Inverse Sine (arcsin) Calculator

Enter a value between -1 and 1 to find the angle whose sine equals that value, in both degrees and radians.

Enter Value

Result

arcsin(0.5)
30°
Angle in Degrees30°
Angle in Radians0.523599 rad
Radians (fraction of π)π/6
cos(angle)0.866025
tan(angle)0.577350

Step-by-Step Solution

arcsin(0.5) = 30° = π/6 rad

Understanding Inverse Sine (arcsin)

The inverse sine function, written as sin-1(x) or arcsin(x), is the reverse of the sine function. Given a value y between -1 and 1, arcsin(y) returns the angle x such that sin(x) = y. The principal value of arcsin lies in the range [-90°, 90°] or [-π/2, π/2] radians.

Common Inverse Sine Values

arcsin(0) = 0°

The angle whose sine is 0.

sin-1(0) = 0

arcsin(0.5) = 30°

The angle whose sine is 1/2.

sin-1(1/2) = π/6

arcsin(1) = 90°

The angle whose sine is 1.

sin-1(1) = π/2

arcsin(-1) = -90°

The angle whose sine is -1.

sin-1(-1) = -π/2

Properties of arcsin

  • Domain: [-1, 1].
  • Range: [-π/2, π/2] radians or [-90°, 90°].
  • arcsin(-x) = -arcsin(x) (odd function).
  • sin(arcsin(x)) = x for all x in [-1, 1].
  • Derivative: d/dx arcsin(x) = 1/√(1-x²).

When to Use Inverse Sine

Use arcsin when you know the ratio of the opposite side to the hypotenuse in a right triangle and need to find the angle. It is also used in physics for calculating angles of incidence, refraction, and in navigation problems.