Quick answer
Inverse sine returns the angle whose sine is x: θ = arcsin(x), for −1 ≤ x ≤ 1. arcsin(0.5) = 30°, or 0.5236 radians.
Enter a value between −1 and 1. The calculator returns the angle in degrees and radians.
Related tools: Sine Calculator: sin of an angle in degrees or radians. Inverse Cosine Calculator: arccos (inverse trigonometric function) with the result in degrees and radians. Inverse Tangent Calculator: arctan (inverse trigonometric function) with the result in degrees and radians.
How to use the Inverse Sine Calculator
- Enter value (between -1 and 1).
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Value | A number | ≥ -1, ≤ 1 |
Formula and method
Formula
θ = arcsin(x)
IEEE-754 double precision; shown to at most 6 decimal places.
Principal value of the inverse function. asin/atan return −90°…90° and acos returns 0°…180°.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Value = 0.5.
| Result | Value |
|---|---|
| arcsin(0.5) | 30° |
| Radians | 0.523599 |
Example 2
Inputs: Value = 1.
| Result | Value |
|---|---|
| arcsin(1) | 90° |
| Radians | 1.570796 |
Example 3
Inputs: Value = -1.
| Result | Value |
|---|---|
| arcsin(-1) | -90° |
| Radians | -1.570796 |
The first line is the headline result (arcsin(0.5)). Below it the inverse Sine Calculator lists radians, which describe the same case from other angles. IEEE-754 double precision; shown to at most 6 decimal places.
Range
The principal value of arcsin is between −90° and 90°. arcsin(1) = 90°, arcsin(0) = 0° and arcsin(−0.5) = −30°. A sine of 0.5 also occurs at 150°, but arcsin returns only 30°.
Uses
Find the angle of a ramp from its rise and length: a rise of 1 m over 2 m of ramp is arcsin(0.5) = 30°. Also used in the law of sines.
Domain, range and a ramp example
arcsin accepts values from −1 to 1 and returns angles from −90° to 90°. arcsin(0.5) = 30° and arcsin(−0.5) = −30°.
For a ramp that rises 1 m over a 4 m slope, the angle is arcsin(1 ÷ 4) ≈ 14.48°.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Value: entering "2" shows "Value must be at most 1." beside that field and no result.
Choosing between the Inverse Sine Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Sine Calculator](/math/sine-calculator/) | angle and unit | sin(30°) |
| [Inverse Cosine Calculator](/math/inverse-cosine-calculator/) | value | arccos(0.5) |
| [Inverse Tangent Calculator](/math/inverse-tangent-calculator/) | value | arctan(1) |
| [Inverse Trigonometry Calculator](/math/inverse-trigonometry-calculator/) | function and value | Angle |
| [Sine Rule Calculator](/math/sine-rule-calculator/) | angle A, side a and angle B | Side b |
Background: trigonometry
In a right triangle, sin θ is the opposite side over the hypotenuse, cos θ the adjacent side over the hypotenuse and tan θ the opposite over the adjacent. Their reciprocals are cosecant, secant and cotangent. At 30° the values are sin = 0.5, cos = 0.866025 and tan = 0.577350.
Check the angle unit before you calculate: degrees convert to radians by multiplying by π/180, so 30° is about 0.5236 rad. Some values are undefined, for example tan 90°, because the cosine is zero there and the ratio would divide by zero.
Assumptions and limitations
- Value must be between -1 and 1
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for value to confirm this.
Frequently asked questions
What is arcsin(1)?
arcsin(1) = 90°, the angle whose sine is 1.
Is sin⁻¹ the reciprocal?
No, it is the inverse function.
Why does 150° have the same sine?
Sine is symmetric about 90°.
Why do I get an error for 1.2?
The sine of a real angle never exceeds 1, so values outside −1 to 1 have no real arcsine.
Is sin⁻¹ the same as 1/sin?
No. sin⁻¹ is the inverse function (arcsin). 1/sin is the cosecant.
What does the Inverse Sine Calculator calculate?
arcsin (inverse trigonometric function) with the result in degrees and radians. It returns arcsin(0.5) and Radians.
What formula does the Inverse Sine Calculator use?
θ = arcsin(x). Principal value of the inverse function.
What do I need to enter in the inverse Sine Calculator?
You enter Value.
How is the result rounded or limited?
IEEE-754 double precision; shown to at most 6 decimal places. Value must be between -1 and 1.
What is an example of a inverse Sine result?
With value 0.5, the arcsin(0.5) is 30°.
Related calculators
- Sine Calculator — sin of an angle in degrees or radians.
- Inverse Cosine Calculator — arccos (inverse trigonometric function) with the result in degrees and radians.
- Inverse Tangent Calculator — arctan (inverse trigonometric function) with the result in degrees and radians.
- Inverse Trigonometry Calculator — Arcsine, arccosine or arctangent in degrees and radians.
- Sine Rule Calculator — Apply the sine rule to find a missing side or angle of a triangle from an angle and its opposite side.
- Trigonometry Calculator — All six trigonometric functions of an angle.
Sources
- Standard trigonometric definitions on the unit circle (NIST DLMF §4.14)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.