Questions tagged [inverse-trigonometric-functions]

The inverse (circular) trigonometric functions are the inverse functions of the circular trigonometric functions: sine($\sin$), cosine($\cos$), tangent($\tan$), cotangent($\cot$), secant($\sec$) and cosecant($\csc$) under suitably restricted domains.

They are denoted by a superscript of $-1$ on the corresponding trigonometric functions or by a prefix $\text{arc-}$ before its name. Example: the inverse sine function can be denoted as $\sin^{-1}x$ or $\arcsin x$.

The intuition for the prefix $\text{arc-}$ is that the inverse trigonometric function$\text{(say}$ $f^{-1}(x)$$\text{)}$ of a value $x$ yields the numerical value of the length of the arc in the unit circle whose $f$ value is $x$. Example, the inverse tangent of $1$ is $\frac{\pi}4$ means that the tangent of the numerical value of an arc of length $\frac{\pi}4$ in the unit circle would be $1$.

Their domains and ranges(principal value branches) are as follows:

$$\sin^{-1}x:[-1,1]\to\left[\frac{-\pi}2,\frac{\pi}2\right]$$ $$\cos^{-1}x:[-1,1]\to[0,\pi]$$ $$\tan^{-1}x:\mathbb R\to\left(\frac{-\pi}2,\frac{\pi}2\right)$$ $$\cot^{-1}x:\mathbb R\to\left(0,\pi\right)$$ $$\csc^{-1}x:\mathbb R-[-1,1]\to\left[\frac{-\pi}2,\frac{\pi}2\right]-\text{{0}}$$ $$\sec^{-1}x:\mathbb R-[-1,1]\to[0,\pi]-\left\{\frac{\pi}2\right\}$$

Some basic identities:

$$\sin^{-1}x+\cos^{-1}x=\frac{\pi}2\ \forall\ x\in[-1,1]$$ $$\tan^{-1}x+\cot^{-1}x=\frac{\pi}2\ \forall\ x\in\mathbb R$$ $$\sec^{-1}x+\csc^{-1}x=\frac{\pi}2\ \forall\ x\in\mathbb R-[-1,1]$$ $$\csc^{-1}x=\sin^{-1}\frac1{x}\ \forall\ x\in\mathbb R-[-1,1]$$ $$\sec^{-1}x=\cos^{-1}\frac1{x}\ \forall\ x\in\mathbb R-[-1,1]$$ $$\cot^{-1}x=\begin{cases}\tan^{-1}\frac1{x}\ \forall\ x>0\\ \tan^{-1}\frac1{x}+\pi\ \forall\ x<0\end{cases}$$

In complex analysis, their domains are extended to $\mathbb C$. In this case, they can be defined in terms of their hyperbolic analogues:

$$\sin^{-1}x=-i\sinh^{-1}ix$$ $$\cos^{-1}x=i(\text{sgn}(x^2-1))\cosh^{-1}x$$ $$\tan^{-1}x=-i\tanh^{-1}ix$$

Their derivatives are particularly useful in integrating some algebraic functions(these formulae are applicable $\forall\ x\in\mathbb C$):

$$\frac{\mathrm d}{\mathrm dx}\sin^{-1}x=\frac1{\sqrt{1-x^2}}$$ $$\frac{\mathrm d}{\mathrm dx}\cos^{-1}x=\frac{-1}{\sqrt{1-x^2}}$$ $$\frac{\mathrm d}{\mathrm dx}\tan^{-1}x=\frac1{1+x^2}$$ $$\frac{\mathrm d}{\mathrm dx}\cot^{-1}x=\frac{-1}{1+x^2}$$ $$\frac{\mathrm d}{\mathrm dx}\sec^{-1}x=\frac1{x^2\sqrt{1-\frac1{x^2}}}$$ $$\frac{\mathrm d}{\mathrm dx}\csc^{-1}x=\frac{-1}{x^2\sqrt{1-\frac1{x^2}}}$$

Reference: Wikipedia