Inverse Sine Calculator — arcsin(x)
Enter a sine value between −1 and 1 to find the angle (in degrees, radians or gradians) whose sine equals that value. The result is the principal value of arcsin.
Output unit
Principal value: the angle whose sine equals the input
- 1
arcsin(x) in radians
arcsin(0.5) = 0.523599Principal value in [−π/2, π/2] — the angle whose sine equals x. - 2
Convert to degrees
0.523599 × 180 ÷ π = 30
How does this calculator work?
arcsin(x) returns the angle θ in [−90°, 90°] whose sine equals x (x must be between −1 and 1). Enter the sine value and choose degrees, radians or gradians; the calculator also shows the complementary arccos angle and equivalents in all three units.
Formula
How this is calculated
The inverse sine function, written arcsin(x) or sin⁻¹(x), reverses the sine operation: given a value x, it returns the angle θ such that sin(θ) = x. Because sine repeats every 360°, many angles share the same sine. The principal-value branch of arcsin resolves this by returning the unique angle in [−90°, 90°] (equivalently [−π/2, π/2] radians). For angles outside this range, the general solution is θ = arcsin(x) + 360°·n or θ = 180° − arcsin(x) + 360°·n for any integer n.
The domain is restricted to [−1, 1] because sine never exceeds 1 or falls below −1 for real angles. Exact values you can verify: arcsin(0) = 0°, arcsin(1) = 90°, arcsin(−1) = −90°, arcsin(0.5) = 30°, arcsin(√2/2) = 45°, arcsin(√3/2) = 60°.
Conversions: 1 radian = 180°/π ≈ 57.296°; gradians (grad) use 400 per full circle (1 grad = 0.9°). The complementary relationship arcsin(x) + arccos(x) = 90° means you can derive one from the other instantly.
Frequently asked questions
Arcsin is the inverse of sine restricted to [−90°, 90°], where sine is strictly increasing and one-to-one. Extending to a wider range would make one input map to multiple outputs, which is not a function.
If arcsin(x) = α, the full set of solutions is α + 360°·n and (180° − α) + 360°·n for any integer n. For example arcsin(0.5) = 30°, so 150°, 390°, −210° etc. also have sine = 0.5.
They are the same function. The notation sin⁻¹ is common on calculators; arcsin is the standard mathematical notation. Neither means 1/sin(x), which is cosecant (csc).
Also known as
TG we-Calculate Editorial Team. (2026). Inverse Sine Calculator — arcsin(x) [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/inverse-sine-calculator
TG we-Calculate Editorial Team. "Inverse Sine Calculator — arcsin(x)." TG we-Calculate. 2026. https://we-calculate.com/calculator/inverse-sine-calculator.
TG we-Calculate Editorial Team, "Inverse Sine Calculator — arcsin(x)," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/inverse-sine-calculator
@misc{wecalculate_inverse_sine_calculator, title = {Inverse Sine Calculator — arcsin(x)}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/inverse-sine-calculator}}, year = {2026}, note = {TG we-Calculate} }
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