Our goal is to find the exact value of the expression sin1(sin(4165π))\displaystyle {{\sin}^{{-{1}}}{\left({\sin{{\left(\frac{{41}}{{65}}\pi\right)}}}\right)}}.

First, what is the range of the inverse sin function?
Next, we must look at the input of the inner-most function. Is this value in the range of the inverse sin function?
Now, which value in the range of the inverse sin function has the same output in the sin function as 4165π\displaystyle \frac{{41}}{{65}}\pi?
 

Using this information, evaluate:
sin1(sin(4165π))=\displaystyle {{\sin}^{{-{1}}}{\left({\sin{{\left(\frac{{41}}{{65}}\pi\right)}}}\right)}}=