What Is the Inverse Sine (Arcsin) Function?
The arcsine function (arcsin, also written sin⁻¹ or asin) is the inverse of the sine function. Give it a value between −1 and 1 and it returns the angle whose sine equals that value. The answer is the principal value, always in the range [−90°, 90°] or [−π/2, π/2] radians.
sin(θ) = x then arcsin(x) = θ, with x ∈ [−1, 1] and θ ∈ [−90°, 90°].Key Properties
Domain [−1, 1]
Range [−90°, 90°]
How to Use the Arcsin Calculator
Enter a Value in [−1, 1]
Type a value between −1 and 1 into the input field. Anything outside this range shows "Value must be between -1 and 1", since it has no real arcsine.
Choose DEG or RAD
Select DEG to see the result in degrees, or RAD for radians.
Read the Angle Instantly
The angle is calculated automatically as you type. The result appears in the active unit, with the equivalent in the other unit shown alongside.
Verify the Result
The calculator confirms the answer with the check sin(result) = input, so you can trust the computed angle.
Calculator Features
Inverse Sine of Any Valid Value
Computes arcsin for any input within the domain [−1, 1].
Degrees & Radians
Shows the resulting angle in the active unit and the equivalent in the other unit at the same time.
Domain Validation
Checks input against the valid range [−1, 1] and warns when a value falls outside it.
Built-In Verification
Displays the check sin(result) = input, recalculated in real time as you type.
Common Arcsine Values
| Input | arcsin result | Radians |
|---|---|---|
| arcsin(0) | 0° | 0 |
| arcsin(0.5) | 30° | π/6 |
| arcsin(√2/2) | 45° | π/4 |
| arcsin(√3/2) | 60° | π/3 |
| arcsin(1) | 90° | π/2 |
Frequently Asked Questions
What is arcsin and how does it work?
Arcsin is the inverse of sine. Where sine turns an angle into a ratio, arcsin turns a ratio (a value from −1 to 1) back into the angle. If sin(θ) = x, then arcsin(x) = θ.
Why must the input be between −1 and 1?
Because the sine function only outputs values in the range [−1, 1]. No angle has a sine greater than 1 or less than −1, so arcsine cannot accept values outside that range — the calculator flags them as invalid.
What is arcsin(0.5)?
arcsin(0.5) = 30° or π/6 radians, because sin(30°) = 0.5. Other common values: arcsin(√2/2) = 45°, arcsin(√3/2) = 60°, and arcsin(1) = 90°.
Is sin⁻¹ the same as 1/sin?
No. sin⁻¹(x) is the inverse sine function (the same as arcsin), not the reciprocal. The reciprocal of sine is the cosecant, written csc(x) = 1/sin(x).
Why is the range limited to [−90°, 90°]?
Each inverse trig function uses a restricted range so it returns a single, unique angle. For arcsine, [−90°, 90°] covers every possible sine value exactly once.
How do I switch the result between degrees and radians?
Use the DEG / RAD toggle above the input. In RAD mode the angle is formatted as a π-fraction where it matches a standard value (for example π/6 for arcsin(0.5)), and the equivalent degree value is shown alongside.
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