Inverse Trigonometric Functions

2021-03-26
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The inverse functions of the sine, cosine, and tangent are defined as below: sin−1⁡x=y⟺sin⁡y=x     (−π2≤y≤π2)cos−1⁡x=y⟺cos⁡y=x     (0≤y≤π)tan−1⁡x=y⟺tan⁡y=x     (−π2<y<π2)

The trigonometric functions are not one-to-one, therefore they do not have inverse functions.

However, it is possible to restrict the domain of trigonometric functions so that they become one-to-one and the inverse functions of these restricted trigonometric functions can be defined.

Inverse Trigonometric FunctionsDomainRange
Inverse sine function y=sin−1⁡x−1≤x≤1−π2≤y≤π2
Inverse cosine function y=cos−1⁡x−1≤x≤10≤y≤π
Inverse tangent function y=tan−1⁡x−∞<x<∞−π2 < y < π2

Symmetry

sin−1⁡(−x)=−sin−1⁡x
tan−1⁡(−x)=−tan−1⁡x

Notice：cos−1⁡(−x)≠−sin−1⁡x

Derivatives

ddx(sin−1⁡x)=11−x2
ddx(cos−1⁡x)=−11−x2
ddx(tan−1⁡x)=11+x2
ddx(sin−1⁡f(x))=f′(x)1−f(x)2
ddx(cos−1⁡f(x))=−f′(x)1−f(x)2
ddx(tan−1⁡f(x))=f′(x)1+f(x)2

Integration

∫11−x2 dx=sin−1⁡x+C
∫11−x2 dx=−cos−1⁡x+C
∫11+x2 dx=tan−1⁡x+C
∫1a2−(px+q)2 dx=1psin−1⁡px+qa+C
∫1a2−(px+q)2 dx=−1pcos−1⁡px+qa+C
∫1a2+(px+q)2 dx=1aptan−1⁡px+qa+C

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