# How do you evaluate sin^-1(sin(pi/6)) without a calculator?

Aug 26, 2016

$\frac{\pi}{6}$

#### Explanation:

Use ${f}^{- 1} f \left(x\right) = x$ that is valid for any function y = f(x) that is

bijective in an (infinitesimal) $\in$- neighborhood $\left(x - \in , x + \in\right)$ of x.

Here, it is ${\sin}^{- 1} \sin \left(\frac{\pi}{6}\right) = \frac{\pi}{6}$.

It is important to note that $\sin \left(\frac{\pi}{6}\right) = \sin \left(\frac{13}{6} \pi\right) = \frac{1}{2}$ and, while

${\sin}^{- 1} \left(\frac{1}{2}\right)$ = the conventional principal value $\frac{\pi}{6}$,,

${\sin}^{- 1} \sin \left(\frac{\pi}{6}\right) = \frac{\pi}{6}$ and

${\sin}^{- 1} \sin \left(\frac{13}{6} \pi\right) = 13 \frac{\pi}{6}$.