nLab
inverse trigonometric function

Contents

Contents

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\cos^{-1} might easily be confused with the reciprocal? of the cosine function, inasmuch as the notation cos n\cos^n is commonly used to denote an n thn^{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 00 as a fixed point. This leads to convenient power series representations in neighborhoods of the origin, for example

arctan(x)=xx 33+x 55\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)=π2arcsin(x),arccot(x)=π2arctan(x)\arccos(x) = \frac{\pi}{2} - \arcsin(x), \qquad \arccot(x) = \frac{\pi}{2} - \arctan(x)

while we also have

arcsec(x)=arccos(1x),arccsc(x)=arcsin(1x).\arcsec(x) = \arccos(\frac1{x}), \qquad \arccsc(x) = \arcsin(\frac1{x}).

Example

Last revised on May 12, 2019 at 21:30:20. See the history of this page for a list of all contributions to it.