sine function induces a bijective, strictly increasing function
![{\displaystyle [-\pi /2,\pi /2]\longrightarrow [-1,1],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/527f8bc31212ec82e3778cde36b1ed32192bf657)
and the real cosine function induces a bijective, strictly decreasing function
![{\displaystyle [0,\pi ]\longrightarrow [-1,1].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8dbd9728b307c63938254cd4e4aff49bc8e2f2a7)
Proof
See exercise.
The real tangent function induces a bijective, strictly increasing function
![{\displaystyle ]-\pi /2,\pi /2[\longrightarrow \mathbb {R} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f54b80848470e2b293193379bd774648a7d69df6)
and the real cotangent function induces a bijective strictly decreasing function
![{\displaystyle [0,\pi ]\longrightarrow \mathbb {R} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/014d0207862d72a4533c62948f6bae80d18e9964)
Proof
See exercise.
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
![{\displaystyle [-1,1]\longrightarrow [-{\frac {\pi }{2}},{\frac {\pi }{2}}],x\longmapsto \arcsin x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32e269068b1687b0c1f9ad4b5014b4e7cb54b611)
The inverse function of the real cosine function is
![{\displaystyle [-1,1]\longrightarrow [0,\pi ],x\longmapsto \arccos x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ecd68470a9279fb3e2ea542385322310d9b0f7f)
The inverse function of the real tangent function is
![{\displaystyle \mathbb {R} \longrightarrow ]-{\frac {\pi }{2}},{\frac {\pi }{2}}[,x\longmapsto \arctan x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a599a936531a4aaefac1af5be86d23f98e145ce7)
The inverse function of the real cotangent function is
![{\displaystyle \mathbb {R} \longrightarrow ]0,\pi [,x\longmapsto \operatorname {arccot} x,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f643c1b07952f1bae062cf8e79a6d782e9011f)
