Angle from Ratio (Inverse Trig) Calculator
Find an angle from its sine, cosine or tangent value using arcsin, arccos or arctan — with domain checking for asin and acos.
How it works
θ = arcsin(x), arccos(x) or arctan(x) · x must be in [-1, 1] for arcsin and arccos
Inverse trigonometric functions run the ratios backwards: given a ratio, they return the angle that produces it. arcsin and arccos only accept inputs between -1 and 1, because sine and cosine never leave that range, so any value outside it has no angle. arctan accepts any real number. The functions return their principal value — arcsin and arctan in the range -90° to 90°, and arccos in 0° to 180°.
Worked example
arcsin(0.5) = 30° (0.5236 rad), because sin 30° = 0.5. Likewise arccos(0.5) = 60° and arctan(1) = 45°.
Frequently asked questions
Why must the input to arcsin be between -1 and 1?
Sine of any angle always lies between -1 and 1, so a value like 2 is never the sine of a real angle. The calculator rejects out-of-range inputs instead of returning an invalid result.
What is a principal value?
Many angles share the same sine or tangent, so each inverse function returns one standard answer: arcsin and arctan give -90° to 90°, and arccos gives 0° to 180°. Add full rotations or reflections to find other angles with the same ratio.
How is this different from the trig ratios calculator?
The trig ratios calculator goes from angle to ratio (sin 30° = 0.5); this one goes from ratio back to angle (arcsin 0.5 = 30°). They are inverse operations.
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.