Sin⁻¹ Calculator — Inverse Sine Angle
Given a sine value x between −1 and 1, find the angle θ such that sin(θ) = x. The result is the principal value in [−90°, 90°], returned in degrees, radians, and as a multiple of π.
Principal angle θ in [−90°, 90°] such that sin(θ) = x
- 1
Apply inverse sine (arcsin)
arcsin(0.707) = 0.785247arcsin returns the principal angle in [−π/2, π/2] radians. - 2
Convert radians to degrees
0.785247 × 180 ÷ π = 44.9913
How does this calculator work?
sin⁻¹(x) returns the angle θ in [−90°, 90°] whose sine equals x. Enter any x between −1 and 1 to get θ in degrees (θ = asin(x) × 180/π) and radians. The complementary angle is arccos(x) = 90° − θ. Values outside [−1, 1] have no real solution.
Formula
How this is calculated
The inverse sine function sin⁻¹(x) — also written arcsin(x) — answers: "what angle has a sine equal to x?" Because the sine function is periodic (it repeats every 360°), infinitely many angles share the same sine value, so a unique answer requires restricting the output to the principal branch [−90°, 90°]. On this interval sine is strictly increasing and maps [−1, 1] to [−90°, 90°] exactly once.
The radian result is computed via the IEEE 754 Math.asin() routine. To convert to degrees, multiply by 180/π. The complementary angle arccos(x) follows from the identity arcsin(x) + arccos(x) = 90° (which holds for all x in [−1, 1]), reflecting the fact that in a right triangle the two acute angles sum to 90°.
Inputs outside [−1, 1] have no real inverse sine — the sine function never reaches those values — and the calculator returns a warning. For general-solution angles beyond the principal branch, the full set of solutions is θ + 360°·k and (180° − θ) + 360°·k for any integer k.
Frequently asked questions
sin⁻¹(x) is the inverse sine (arcsin), not 1/sin. The notation f⁻¹ means "undo the function f", so sin⁻¹(x) returns the angle whose sine is x. The reciprocal 1/sin(θ) is called cosecant (csc), a completely different function.
Sine is not one-to-one over its full domain — the same sine value occurs at infinitely many angles. The inverse is defined on the principal branch [−90°, 90°], where sine is monotone, so there is exactly one output per input and the function is well-defined.
From the principal value θ = sin⁻¹(x), the general solutions are θ + 360°·k and (180° − θ) + 360°·k for any integer k. For example, sin⁻¹(0.5) = 30°, and 150°, 390°, 510°, −210°, … all have a sine of 0.5.
Also known as
TG we-Calculate Editorial Team. (2026). Sin⁻¹ Calculator — Inverse Sine Angle [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/sin-1-calculator
TG we-Calculate Editorial Team. "Sin⁻¹ Calculator — Inverse Sine Angle." TG we-Calculate. 2026. https://we-calculate.com/calculator/sin-1-calculator.
TG we-Calculate Editorial Team, "Sin⁻¹ Calculator — Inverse Sine Angle," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/sin-1-calculator
@misc{wecalculate_sin_1_calculator, title = {Sin⁻¹ Calculator — Inverse Sine Angle}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/sin-1-calculator}}, year = {2026}, note = {TG we-Calculate} }
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