About

The Inverse Sine function, arcsin(x), determines the angle of a right triangle given the ratio of the opposite side to the hypotenuse. Unlike arccos, the principal value range of arcsin is from -π/2 to π/2 (-90° to 90°). This reflects the function's symmetry around the origin in the Cartesian plane.

This tool is particularly useful in fields like optics (calculating refraction angles), architecture (roof slopes), and game development (calculating trajectories). The visualizer below helps demonstrate how the input value (Sine) corresponds to the vertical component (Y-axis) on the unit circle.

Formulas

Standard definition:

θ = arcsin(opposite / hypotenuse)

Domain and Range:

−1 ≤ x ≤ 1

−π/2 ≤ θ ≤ π/2

Reference Data

Input (Sine)	Angle (Degrees)	Angle (Radians)	Quadrant
1.0	90°	π / 2	Positive Y-Axis
0.866	60°	π / 3	Q1
0.707	45°	π / 4	Q1
0.5	30°	π / 6	Q1
0	0°	0	Positive X-Axis
-0.5	-30°	-π / 6	Q4
-0.707	-45°	-π / 4	Q4
-1.0	-90°	-π / 2	Negative Y-Axis

Frequently Asked Questions

The arcsin function is defined in the 1st and 4th quadrants of the unit circle. A positive input returns an angle in the 1st quadrant (0° to 90°), while a negative input returns an angle in the 4th quadrant (-90° to 0°), commonly expressed as a negative number.

No. In a right triangle, the opposite side can never be longer than the hypotenuse. Therefore, their ratio (sine) cannot exceed 1. Inputs outside [-1, 1] will result in a mathematical error.

Carpenters use it to cut rafters (pitch angles). Machinists use it for sine bars to set precise angles. Surveyors use it to calculate elevation changes.