Inverse Trigonometric Functions
If trigonometric functions take an angle and return a ratio, then inverse trigonometric functions do the opposite — they take a ratio and return an angle.
🔄 The Big Idea
Consider the equation:
$\sin(\theta) = x$
If we know $x$ (the ratio) and want to find $\theta$ (the angle), we write:
$\theta = \sin^{-1}(x) = \arcsin(x)$
⚠️ Important: $\sin^{-1}(x)$ does NOT mean $\frac{1}{\sin(x)}$. The superscript $-1$ denotes the inverse function, not a reciprocal.
📋 The Three Main Inverse Trig Functions
| Function | Notation | Domain | Range (Principal Value) |
|---|---|---|---|
| Inverse Sine | $\sin^{-1}(x)$ or $\arcsin(x)$ | $[-1, 1]$ | $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ |
| Inverse Cosine | $\cos^{-1}(x)$ or $\arccos(x)$ | $[-1, 1]$ | $[0, \pi]$ |
| Inverse Tangent | $\tan^{-1}(x)$ or $\arctan(x)$ | $(-\infty, \infty)$ | $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ |
🤔 Why Do We Need Restricted Ranges?
Trigonometric functions are periodic — they repeat their values. For example, $\sin(30°) = \sin(150°) = 0.5$. So if we ask "what angle has a sine of 0.5?", there are infinitely many answers!
To make the inverse a proper function (one output for each input), we restrict the range to a principal value branch. This ensures each input gives exactly one output.
$\arcsin$ returns angles in Q1 or Q4
$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ or $[-90°, 90°]$
$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ or $[-90°, 90°]$
$\arccos$ returns angles in Q1 or Q2
$[0, \pi]$ or $[0°, 180°]$
$[0, \pi]$ or $[0°, 180°]$
$\arctan$ returns angles in Q1 or Q4
$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ or $(-90°, 90°)$
$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ or $(-90°, 90°)$