nLab inverse trigonometric function

Idea

An inverse trigonometric function is a right inverse to one of the six standard trigonometric functions, or more precisely a continuous right inverse to the epimorphism part of the (epi, mono) factorization of a trigonometric function.

Inverse functions are often named using the prefix “arc”, so that for example “arcsine” is another name for the inverse sine. This naming arises because the values of an inverse trigonometric function are considered as arc lengths, just as trigonometric functions themselves are defined in terms of arc length parametrizations. Another possible reason for using the arc- prefix is to avoid a notational confusion, where for example $\cos^{-1}$ might easily be confused with the reciprocal of the cosine function, inasmuch as the notation $\cos^n$ is commonly used to denote an $n^{th}$ power of the cosine.

Since many such right inverses are possible, the choice of right inverse is a matter of convention. For the arcsine and arctangent, the choice is dictated by having $0$ as a fixed point. This leads to convenient power series representations in neighborhoods of the origin, for example

$$\arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \ldots$$

For other inverse trigonometric functions, the choice is dictated by convenient formulas such as

$$\arccos(x) = \frac{\pi}{2} - \arcsin(x), \qquad \arccot(x) = \frac{\pi}{2} - \arctan(x)$$

while we also have

$$\arcsec(x) = \arccos(\frac1{x}), \qquad \arccsc(x) = \arcsin(\frac1{x}).$$

Example

• arccos

• arcsin

• arctan