Sin Inverse Calculator

May 11, 2026

When you know the sine of an angle but need to determine the specific angle itself, you use the inverse sine function, commonly known as arcsin. This function acts as the undo button for sine, allowing you to derive the angle from a given ratio (the length of the opposite side divided by the hypotenuse).

Sin⁻¹ CalculatorInput value (x) Enter a ratio between −1 and 1 (opposite side ÷ hypotenuse)

Primary output unit

Degrees Radians

Unit circle visualization

Common values reference

Value (x)	Angle (degrees)	Angle (radians)
−1	−90°	−π/2
−√3/2 ≈ −0.866	−60°	−π/3
−√2/2 ≈ −0.707	−45°	−π/4
−0.5	−30°	−π/6
0	0°	0
0.5	30°	π/6
√2/2 ≈ 0.707	45°	π/4
√3/2 ≈ 0.866	60°	π/3
1	90°	π/2

Disclaimer

This calculator provides mathematical estimations for educational purposes. The principal value range for arcsin is [−π/2, π/2] radians or [−90°, 90°]. Verify critical calculations against geometric definitions.

This calculator is designed for educational purposes and provides mathematical estimations; always verify critical calculations against official geometric definitions.

Understanding the inverse sine function

The inverse sine function, denoted as $\sin^{-1}(x)$ or $\arcsin(x)$, is one of the fundamental inverse trigonometric functions. While the standard sine function map an angle to a number between -1 and 1, the arcsine function maps a number in the interval $[-1, 1]$ back to an angle.

For example, if you are looking at a right-angled triangle where the ratio of the opposite side to the hypotenuse is 0.5, calculating $\arcsin(0.5)$ tells you that the angle is 30 degrees (or $\pi/6$ radians).

How to use the calculator

To find the angle from a sine value, follow these steps:

Enter the sine value: Type the value into the input field. The value must be between -1 and 1.
Select output units: Choose whether you require the final result in degrees or radians. This choice is critical depending on whether your work follows a geometry curriculum (degrees) or a calculus/advanced mathematics curriculum (radians).
Process: The calculator instantly outputs the resulting angle.

Why range and domain matter

The function is restricted to specific boundaries to ensure that each input results in a single, well-defined output:

Domain: The input value must fall within the range of $[-1, 1]$. If you enter a value like 1.5, the calculation is undefined in the set of real numbers because a side of a triangle cannot be longer than its hypotenuse.
Range: The output for arcsine is restricted to the interval $[-\pi/2, \pi/2]$ in radians, which corresponds to $[-90^\circ, 90^\circ]$ in degrees. If you require an angle outside of this principal value range, you must apply trigonometric identities based on the specific quadrant of the unit circle you are working in.

Common examples and applications

Inverse sine is widely used in physics, engineering, and computer graphics when determining vectors or angular positions based on coordinate ratios.

Finding angles in right triangles: If you have a triangle with an opposite side of 3 and a hypotenuse of 5, the sine is $3/5 = 0.6$. Calculating $\arcsin(0.6)$ gives you approximately $36.87^\circ$.
Graphing and Waves: Engineers use arcsin to analyze the behavior of waves, such as electromagnetic signals or sound waves, specifically to determine time points where the signal reaches a specific amplitude.

Frequently Asked Questions

What is the difference between sin and arcsin?

The sine function takes an angle as input and returns the ratio of the opposite side to the hypotenuse in a right triangle. The arcsin function, or inverse sine, does the opposite: it takes that ratio as input and provides the angle corresponding to it.

Why is there no result for inputs greater than 1 or less than -1?

In a right triangle, the hypotenuse is always the longest side. Since the sine is calculated as (Opposite / Hypotenuse), the maximum value is 1 and the minimum is -1. Any input outside this range has no real-number solution for the sine function.

How do I switch between degrees and radians?

Degrees and radians are simply two different units for measuring angles. Most scientific calculators allow you to toggle between them. Remember to ensure your setting matches the units your specific problem requires to avoid errors.

Is arcsin the same as 1/sin?

No. The notation $\sin^{-1}(x)$ means the inverse sine function (arcsine). It is not the same as $(\sin(x))^{-1}$, which equals $1/\sin(x)$ or the cosecant function ($\csc(x)$).