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

1 Answer
Sep 2, 2016

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#.