What Are the Domains and Ranges of Arcsin, Arccos, and Arctan?

The domains and ranges come from restricting sine, cosine, and tangent so that each is one-to-one. In radians, the principal-value ranges are fixed intervals; inverse trigonometric functions return only angles within those intervals.

Why These Restrictions Are Necessary

An inverse function exists only when the original function is one-to-one. Since sine, cosine, and tangent are periodic, their unrestricted graphs repeat output values and therefore fail the horizontal line test.

For $\arcsin$, sine is restricted to $\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, where it increases from to and takes each value once. Thus, its domain is , and the restricted angle interval becomes its range.

For $\arccos$, cosine is restricted to $[0,\pi]$, where it decreases from $1$ to $-1$. Its domain is $[-1,1]$, and its range is .

For $\arctan$, tangent is restricted to $\left]-\frac{\pi}{2},\frac{\pi}{2}\right[$. It approaches vertical asymptotes at the endpoints, so they are excluded. Across this interval, tangent takes every real value; therefore, the domain of is .

Worked Examples

Examples include

$$\arcsin(1)=\frac{\pi}{2},\qquad \arccos(-1)=\pi,\qquad \arctan(1)=\frac{\pi}{4}.$$

Graphically, each inverse graph is the reflection of its restricted original graph in the line $y=x$. This explains why domain and range swap. Notice that $\arctan x$ can accept any real input, but it can never output $\pm\frac{\pi}{2}$, because tangent is undefined at those angles. By contrast, arcsin and arccos include their endpoint outputs.

An inverse exchanges inputs and outputs. Therefore, $\sin(\arcsin x)=x$ for $-1\leq x\leq1$. However, $\arcsin(\sin x)=x$ only when lies in the principal range of ; outside it, an equivalent principal angle is returned.

A common misconception is that $\sin^{-1}x$ means $1/\sin x$. It does not: $\sin^{-1}x=\arcsin x$, while the reciprocal of sine is .

IB Exam Technique

This is Mathematics: Analysis and Approaches HL, sub-topic AHL 3.9, and may be assessed on Paper 1 or Paper 2. Use radians unless degrees are specified, state included and excluded endpoints carefully, and check that an input belongs to the inverse function's domain before evaluating it.