Why arcsin Gives One Angle

Sine turns an angle into a number; arcsin goes back. arcsin(1/2) asks which angle has sine 1/2, and many do: 30°, 150°, 390°, and more every full turn. A function gives one output for each input, so arcsin answers only from [−90°, 90°].

The line at height 1/2 meets the wave over and over. Only 30° lies in [−90°, 90°].

On that stretch sine climbs from −1 to 1 and meets each value exactly once, so the choice is never in doubt: arcsin(1/2) = 30°, never 150°. In radians the range is [−π/2, π/2]. Calculators label arcsin as sin⁻¹; the −1 means the inverse, not one over sine.