Composing Functions and Inverses

sin(arcsin x) always returns x, provided x lies between −1 and 1. That direction is safe.

The other order

arcsin(sin θ) returns θ only when θ already lies in the restricted range, −90° to 90°.

A concrete failure

arcsin(sin 150°) is not 150°. The sine is 0.5, and arcsin sends it back to 30°.

The three ranges

arcsin runs from −90° to 90°. arccos runs from 0° to 180°. arctan runs from −90° to 90°.

Mixed compositions

For something like cos(arcsin 0.6), draw a right triangle with opposite 0.6 and hypotenuse 1. The adjacent side is 0.8, so the answer is 0.8.

Sometimes they cancel and sometimes they do not

sin(sin⁻¹ x) = x for every x in [−1, 1]. But sin⁻¹(sin x) = x only when x is already in the principal range; otherwise it returns the coterminal or reflected angle that is. The two compositions are not symmetric.

A worked case

sin⁻¹(sin(3π/4)) is not 3π/4, because that lies outside [−π/2, π/2]. Since sin(3π/4) = √2/2, the answer is π/4. Always check whether the input falls in the principal range before cancelling.

Mixed compositions need a triangle

For cos(sin⁻¹ x), draw a right triangle with opposite x and hypotenuse 1. Pythagoras gives adjacent √(1 − x²), so the answer is √(1 − x²). Drawing the triangle converts an intimidating expression into a routine one.

Watch the domains

sin⁻¹ accepts only inputs between −1 and 1, so any composition must respect that. An expression that produces a value outside the range simply has no solution, and saying so is the correct answer.