Question

What is the domain and range of `y = arcsin x`?

Answer 1

Range: `[-pi/2,pi/2]`

Domain: `[-1,1]`

Explanation:

The following is a fragment from my lecture about `y=arcsin x` presented on UNIZOR.COM. If you go to this very useful Web site, click Trigonometry - Inverse Trigonometric Functions - y=arcsin(x).

The original sine function defined for any real argument does not have an inverse function because it does not establish a one-to-one correspondence between its domain and a range.

To be able to define an inverse function, we have to reduce the original definition of a sine function to an interval where this correspondence does take place. Any interval where sine is monotonic and takes all values in its range would fit this purpose.

For a function `y=sin(x)` an interval of monotonic behavior is usually chosen as `[−π/2,π/2]`, where the function is monotonously increasing from `−1` to `1`.

This variant of a sine function, reduced to an interval where it is monotonous and fills an entire range, has an inverse function called `y=arcsin(x)`.

It has range `[−π/2,π/2]` and domain from `-1` to `1`.