calculus - Why is $sin^{-1}(sin(frac{5pi}{8}))ne frac{5pi}{8}$?

I am defining a function and making sure that it works. I thought $\sin^{-1}(\sin(x))=x$ but if I put it into a calculator I get $\sin^{-1}(\sin(\frac{5\pi}{8}))\approx1.178$; which is not $\frac{5\pi}{8}\approx1.96$.

What is the reason for this?

Answer

If you translate the statement what is $\sin^{-1}(\sin(\frac{5\pi}{8}))$ into a statement in English it would say the following:

What angle between $-\frac{\pi}{2}$ and $\frac{\pi}{2}$ has the same sine value as the angle $\frac{5\pi}{8}$?

If you draw the unit circle and the angle $\frac{5\pi}{8}$ and then draw a horizontal line through the point where the angle intersects the unit circle, I believe you will see that the horizontal line also intersects the unit circle in quadrant I at the point of intersection with the angle $\frac{3\pi}{8}$. So the answer is $\frac{3\pi}{8}$.