Arcsin of −√2/2

Arcsin, Arccos, Arctan, Arccot

What is arcsin(−√2/2)?

Answer

arcsin(−√2/2) =

−45° = −π/4

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

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

arcsin(−√2/2) = −arcsin(√2/2) = −45°

sin(−45°) = −√2/2 ✓

α = arcsin(−√2/2) = −45°

P(cos(−45°), sin(−45°)) = (√2/2, −√2/2)

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

arcsin(−√2/2) = −45° = −π/4

arcsin is an odd function: arcsin(−x) = −arcsin x. So arcsin(−√2/2) = −arcsin(√2/2) = −45°, and this angle lies in the range [−90°, 90°].

The calculator is in radian mode (RAD) and gives the angle in radians: arcsin(−√2/2) = −π/4 ≈ −0.7854 rad. In degrees (DEG) it is −45°.