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Arcsin(x), written sin⁻¹(x), is the inverse of the sine function: it answers "what angle has this sine value?" Since sine only ever produces values between -1 and 1, arcsin is only defined on that range, and by convention it always returns an angle between -90° and 90° (its principal value range). Arcsin turns up whenever a known ratio needs to become an angle: finding the angle of elevation or depression in navigation and surveying, computing a projectile's launch angle from its velocity components in physics, working out a joint angle in inverse kinematics for robotics or animation, or determining the angle of incidence in optics from Snell's Law.
How it works
Sine takes an angle and returns the ratio of the opposite side to the hypotenuse in a right triangle (or, more generally, the y-coordinate on the unit circle). Arcsin reverses that: given a ratio x between -1 and 1, it finds the unique angle θ in the range -90° to 90° such that sin(θ) = x. Sine is one-to-one (strictly increasing) over that -90°-to-90° interval, which is exactly why that range was chosen as arcsin's principal value range — it guarantees a single unambiguous answer.
- Enter value (between -1 and 1).
- Click Calculate to see your results.
Examples
arcsin(0.5)
arcsin(0.5) = 30°, because sin(30°) = 0.5.
arcsin(-1)
arcsin(-1) = -90°, the minimum possible arcsin output, since sin(-90°) = -1.
arcsin(0.7071)
arcsin(0.7071) ≈ 45°, since sin(45°) = √2/2 ≈ 0.7071.
Who should use it
- Finding an angle of elevation or depression from a height-to-distance ratio.
- Computing a launch angle from velocity components in projectile motion.
- Working out the angle of incidence in Snell's Law problems.
Industry applications
- Physics and engineering
- Navigation and surveying
- Trigonometry education
Advantages
- Instant, exact results in both degrees and radians.
- Clear, unambiguous principal-range convention.
Limitations
- Only returns the principal value (-90° to 90°) — cannot by itself distinguish an obtuse angle with the same sine.
Common mistakes to avoid
- Entering a value outside [-1, 1], which has no valid real-number arcsin.
- Confusing arcsin(x) with cosecant (1/sin(x)) — they are unrelated operations despite similar-looking notation.
- Forgetting that a sine ratio corresponds to two possible angles in a full circle (e.g., both 30° and 150° have sine 0.5) — arcsin only ever reports the principal one.
Best practices
- If your original problem involves an obtuse angle, check whether 180° minus the arcsin result is the actual angle you need, since arcsin only reports the -90° to 90° branch.
- Use the co-function shortcut arcsin(x) + arccos(x) = 90° as a quick cross-check.
- Keep track of whether your downstream formula expects degrees or radians.
Tips
- Need the inverse cosine or inverse tangent instead? Use the Arccos Calculator or Arctan Calculator.