Inverse Trig Functions Calculator (arcsin, arccos, arctan)

Ratio → angle: the reverse of trig functions

A trig function takes an angle and returns a ratio: $\sin 30° = 0.5$. An inverse trig function runs that backward — it takes the ratio and returns the angle:

$\text{if } \sin\theta = x \text{ then } \theta = \arcsin x \; (= \sin^{-1}x)$

The same pattern holds for all six functions: $\arccos$, $\arctan$, $\arccot$, $\arcsec$, $\arccsc$ each undo their matching trig function.

The six inverse functions and their principal range

Because sine, cosine, and the rest repeat forever — $\sin 30° = \sin 150° = 0.5$ — an inverse function has to pick exactly one angle to return, or it wouldn't be a function at all. Each one is restricted to a principal range:

So even though sin 150° = 0.5 is a true statement, arcsin(0.5) reports 30°, not 150° — 30° is the one representative that lives inside arcsin's principal range.

Worked examples

$\arcsin(0.5) = 30° = \frac{\pi}{6} \text{ rad}$

$\arccos(0.5) = 60° = \frac{\pi}{3} \text{ rad}$

$\arctan(1) = 45° = \frac{\pi}{4} \text{ rad}$

$\arccos(-1) = 180° = \pi \text{ rad}$

Each of these lands on a special angle (a multiple of 30° or 45°), so the calculator reports the exact degree and exact π-fraction alongside the decimal radians — not just a rounded approximation.

Why some inputs have no answer (domain guards)

Every real angle's sine and cosine sit somewhere in $[-1, 1]$ — the unit circle's y- and x-coordinates never leave that band. That means:

• arcsin(2) and arccos(-1.5) have no real answer — no angle's sine or cosine ever reaches 2 or −1.5.
• arcsec(0.5) and arccsc(0.3) also fail, for the mirror-image reason: $\sec\theta = 1/\cos\theta$ and $\csc\theta = 1/\sin\theta$ are reciprocals of numbers in $[-1,1]$, so they can never land strictly between −1 and 1. arcsec/arccsc only accept $|x| \ge 1$.
• arctan and arccot have no such gap — tangent and cotangent range over every real number, so any x is valid.

Reading the unit-circle diagram

The diagram highlights the selected function's principal range as a colored arc along the circle — the "allowed zone" the answer must fall in — then plots the terminal point for the resolved angle θ and draws a small arc from the positive x-axis out to θ. For arcsin, that highlighted zone is the right half of the circle (−90° to 90°); for arccos, it's the top half (0° to 180°) — matching the table above.

Common mistakes

• Confusing $\sin^{-1}x$ with $1/\sin x$. They're different operations entirely — one returns an angle, the other returns $\csc x$.
• Expecting multiple answers. arcsin, arccos, etc. always return exactly one angle (the principal one), even though infinitely many coterminal angles share that same ratio.
• Entering a value outside the domain, like arcsin(2) or arcsec(0.5) — these simply have no real-number answer.
• Forgetting which unit the answer is in. This calculator's headline is always in degrees; radians and the exact π-fraction (when one exists) are shown alongside it.

Where this shows up

• Solving triangles — Law of Sines and Right Triangle both need an inverse trig step whenever you know two sides but want an angle.
• Navigation and surveying — finding a bearing or angle of elevation from a measured slope or distance ratio.
• Graphics and robotics — converting a direction vector's components back into a rotation angle.

This calculator is the reverse of Trigonometric Functions (angle → ratio via the unit circle) and complements Trigonometric Ratios (SOH-CAH-TOA from two right-triangle sides) — all three share the same underlying six-function engine.