# How do you solve and find the value of sin^-1(tan(pi/4))?

Feb 14, 2017

${\sin}^{- 1} \left(\tan \left(\frac{\pi}{4}\right)\right) = \textcolor{g r e e n}{\frac{\pi}{2} + k \cdot 2 \pi , \forall k \in \mathbb{Z}}$

#### Explanation:

The angle $\frac{\pi}{4}$ is one of the standard angles with
$\textcolor{w h i t e}{\text{XXX}} \tan \left(\frac{\pi}{4}\right) = 1$

So ${\sin}^{- 1} \left(\tan \left(\frac{\pi}{4}\right)\right) = {\sin}^{- 1} \left(1\right)$

If we restrict $\theta$ to the range $\left[0 , 2 \pi\right)$
the only value of $\theta$ for which $\sin \left(\theta\right) = 1$ is $\theta = \frac{\pi}{2}$

For the unrestricted case, this value will repeat with every complete rotational cycle,
so $\theta = \frac{\pi}{2} + k \cdot 2 \pi , \forall k \in \mathbb{Z}$