Explain why, even though sin x and arcsin x are inverse functions, arcsin(sin(x)) ≠ x for all x ?

Question

A complete response will reflect an understanding of the periodic nature of the sine function as well as the domain and range of the arcsine function.

Answer 1

I like to envision the Unit Circle. The sin(45°) in quadrant I = sin(135°) in quadrant II

But taking the arcsine(135°) yields 45°. This correlates to the answer answered previously of choosing outside the interval of the arcsine.

But a visual person might tend to remember the Unit Circle workings more than the theoretical side.

Answer 2

The range of the Arcsine function is [-π/2, π/2], which is also the domain of the restricted sine function that is used in defining the Arcsine function.

So, if we choose a value of x that lies outside of that interval then Arcsin(sinx) ≠ x.

For example, if x = π, than Arcsin(sinπ) = Arcsin0 = 0. So, Arcsin(sinπ) ≠ π.