Formula and Method for the Inverse Sine (arcsin)
The inverse sine function, written arcsin(x) or sin⁻¹(x), answers the question: "which angle has a sine equal to x?" It is the inverse of the sine function, restricted to a single output range so that it behaves as a true function. Because sine repeats every 360° (2π radians) and is symmetric within each period, infinitely many angles technically share the same sine value — arcsin resolves that ambiguity by always returning the one angle in the principal range of -90° to 90° (-π/2 to π/2 radians).
How the calculation works
Enter a value x between -1 and 1 — this is the sine ratio (opposite ÷ hypotenuse in a right triangle, or the y-coordinate on the unit circle). The calculator computes θ = arcsin(x) using the principal branch, then converts between degrees and radians with radians = degrees × π/180. It also reports the related angle cos(θ) = √(1 - x²), which follows directly from the Pythagorean identity sin²(θ) + cos²(θ) = 1, and the second angle within one full turn that also satisfies sin(θ) = x, found as 180° - θ (normalized to the 0°-360° range).
Domain, range, and multiple solutions
The domain of arcsin(x) is -1 ≤ x ≤ 1, because no real angle produces a sine outside that interval. Its range — the set of possible outputs — is limited to -90° to 90° (-π/2 to π/2 rad), the principal value. Beyond that single window, sine's periodicity means every valid x actually corresponds to a full family of angles: θ = arcsin(x) + 360°n and θ = 180° - arcsin(x) + 360°n for any integer n. The calculator's "Other Solution" result shows the second family member within one turn (0° to 360°).
Common mistakes
- Confusing arcsin(x) with 1/sin(x): arcsin returns an angle; 1/sin(x) (cosecant) returns a ratio. They are not related operations.
- Entering a value outside [-1, 1]: sine can never exceed 1 or go below -1, so arcsin is undefined for any x outside that range — there is no real-number answer.
- Assuming arcsin(sin θ) always returns θ: this only holds when θ is already inside the principal range [-90°, 90°]; outside it, arcsin "wraps" the angle back into that window.
- Mixing degrees and radians: always confirm which unit a downstream formula expects before plugging in the result.
Real-world applications
- Physics: finding launch or incidence angles, such as the critical angle in optics via Snell's law (θ = arcsin(n₂/n₁)).
- Engineering: solving for phase angles in AC circuits and mechanical vibration analysis.
- Navigation and surveying: recovering an unknown angle in a right triangle when the opposite side and hypotenuse are known.
- Computer graphics and robotics: converting a normalized direction component back into a rotation angle.