Inverse Trigonometric Functions

sin θ = 0.5 asks which angle has that sine. The answer is written arcsin 0.5.

There is a difficulty

Infinitely many angles have sine 0.5: 30°, 150°, 390°, and so on forever.

So the domain is restricted

arcsin returns only angles between −90° and 90°, which makes it a genuine function.

The others still exist

The calculator gives one. In a problem you must decide whether another angle also fits.

The domain has to be restricted

Sine takes every value in its range infinitely often, so it is not one-to-one and has no inverse. Restricting it to −π/2 to π/2 makes it one-to-one, and that restricted function is what sin⁻¹ inverts.

The three principal ranges

sin⁻¹ returns values in [−π/2, π/2]; cos⁻¹ in [0, π]; tan⁻¹ in (−π/2, π/2). These are conventions chosen to make each function one-to-one, and a calculator will never return anything outside them.

The notation is genuinely ambiguous

sin⁻¹ x means the inverse function; (sin x)⁻¹ means the reciprocal. The superscript −1 means different things in the two positions. Many texts write arcsin to avoid the collision, and it is a reasonable habit.

You get one answer out of many

sin⁻¹(0.5) returns 30°, but sine also equals 0.5 at 150°, 390° and infinitely many others. The calculator gives the principal value only, and recovering the rest is what the next lesson is about.