Compute the inverse sine (arcsine) of any number in the domain [-1, 1]. Choose output in radians or degrees.Visualize the arcsin function curve, see exact point on graph, and explore detailed mathematical properties, derivations, and real‑world applications.
The inverse sine function, denoted arcsin(x) or sin⁻¹(x), returns the angle whose sine is x. Because the sine function is periodic and not one‑to‑one, we restrict its domain to [−π/2, π/2] (the principal branch) to obtain a well‑defined inverse. Thus, for any x ∈ [−1, 1], arcsin(x) ∈ [−π/2, π/2] (or [−90°, 90°] in degrees).
If θ = arcsin(x) then sin(θ) = x and θ ∈ [−π/2, π/2]
Derivative: d/dx arcsin(x) = 1/√(1−x²) Integral: ∫ arcsin(x) dx = x arcsin(x) + √(1−x²) + C
The concept of inverse trigonometric functions emerged in the works of European mathematicians during the 17th and 18th centuries. John Bernoulli, Leonhard Euler, and others contributed to their formalization. The notation "arcsin" (meaning "the arc whose sine is") was introduced by the French mathematician Joseph‑Louis Lagrange. The arcsin function is fundamental in integral calculus (e.g., ∫ dx/√(1−x²) = arcsin(x) + C), in solving differential equations, and in countless physics and engineering problems such as pendulum motion, wave propagation, and navigation (great‑circle distance calculations).
Given x, arcsin(x) can be computed using the series expansion (for |x| ≤ 1):
arcsin(x) = x + (1/6)x³ + (3/40)x⁵ + (5/112)x⁷ + … (valid for |x| ≤ 1).
Our calculator uses the high‑precision Math.asin() function built into JavaScript, which typically implements highly optimized numerical algorithms (such as the CORDIC algorithm or minimax polynomial approximations) to ensure both speed and sub‑1e‑15 relative error. This guarantees results with high accuracy. Additionally, we provide unit conversion (radians ↔ degrees) and verify the result by recomputing sin(θ).
| Property | Formula |
|---|---|
| Symmetry | arcsin(−x) = −arcsin(x) (odd function) |
| Complementary relation | arcsin(x) + arccos(x) = π/2 |
| Sine of arcsin | sin(arcsin(x)) = x |
| Arcsin of sine (restricted) | arcsin(sin(θ)) = θ for θ ∈ [−π/2, π/2] |
| Derivative | d/dx arcsin(x) = 1/√(1−x²) |
| Indefinite integral | ∫ arcsin(x) dx = x arcsin(x) + √(1−x²) + C |
In optics, Snell's law states n₁ sin(θ₁) = n₂ sin(θ₂). When light passes from water (n₁ ≈ 1.33) to air (n₂ = 1.0) with an incident angle θ₁ = 45°, the transmitted angle θ₂ = arcsin((n₁/n₂) sin(θ₁)) = arcsin(1.33 × 0.7071) = arcsin(0.940). Our calculator gives θ₂ ≈ 1.222 rad (70.0°). This critical computation helps design lenses, fiber optics, and underwater cameras. The arcsin function is indispensable for solving for angles in any trigonometric relationship.
In digital signal processing and communications, the inverse sine function is crucial for recovering the phase of a signal. Consider a quadrature signal with in‑phase (I) and quadrature (Q) components. The phase φ of the signal can be extracted using φ = arcsin(Q / A) in certain quadrants, where A = √(I² + Q²) is the amplitude. More robustly, the four‑quadrant arctangent function (atan2) is used, which itself relies on arcsin and arccos. This application is foundational in software‑defined radio, radar, and any system that modulates information in the phase of a carrier wave.
| x | arcsin(x) (radians) | arcsin(x) (degrees) |
|---|---|---|
| -1 | -π/2 ≈ -1.5708 | -90° |
| -√3/2 ≈ -0.8660 | -π/3 ≈ -1.0472 | -60° |
| -√2/2 ≈ -0.7071 | -π/4 ≈ -0.7854 | -45° |
| -0.5 | -π/6 ≈ -0.5236 | -30° |
| 0 | 0 | 0° |
| 0.5 | π/6 ≈ 0.5236 | 30° |
| √2/2 ≈ 0.7071 | π/4 ≈ 0.7854 | 45° |
| √3/2 ≈ 0.8660 | π/3 ≈ 1.0472 | 60° |
| 1 | π/2 ≈ 1.5708 | 90° |
This tool's content and implementation adhere to standard practices in mathematics and engineering, and are informed by the following authoritative sources: