# How do you find the exact values of sintheta and tantheta when costheta=0?

Jan 30, 2017

$s i m \theta = \pm 1$ and $\tan \theta$ is not defined.

#### Explanation:

As $\cos \theta = 0$, we have

$\sin \theta = \sqrt{1 - {\cos}^{2} \theta} = \sqrt{1 - 0} = \sqrt{1} = \pm 1$ and

$\theta = \pm \frac{\pi}{2}$

But, $\tan \theta = \sin \frac{\theta}{\cos} \theta$ and as $\sin \theta \ne 0$, but $\cos \theta = 0$, $\tan \theta$ is not defined.

However, as $\theta \to \frac{\pi}{2}$ from left (on real number line), $\tan \theta \to \infty$ and as $\theta \to \frac{\pi}{2}$ from right (on real number line), $\tan \theta \to - \infty$.

Similarly, as $\theta \to - \frac{\pi}{2}$ from left (on real number line), $\tan \theta \to \infty$ and as $\theta \to - \frac{\pi}{2}$ from right (on real number line), $\tan \theta \to - \infty$.
graph{tanx [-10, 10, -5, 5]}