# What is the inverse sine of x?

Finds arcsin x, the angle from −90° to 90° whose sine is x, in degrees and radians, exact as a multiple of π for the special values.

- Page: https://www.acalculator.org/math/inverse-sine-calculator
- JSON spec: https://www.acalculator.org/math/inverse-sine-calculator.json
- Version: 30070409c0b6

## Default answer

Example with the default inputs (Value x 0.5): arcsin(0.5) = 30°.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| x | Value x | A number from −1 to 1: the sine of the angle. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| angle | arcsin x | The principal angle whose sin is x, in degrees. |
| radians | In radians | The same angle in radians. |
| exact | Exact angle | The angle as a multiple of π, when x is the value at a multiple of 15°. |

## Method

arcsin x is the angle θ from −90° to 90° (−π/2 to π/2) with sin θ = x. When x is within 10⁻¹⁴ of the sine of a multiple of 15° in that range, the angle is that multiple exactly; otherwise it is computed in 64-bit floating point.

## Assumptions

- x is from −1 to 1; the sine of a real angle is never outside that range.
- The answer is the principal value. Other angles with the same sine are 180° − θ and those angles plus whole turns.

## Worked examples

1. x = 0.5 gives angle = 0.523599, radians = 0.523599, exact = π/6 = 30° (sin 30° = 1/2). Source: OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arcsin: domain [−1, 1], range [−π/2, π/2]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions.
2. x = -1 gives angle = -1.570796, exact = -π/2 = -90° (sin -90° = -1). Source: OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arcsin: domain [−1, 1], range [−π/2, π/2]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions.
3. x = 0.707107 gives angle = 0.785398, exact = π/4 = 45° (sin 45° = √2/2). Source: OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arcsin: domain [−1, 1], range [−π/2, π/2]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions.
4. x = 0.3 gives angle = 0.304693, radians = 0.304693. Source: NIST DLMF §4.23 (inverse trigonometric functions, principal values), https://dlmf.nist.gov/4.23.

## FAQ

### What is inverse sine?

Inverse sine, written arcsin x or sin⁻¹ x, undoes the sine: it gives the angle whose sine is x. Because many angles share a sine, it gives the principal value, the one from −90° to 90°.

### What is arcsin 0.5?

arcsin 0.5 = 30° = π/6 radians, because sin 30° = 1/2. Other angles with sine 1/2, such as 150°, are not principal values.

### Why does arcsin 2 give an error?

The sine of a real angle is always from −1 to 1, so no angle has sine 2. The calculator accepts x from −1 to 1 only.

### Is sin⁻¹ x the same as 1 ÷ sin x?

No. sin⁻¹ x is the inverse function, arcsin x. The reciprocal 1 ÷ sin x is the cosecant, csc x.

### How do I find all the angles with a given sine?

Take θ = arcsin x. The angles with sine x are θ + 360°k and 180° − θ + 360°k for any whole number k. For x = 1/2: 30°, 150°, 390°, 510°, and so on.

### How precise is the answer?

For the special values (0, ±1/2, ±√2/2, ±√3/2, ±1 and the 15° values) the angle is exact. Typing √2/2 as 0.7071067811865476 counts, because it is within 10⁻¹⁴ of √2/2. Other values are computed to about 16 significant digits.

## Sources

- OpenStax, Algebra and Trigonometry 2e, §8.3 Inverse Trigonometric Functions (arcsin: domain [−1, 1], range [−π/2, π/2]). https://openstax.org/books/algebra-and-trigonometry-2e/pages/8-3-inverse-trigonometric-functions (retrieved 2026-10-01)
- OpenStax, Algebra and Trigonometry 2e, §7.3 Unit Circle (special angles). https://openstax.org/books/algebra-and-trigonometry-2e/pages/7-3-unit-circle (retrieved 2026-10-01)
- NIST Digital Library of Mathematical Functions, §4.23 Inverse Trigonometric Functions (principal values). https://dlmf.nist.gov/4.23 (retrieved 2026-10-01)