Inverse Trig Calculator
Enter a value and select an inverse trig function to find the corresponding angle. Results are given as the principal value in both degrees and radians.
The Six Inverse Trig Functions
The inverse trig functions "undo" the trig functions — given a ratio, they return an angle. Because trig functions repeat every 360°, a unique answer requires restricting the output to a principal value range.
| Function | Notation | Domain (valid inputs) | Principal value range |
|---|---|---|---|
| arcsin | $\sin^{-1}(x)$ | $-1 \le x \le 1$ | $[-90°,\, 90°]$ |
| arccos | $\cos^{-1}(x)$ | $-1 \le x \le 1$ | $[0°,\, 180°]$ |
| arctan | $\tan^{-1}(x)$ | all real numbers | $(-90°,\, 90°)$ |
| arccsc | $\csc^{-1}(x)$ | $|x| \ge 1$ | $[-90°, 0) \cup (0°, 90°]$ |
| arcsec | $\sec^{-1}(x)$ | $|x| \ge 1$ | $[0°, 90°) \cup (90°, 180°]$ |
| arccot | $\cot^{-1}(x)$ | all real numbers | $(0°,\, 180°)$ |
Useful Identities
$$\arcsin(x) + \arccos(x) = 90° \quad (\text{for } -1 \le x \le 1)$$ $$\arctan(x) + \arctan\!\left(\tfrac{1}{x}\right) = 90° \quad (\text{for } x > 0)$$ $$\arcsin(-x) = -\arcsin(x) \qquad \arccos(-x) = 180° - \arccos(x)$$To compute the trig functions (forward direction), use the Sin Cos Tan calculator. To understand where angles fall on the circle geometrically, try the unit circle calculator.
Frequently Asked Questions
Why does arcsin only return values between −90° and 90°?
The sine function is not one-to-one over all real numbers — e.g. $\sin(30°) = \sin(150°) = 0.5$. To define an inverse, we restrict the range to $[-90°, 90°]$ where sine is strictly increasing. This gives the principal value. If your problem requires an angle outside this range, use the symmetry identities or the context of the problem to find the other solution.
Is arctan(x) the same as 1/tan(x)?
No. $\arctan(x)$ is the inverse function (returns an angle), while $1/\tan(x) = \cot(x)$ is the reciprocal function (returns a ratio). This is the same distinction as $\sin^{-1}$ vs $1/\sin = \csc$.
What does "domain error" mean?
The input you entered is outside the domain of the selected function. For example, $\arcsin(2)$ has no solution because the sine function never reaches 2 — its maximum is 1. The calculator will explain the valid range when this happens.
