Arcsin Calculator - Inverse Sine Function
Calculate the inverse sine (arcsin) of any value in [-1, 1] and get the result in degrees or radians.
Enter a sine value between -1 and 1, choose degrees or radians, and get the angle whose sine equals that value.
Arcsin Calculator - Inverse Sine Function
Calculate the inverse sine (arcsin) of any value in [-1, 1] and get the result in degrees or radians.
Valid domain: -1 ≤ x ≤ 1
About the arcsin calculator
The arcsine function, written arcsin(x) or sin⁻¹(x), is the inverse of the sine function. Given any value x in the interval [-1, 1], arcsin(x) returns the angle θ whose sine equals x. The output is in the principal-value range of -90° to 90° (or -π/2 to π/2 radians), which is chosen by convention to make the inverse function unique and well-defined.
Arcsine is one of the six inverse trigonometric functions and is arguably the most commonly used after arctangent. It appears whenever you know the ratio of the opposite side to the hypotenuse in a right triangle and want to find the angle. For a triangle where the opposite side has length 3 and the hypotenuse 5, the angle is arcsin(3/5) = arcsin(0.6) ≈ 36.87°.
The domain [-1, 1] reflects the fact that the sine function always produces values in this range for real angles. Attempting to compute arcsin of a value outside this range, such as arcsin(2), has no real solution. In engineering and physics, domain violations often indicate a modelling error — for example, an overstated ratio — and should trigger a review of the input data rather than a creative interpretation.
Keyboard reference values: arcsin(0) = 0°, arcsin(0.5) = 30°, arcsin(√2/2) ≈ 45°, arcsin(√3/2) ≈ 60°, arcsin(1) = 90°. The negative mirror: arcsin(-0.5) = -30°, arcsin(-1) = -90°. These values arise from the standard right triangles and should be committed to memory for quick mental checks.
In physics, arcsin is used in optics to apply Snell's law: the refraction angle is arcsin(n₁ × sin(θ₁) / n₂). In projectile motion, the launch angle for maximum range on a flat surface is arcsin(1/√2) = 45°. In electrical engineering, the power factor angle is found using the arcsin of the reactive-to-apparent power ratio.
In statistics and probability, the arcsin transformation (also called the angular transformation or arcsine square root transformation) is used to stabilize the variance of proportions, making statistical tests more reliable for data near 0 or 1. This application uses arcsin(√p), where p is a proportion, and produces values in [0°, 90°].
This calculator returns the principal value, which lies in [-90°, 90°]. If your problem requires an angle outside this range — for example, you know the sine is 0.5 and you expect the angle to be 150° rather than 30° — you need to apply additional logic: 150° = 180° - 30° = 180° - arcsin(0.5).
Arcsin examples
Key reference values for the arcsine function from the standard right triangles.
| Input | arcsin(x) | Explanation |
|---|---|---|
| arcsin(0.5) | 30° ≈ 0.5236 rad | The sine of 30° equals 0.5, so arcsin(0.5) = 30°. Classic 30-60-90 triangle result. |
| arcsin(0.7071) | 45° ≈ 0.7854 rad | arcsin(√2/2) = 45°. Comes from the 45-45-90 triangle where both legs equal 1 and hypotenuse equals √2. |
| arcsin(0) | 0° = 0 rad | The sine of 0° is 0. arcsin(0) = 0, the midpoint of the arcsin range. |
| arcsin(-0.5) | -30° ≈ -0.5236 rad | Negative input gives a negative output angle. arcsin(-0.5) = -30°, demonstrating the odd-function symmetry of arcsin. |
How to use the arcsin calculator
- Enter the sine value in the Input Value field. It must be a number between -1 and 1 inclusive.
- Select the Output Unit: click Degrees to receive the answer in degrees, or Radians for the radian equivalent.
- Click Calculate Arcsin to compute the result. The output angle is in the range [-90°, 90°] for degrees or [-π/2, π/2] for radians.
- Read the result in the display area, which shows the full expression arcsin(x) = result.
- Click Reset to clear the input and start over with a new calculation.
Arcsin calculator FAQ
Why is the domain of arcsin limited to [-1, 1]?
The sine function maps real angles to the interval [-1, 1]. No real angle has a sine greater than 1 or less than -1, so the inverse function is only defined on this interval. Entering a value outside [-1, 1] would correspond to a complex angle, which is beyond the scope of this real-valued calculator.
What is the output range of arcsin?
The principal value of arcsin lies in [-90°, 90°] for degrees or [-π/2, π/2] for radians. This range is chosen to make arcsin a proper single-valued function. If you need an angle outside this range, add or subtract 180° (or π) as appropriate for your specific problem.
How does arcsin differ from arccos?
arcsin returns angles in [-90°, 90°] while arccos returns angles in [0°, 180°]. They are related by arcsin(x) + arccos(x) = 90° for any x in [-1, 1]. Use arcsin when you know the opposite-side-to-hypotenuse ratio and arccos when you know the adjacent-side-to-hypotenuse ratio.
Is arcsin the same as 1/sin?
No. arcsin(x) means the inverse function — the angle whose sine is x. The notation sin⁻¹(x) can be misleading because in most other contexts the superscript -1 means reciprocal. To express 1/sin(x), write csc(x) (cosecant) rather than sin⁻¹(x), to avoid confusion with the inverse function.
Why does arcsin(-x) = -arcsin(x)?
Arcsin is an odd function, meaning arcsin(-x) = -arcsin(x). This follows from the fact that the sine function is also odd: sin(-θ) = -sin(θ). So if arcsin(x) = θ, then sin(θ) = x, and sin(-θ) = -x, meaning arcsin(-x) = -θ.
How is arcsin used in Snell's law?
Snell's law states that n₁ × sin(θ₁) = n₂ × sin(θ₂), where n₁ and n₂ are the refractive indices and θ₁ and θ₂ are the angles of incidence and refraction. To find the refraction angle, rearrange to sin(θ₂) = (n₁ / n₂) × sin(θ₁) and apply arcsin: θ₂ = arcsin((n₁/n₂) × sin(θ₁)). This calculator handles that final arcsin step directly.