Inverse Trigonometric Functions — Formulas & Graphs

Evaluating an inverse trig expression is a four-step habit: match the function, check the domain, choose the angle from the principal range, and verify. Inverse trig functions return an angle from a trig value, and $\arcsin x$, $\arccos x$, and $\arctan x$ each return one standard angle — the principal value — not every angle that works.

That restriction is what makes the inverse single-valued. Sine, cosine, and tangent repeat values across their full graphs, so they only have inverses after we limit them to intervals where each output comes from exactly one angle.

When this method applies

Use inverse trig whenever you know a ratio and need the angle back: right-triangle geometry, navigation, slope and direction, vector components, and triangle modeling. They also appear in calculus — in derivatives, in antiderivatives such as $\int \frac{1}{1+x^2}\,dx = \arctan x + C$, and in trig substitutions. The method only returns the standard angle, so reading the principal range correctly is part of using it at all.

The procedure, step by step

1. Match the function. Decide whether the value should be read with $\arcsin$, $\arccos$, or $\arctan$.
2. Check the domain. Confirm the input is allowed before evaluating. The defining ranges:

3. Use the principal range. Choose the standard output angle from that restricted range, not just any angle with the same trig value.
4. Verify. Apply sine, cosine, or tangent to your result to confirm it returns the original input.

A full run-through

Evaluate

Step 1: the value sits with cosine, so use $\arccos$. Step 2: $-\tfrac{1}{2}$ is inside $[-1, 1]$, so it is allowed. Step 3: many angles satisfy $\cos y = -\tfrac{1}{2}$, but $\arccos$ must return the one in $0 \le y \le \pi$, which is $y = \tfrac{2\pi}{3}$:

Step 4: $\cos\tfrac{2\pi}{3} = -\tfrac{1}{2}$ confirms it. The habit to build: do not ask for any angle that works — ask for the angle in the correct range.

How the graphs fit this

Inverse trig graphs are reflections across $y = x$, but only of the restricted original. For example, $y = \arcsin x$ reflects $y = \sin x$ on $-\tfrac{\pi}{2} \le x \le \tfrac{\pi}{2}$; similarly $\arccos$ pairs with cosine on $[0, \pi]$ and $\arctan$ with tangent on $\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)$. Never reflect the full repeating graph — it fails the horizontal line test and has no inverse.

Where each step trips people up

• At "match the function," confusing inverse with reciprocal trig. $\arcsin x$ is not $\csc x$, and $\sin^{-1} x$ means inverse sine, not $1/\sin x$.
• At "use the principal range," ignoring it. $\sin\tfrac{5\pi}{6} = \tfrac{1}{2}$, yet $\arcsin\tfrac{1}{2} = \tfrac{\pi}{6}$, because $\tfrac{\pi}{6}$ is the allowed angle.
• At "check the domain," forgetting it. $\arcsin 2$ and $\arccos(-3)$ are not real-valued, since sine and cosine never leave $[-1, 1]$. Self-check: ask which function matches the value, then which angle in its principal range fits.

FAQ

Run the steps on $\arcsin\left(-\tfrac{\sqrt{2}}{2}\right)$ and $\arctan(1)$. Pin down the principal range first and both answers fall into place quickly.