Arcsine Calculator

The arcsine function, written arcsin(x) or sin⁻¹(x), is the inverse of the sine function. Given a value x in the interval [−1, 1], arcsin(x) returns the unique angle θ in [−π/2, π/2] (equivalently [−90°, 90°]) such that sin(θ) = x. Because the sine function is periodic and repeats every 360°, the inverse needs a restricted range to produce a single-valued output — that restricted range is called the principal value. Arcsine is an odd function: arcsin(−x) = −arcsin(x). Its complementary identity arcsin(x) + arccos(x) = π/2 holds for all x in the domain. Derivative: d/dx arcsin(x) = 1/√(1 − x²).

Sine takes an angle and returns a ratio between −1 and 1 (the opposite side over the hypotenuse in a right triangle). Arcsine does the opposite: you give it a ratio, and it returns the angle. For example, sin(30°) = 0.5, so arcsin(0.5) = 30°. In calculator notation arcsin is often labeled sin⁻¹ — this is not an exponent, it denotes the inverse function.

The sine of any real angle is always between −1 and 1 inclusive — that is its entire range. So only values in [−1, 1] correspond to a real angle. If x is outside this interval (for example x = 2 or x = −1.5), there is no real angle whose sine equals x, and the answer is undefined in real numbers. To compute arcsin for |x| > 1 you need complex numbers, which is outside the scope of this calculator.

Because sin(θ) repeats every 360° and is symmetric, infinitely many angles share the same sine value. To make arcsin a proper function (one output per input), mathematicians restrict its output to the principal branch [−π/2, π/2], i.e. [−90°, 90°]. If you need every angle whose sine equals x, use θ = arcsin(x) + 360°·k and θ = 180° − arcsin(x) + 360°·k for integer k.

Arcsine shows up whenever you know a ratio and need the angle. Common uses:

• Right-triangle trigonometry: angle = arcsin(opposite / hypotenuse). A ramp 3 m long rising 1.5 m has angle arcsin(1.5/3) = 30°.
• Physics: Snell's law for refraction, angle of incidence problems, projectile launch angles.
• Surveying and navigation: elevation, bearing, and latitude calculations.
• Engineering and signal processing: phase angles in AC circuits and oscillations.
• Statistics: the arcsine variance-stabilizing transformation for proportions.

Use the toggle above to switch instantly. Manually: radians = degrees × π/180, and degrees = radians × 180/π. Quick reference: 180° = π rad ≈ 3.14159, 90° = π/2 rad ≈ 1.57080, 60° = π/3 rad ≈ 1.04720, 45° = π/4 rad ≈ 0.78540, 30° = π/6 rad ≈ 0.52360. Radians are the standard in calculus, physics, and programming (JavaScript's Math.asin returns radians).

They are complementary: arcsin(x) + arccos(x) = π/2 = 90° for every x in [−1, 1]. So if arcsin(0.5) = 30°, then arccos(0.5) = 60°. This mirrors the fact that in a right triangle the two non-right angles sum to 90°. Arcsine is an odd function (symmetric about the origin), while arccosine is neither odd nor even.