Medium

The Inverse Sine Function and Its Graph

Simple Explanation

arcsin(x) (also written sin⁻¹x) gives the ONE angle, restricted to the range [−π/2, π/2], whose sine equals x. Its domain is limited to [−1,1], since no angle has a sine outside that range.

Why Do We Need It?

Sine is not one-to-one over all real numbers (many angles share the same sine value) — restricting the range is what makes arcsin an actual function, with exactly one output per input.

See It

The graph of y = arcsin(x)

Formula

The Inverse Sine Function

y = arcsin(x), domain [−1,1], range [−π/2, π/2]

arcsin(x) gives the ONE angle, within the restricted range [−π/2,π/2], whose sine equals x.

x: — a value between −1 and 1

When to use it: Whenever an angle needs to be recovered from a known sine value.

Worked Example

Evaluate the inverse sine function

Find arcsin(0.5).

Why Does This Work?

Although infinitely many angles have a sine of 0.5 (like π/6, 5π/6, π/6+2π, and so on), the restricted range [−π/2,π/2] contains exactly ONE of them — π/6 — making arcsin well-defined as a proper function.

Real-Life Example

Finding an angle from a known ratio in navigation

A navigator knows the ratio of a ship's vertical displacement to its total distance traveled (essentially a sine ratio) and needs the actual angle of travel.

arcsin recovers this angle directly from the known ratio.

Practice

Find arcsin(1), in degrees.

Medium

Common mistake

Giving an answer outside the restricted range [−π/2,π/2] — even though many angles share the same sine value, arcsin always returns the one within this specific range.

Quick Review

arcsin(x): domain [−1,1], range [−π/2, π/2].
Gives the ONE angle (within the restricted range) whose sine is x.
The restricted range is what makes arcsin a genuine function.