Inverse trigonometric functions reverse the basic trigonometric functions. For example,
\(\arcsin(0.5)\) asks: “Which principal angle has sine equal to \(0.5\)?”
1. Why inverse trig functions need restricted ranges
Sine, cosine, and tangent are not one-to-one over all real numbers. The same output value can come
from many different angles. To make an inverse function, we restrict the output angle to one principal range.
2. Inverse sine
\[
\arcsin x=\theta
\quad\Longleftrightarrow\quad
\sin\theta=x
\]
Domain and range:
\[
-1\le x\le 1
\]
\[
-\frac{\pi}{2}\le \theta\le \frac{\pi}{2}
\]
Example:
\[
\arcsin\left(\frac12\right)=\frac{\pi}{6}
\]
3. Inverse cosine
\[
\arccos x=\theta
\quad\Longleftrightarrow\quad
\cos\theta=x
\]
Domain and range:
\[
-1\le x\le 1
\]
\[
0\le \theta\le \pi
\]
Example:
\[
\arccos\left(-\frac12\right)=\frac{2\pi}{3}
\]
4. Inverse tangent
\[
\arctan x=\theta
\quad\Longleftrightarrow\quad
\tan\theta=x
\]
Domain and range:
\[
x\in\mathbb{R}
\]
\[
-\frac{\pi}{2}<\theta<\frac{\pi}{2}
\]
Example:
\[
\arctan(-1)=-\frac{\pi}{4}
\]
5. Formula summary
| Function |
Meaning |
Domain |
Principal range |
| \(\arcsin x\) |
\(\sin\theta=x\) |
\(-1\le x\le 1\) |
\(-\dfrac{\pi}{2}\le\theta\le\dfrac{\pi}{2}\) |
| \(\arccos x\) |
\(\cos\theta=x\) |
\(-1\le x\le 1\) |
\(0\le\theta\le\pi\) |
| \(\arctan x\) |
\(\tan\theta=x\) |
\(x\in\mathbb{R}\) |
\(-\dfrac{\pi}{2}<\theta<\dfrac{\pi}{2}\) |
6. Common mistakes
- \(\arcsin(1/2)\) returns \(\pi/6\), not every angle with sine \(1/2\).
- \(\arccos(-1/2)\) returns \(2\pi/3\), not \(-2\pi/3\), because arccos uses the range \([0,\pi]\).
- \(\arctan(-1)\) returns \(-\pi/4\), because arctan uses the range \((-\pi/2,\pi/2)\).
- \(\arcsin x\) and \(\arccos x\) are undefined for real inputs outside \([-1,1]\).
- Do not confuse inverse functions with reciprocals: \(\sin^{-1}x\) often means \(\arcsin x\), not \(1/\sin x\).
