Arcsin of −1

Arcsin, Arccos, Arctan, Arccot

What is arcsin(−1)?

Answer

arcsin(−1) =

−90° = −π/2

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

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

arcsin(−1) = −arcsin(1) = −90°

sin(−90°) = −1 ✓

α = arcsin(−1) = −90°

P(cos(−90°), sin(−90°)) = (0, −1)

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

arcsin(−1) = −90° = −π/2

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

The calculator is in radian mode (RAD) and gives the angle in radians: arcsin(−1) = −π/2 ≈ −1.5708 rad. In degrees (DEG) it is −90°.