Definition: The inverse sine function, written arcsin or sin inverse, returns the angle between negative pi over 2 and pi over 2 whose sine equals a given number.
What does arcsin do?
Sine is not one-to-one over all angles, so we restrict it to -pi/2 to pi/2. On that interval each output between -1 and 1 occurs once, and arcsin x returns that unique angle. Its domain is [-1, 1].
How do you evaluate it?
Ask: which angle in the restricted range has this sine?
Worked example: arcsin(-1/2) = -pi/6, because sin(-pi/6) = -1/2 and -pi/6 lies in the allowed range.
A trap: arcsin(sin(5 pi/6)). First sin(5 pi/6) = 1/2, then arcsin(1/2) = pi/6, not 5 pi/6, since 5 pi/6 is outside the range.
What do students confuse?
The notation sin-1x is not 1/sin x. The reciprocal is cosecant. Mathbot reads sin-1 as “the angle whose sine is.” Also, arcsin 2 does not exist, since no sine value exceeds 1.