Inverse Trigonometric Functions

Regular trig asks: "I know the angle — what's the ratio?" Inverse trig flips the question: "I know the ratio — what's the angle?" For example, $\arcsin(\frac{1}{2})$ asks "which angle has a sine of $\frac{1}{2}$?" and the answer is $\frac{\pi}{6}$ (30°). The input is a number, and the output is an angle — the reverse of normal trig functions.

There's a catch: sine hits the same value at many different angles (for example, $\sin(30°)$ and $\sin(150°)$ both equal $\frac{1}{2}$). So to make inverse trig give one clean answer, we restrict the range — each inverse function only looks at a specific slice of angles. Arcsin returns angles in $[-\frac{\pi}{2}, \frac{\pi}{2}]$, arccos returns $[0, \pi]$, and arctan returns $(-\frac{\pi}{2}, \frac{\pi}{2})$. Always check that your answer is in the right slice.

For problems like $\sin(\arccos(x))$ — where trig and inverse trig are layered together — there's a powerful trick: draw a right triangle. If $\theta = \arccos(x)$, label the adjacent side as $x$ and hypotenuse as $1$, find the missing side with the Pythagorean theorem, then read off whatever ratio you need. This triangle technique works for any composition and avoids sign errors.

One common trap: $\arcsin(\sin(x))$ does not always equal $x$. It only equals $x$ when $x$ is already inside arcsin's restricted range. For example, $\arcsin(\sin(\frac{5\pi}{6})) = \frac{\pi}{6}$, not $\frac{5\pi}{6}$. Always compute the inner function first, then ask: "what angle *in the restricted range* gives this value?"