# How do you evaluate tan^-1(0) without a calculator?

Dec 1, 2016

$\theta = 0$

#### Explanation:

Find ${\tan}^{-} 1 \left(0\right)$ without a calculator.

${\tan}^{-} 1 \left(0\right)$ refers to the ANGLE whose tangent equals zero.

Recall the identity $\tan \theta = \sin \frac{\theta}{\cos} \theta$

If $\tan \theta = 0$, then the numerator of $\sin \frac{\theta}{\cos} \theta$ must also equal zero.

So, sintheta"=0.

Referring to the unit circle, the angles with a sine of zero are
$\theta = 0 , \pi$.

BUT, IT'S A BIT MORE COMPLICATED!

The range of ${\tan}^{-} 1$ (also called arctan) is $- \frac{\pi}{2}$ to $\frac{\pi}{2}$.

In other words, the only "allowed" values of ${\tan}^{-} 1$ fall within this range. So, the the answer $\theta = \pi$ is not allowed, and the answer to the problem ${\tan}^{-} 1 \left(0\right)$ is $\theta = 0$