Arcsin Calculator
Enter a value between −1 and 1 and get its inverse sine — the angle whose sine equals that value — in degrees, radians, multiples of π and gradians.
arcsin(0.5)
30°
- Degrees
- 30°
- Radians
- 0.523599 rad
- Radians as a multiple of π
- 0.166667 × π
- Gradians
- 33.333333 gon
arcsin returns the angle between −90° and 90° (−π/2 and π/2 radians) whose sine equals your value. It is the inverse of sine on that interval.
How to use this calculator
Type the sine value you want to invert — any number from −1 to 1, decimals and negatives included. Pick whether the headline result should show degrees or radians; the breakdown always lists both, plus the radian value written as a multiple of π and the angle in gradians. Negative inputs give negative angles: that is correct, because arcsin of a negative value lands between −90° and 0°. If you enter a value outside −1 to 1 the calculator tells you the input is out of range rather than returning a wrong answer, because no real angle has a sine beyond that interval.
How the calculation works
Sine takes an angle and returns a ratio between −1 and 1; arcsin runs that in reverse. Because sine repeats every 360° and takes each value twice per cycle, it only has a true inverse when restricted to one interval — by convention −90° to 90° (−π/2 to π/2 radians), where sine climbs from −1 to 1 passing through every value exactly once. arcsin(x) is therefore defined as the unique angle θ in [−π/2, π/2] with sin θ = x, the principal value used by every scientific calculator and programming language. Degrees convert from radians by multiplying by 180/π, and gradians (also called gons, where a right angle is 100) by multiplying by 200/π. All other angles sharing the same sine are θ plus whole multiples of 360°, or 180° − θ plus whole multiples of 360°.
Worked example
Take x = 0.5. The angle between −90° and 90° whose sine is 0.5 is 30°, so arcsin(0.5) = 30° — exactly π/6 radians (≈ 0.523599) or 33.333333 gradians. You can verify it in reverse: sin(30°) = 0.5. For a negative input the answer is simply the negative angle: arcsin(−0.5) = −30° = −π/6 radians, because sine is an odd function — sin(−θ) = −sin(θ).
Frequently asked questions
What is arcsin?
Arcsin (also written sin⁻¹ or asin) is the inverse of the sine function. Where sine takes an angle and gives you a ratio, arcsin takes the ratio and gives you back the angle. arcsin(x) returns the unique angle between −90° and 90° (−π/2 and π/2 radians) whose sine equals x. For example, arcsin(0.5) = 30° because sin(30°) = 0.5.
Why does arcsin only accept values between −1 and 1?
Because sine itself only ever produces values in that range — it is the ratio of the opposite side to the hypotenuse in a right triangle (or the y-coordinate on the unit circle), and neither can exceed the hypotenuse or radius. Asking for arcsin(2) means asking "which angle has a sine of 2?", and no real angle does, so the result is undefined.
Why does arcsin return angles between −90° and 90°?
Sine repeats every 360° and satisfies sin(180° − θ) = sin(θ), so infinitely many angles share any given sine value. A function can only return one answer, so arcsin is defined on the interval where sine passes through every value in [−1, 1] exactly once: −90° to 90°. This is called the principal value. Every other solution is either θ + 360°k or (180° − θ) + 360°k for a whole number k. Note this differs from arccos, whose principal range is 0° to 180°.
Why is my arcsin result negative?
Because your input was negative, and that is the correct answer. Sine is an odd function — sin(−θ) = −sin(θ) — so the principal-value angle for a negative sine is a negative angle: arcsin(−0.5) = −30°. If your problem expects an angle between 0° and 360° (a bearing or a rotation, say), add 360° to the negative result, or use 180° − θ for the second-quadrant solution, depending on which quadrant fits your situation.
What are the exact arcsin values worth remembering?
The unit-circle specials: arcsin(−1) = −90°, arcsin(−√3/2) = −60°, arcsin(−√2/2) = −45°, arcsin(−1/2) = −30°, arcsin(0) = 0°, arcsin(1/2) = 30°, arcsin(√2/2) = 45°, arcsin(√3/2) = 60°, and arcsin(1) = 90°. In radians those are −π/2, −π/3, −π/4, −π/6, 0, π/6, π/4, π/3 and π/2.
How do arcsin, arccos and arctan relate?
They are the inverses of the three main trig functions, each restricted to its own principal range: arcsin and arctan return angles in [−90°, 90°], while arccos returns angles in [0°, 180°]. For any x in [−1, 1] they satisfy the identity arcsin(x) + arccos(x) = 90° (π/2 radians), so you can always get one from the other by subtracting from 90°.