Arcsin of −1/2

Arcsin, Arccos, Arctan, Arccot

What is arcsin(−1/2)?

Answer

arcsin(−1/2) =

−30° = −π/6

Range: arcsin x is an angle from −90° to 90°, so the answer is unique.

−1/2 = −0.5.

arcsin(−1/2) is the angle α in [−90°, 90°] with sin α = −1/2.

arcsin(−1/2) = −arcsin(1/2) = −30°

sin(−30°) = −1/2 ✓

α = arcsin(−1/2) = −30°

P(cos(−30°), sin(−30°)) = (√3/2, −1/2)

P lies on the line y = −1/2 in the right half of the circle, where cos α ≥ 0; highlighted: sin α = −1/2.

arcsin(−1/2) = −30° = −π/6

arcsin is an odd function: arcsin(−x) = −arcsin x. So arcsin(−1/2) = −arcsin(1/2) = −30°, and this angle lies in the range [−90°, 90°].

The calculator is in radian mode (RAD) and gives the angle in radians: arcsin(−1/2) = −π/6 ≈ −0.5236 rad. In degrees (DEG) it is −30°.