What this calculator does
Arcsin is the inverse of the sine function. Sine takes an angle and returns a ratio between -1 and 1; arcsin does the reverse, taking that ratio and returning the angle that produced it. It is written sin⁻¹(x) or asin(x) on a calculator, and both mean the same thing as arcsin.
Because sine repeats itself, many different angles share the same sine value: sin(30°) and sin(150°) are both 0.5. Arcsin always returns a single answer, restricted to the range -90° to 90°, rather than every angle that would work. That restricted range is called the principal value, and it is what this calculator and every scientific calculator returns.
The formula
Enter a value between -1 and 1, since no angle has a sine outside that range. The calculator returns the principal value of arcsin(x): the one angle between -90° and 90° whose sine is exactly x, shown in both degrees and radians.
| Term | Meaning |
|---|---|
| arcsin(x) | The angle, between -90° and 90°, whose sine equals x. |
| Principal value | The single angle arcsin returns, out of the infinitely many angles that technically share the same sine. |
| Domain | The set of inputs arcsin accepts: -1 ≤ x ≤ 1, since sine itself never produces a value outside that range. |
The inputs explained
| Field | What to enter |
|---|---|
| Value (the sine ratio, between -1 and 1) | The sine ratio you already have, from -1 to 1. If you have an angle and want its sine instead, this is the wrong direction; use a sine calculation directly. |
When to use it
Solving a right triangle from a side ratio
Given the opposite side and the hypotenuse of a right triangle, their ratio is the sine of the unknown angle; arcsin turns that ratio back into the angle itself.
Working through trigonometric equations
Any equation reduced to the form sin(θ) = x needs arcsin to isolate θ, whether the original problem came from geometry, physics or engineering.
Checking a calculator or spreadsheet result
Spreadsheet functions such as ASIN return radians by default; comparing against this calculator’s degree output is a quick way to confirm a formula is set up correctly.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
What is arcsin of common values?
The angle in degrees and radians for a spread of values from -1 to 1.
| Value (x) | Angle (degrees) | Angle (radians) |
|---|---|---|
| -1.00 | -90.000° | -1.5708 |
| -0.75 | -48.590° | -0.848062 |
| -0.50 | -30.000° | -0.523599 |
| -0.25 | -14.478° | -0.252680 |
| 0.00 | 0° | 0 |
| 0.25 | 14.478° | 0.252680 |
| 0.50 | 30.000° | 0.523599 |
| 0.75 | 48.590° | 0.848062 |
| 1.00 | 90.000° | 1.5708 |
Questions
What is arcsin?
Arcsin is the inverse sine function: given a ratio between -1 and 1, it returns the angle, between -90° and 90°, whose sine is that ratio. It undoes what sine does.
Why can arcsin only accept values from -1 to 1?
Sine itself never produces a value outside that range for any real angle, so there is no angle to return for an input beyond it. A calculator or spreadsheet will show an error for arcsin of, say, 2.
Is arcsin the same as 1/sin(x)?
No. 1/sin(x) is the cosecant of x, a completely different function. Arcsin (sin⁻¹) is the inverse function, working backwards from a ratio to an angle, not the reciprocal of sine.
Why does arcsin only give one answer when several angles share that sine?
By convention, arcsin returns the principal value, restricted to -90° to 90°, so the function always has a single well-defined output. Other angles with the same sine, such as 180° minus the principal value, are valid solutions to a triangle or equation but are not what arcsin itself returns.
For the inverse of tangent instead, see the arctan calculator. To solve a full right-angled triangle from a known angle and side, use the trigonometry calculator.