Arcsin of −√3/2

Arcsin, Arccos, Arctan, Arccot

What is arcsin(−√3/2)?

Answer

arcsin(−√3/2) =

−60° = −π/3

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

arcsin(−√3/2) is the angle α in [−90°, 90°] with sin α = −√3/2.

arcsin(−√3/2) = −arcsin(√3/2) = −60°

sin(−60°) = −√3/2 ✓

α = arcsin(−√3/2) = −60°

P(cos(−60°), sin(−60°)) = (1/2, −√3/2)

P lies on the line y = −√3/2 in the right half of the circle, where cos α ≥ 0; highlighted: sin α = −√3/2.

arcsin(−√3/2) = −60° = −π/3

arcsin is an odd function: arcsin(−x) = −arcsin x. So arcsin(−√3/2) = −arcsin(√3/2) = −60°, and this angle lies in the range [−90°, 90°].

The calculator is in radian mode (RAD) and gives the angle in radians: arcsin(−√3/2) = −π/3 ≈ −1.0472 rad. In degrees (DEG) it is −60°.