Inverse Trigonometric Functions

An ordinary trigonometric function takes an angle and returns a ratio. Its inverse does the reverse — ratio in, angle out. Domain and range simply trade places: what was the range of $\sin$ becomes the domain of $\arcsin$ .

Concept Why the range must be restricted

Infinitely many angles share the same sine — 30°, 150°, 390° and so on. For the inverse to be a genuine function it must return exactly one of them, so each inverse is given a fixed output interval, called its principal range.

Both notations are in use: $\sin^{-1}x$ and $\arcsin x$ mean the same thing. The arc form is safer, because the −1 is easily misread as an exponent.

Graph arcsin x and arccos x

Both take inputs only from −1 to 1, because that is all a sine or cosine can produce. $\arcsin$ rises from −90° to 90°; $\arccos$ falls from 180° down to 0°.

Graph arctan x

Its domain is all real numbers, because the tangent takes every value.

Its range is the open interval from −90° to 90°: the curve flattens toward those two horizontal asymptotes but never reaches them.

Example Asking the question in words

Which angle has a sine of

\tfrac{1}{2}

\arcsin\!\left(\tfrac{1}{2}\right)

⟹ 30°

Which angle has a cosine of

\tfrac{1}{2}

\arccos\!\left(\tfrac{1}{2}\right)

⟹ 60°

Which angle has a tangent of 1?

\arctan(1)

⟹ 45°

Reference Domains and ranges

Function	Domain	Range	Example
arcsin x	−1 to 1	−90° to 90°	arcsin(0.5) = 30°
arccos x	−1 to 1	0° to 180°	arccos(0.5) = 60°
arctan x	all real numbers	between −90° and 90°	arctan(1) = 45°

Summary

An inverse trigonometric function returns the angle that produced a given ratio.
arcsin and arccos accept only inputs from −1 to 1; their ranges are −90°…90° and 0°…180°.
arctan accepts every real number, with two horizontal asymptotes at ±90°.
Prefer the arc notation, so the −1 is never mistaken for a power.