# Why are the tangents for 90 and 270 degrees undefined?

Oct 15, 2015

This is a good question indeed!

#### Explanation:

I'll try to give you a visual explanation.
Consider the trigonometric meaning of tangent of an angle $\alpha$: The tangent of $\alpha$ is equal to the length of the segment $A B$; but when $\alpha$ becomes ${90}^{\circ}$ the length of $A B$ get stretched upwards (or downwards for $\alpha = {270}^{\circ}$) so that we will never meet $A$!!! Hope it helps!

Oct 15, 2015

You can also say that $\tan x = \sin \frac{x}{\cos} x$.

You can take a look at the overlap of $\sin x$ and $\cos x$ below to see what happens to each as we approach ${90}^{o}$ and ${270}^{o}$ from the right or left:

graph{(y - sinx)(y - cosx) = 0 [-0.034, 6.2831, -1.2, 1.2]}

We can establish that:

${\lim}_{x \to {90}^{{o}^{-}}} \sin \frac{x}{\cos} x = - \frac{1}{0} = - \infty$

because $\cos x$ decreases while $\sin x$ increases as $x \to {90}^{o}$ from the left, and

${\lim}_{x \to {90}^{{o}^{+}}} \sin \frac{x}{\cos} x = \frac{1}{0} = \infty$

because both $\cos x$ and $\sin x$ increase as $x \to {90}^{o}$ from the right.

We can also see that:

${\lim}_{x \to {270}^{{o}^{-}}} \sin \frac{x}{\cos} x = - \frac{1}{0} = - \infty$

because $\sin x$ decreases but $\cos x$ increases as $x \to {270}^{o}$ from the left, and

${\lim}_{x \to {270}^{{o}^{+}}} \sin \frac{x}{\cos} x = \frac{1}{0} = \infty$

because both $\sin x$ and $\cos x$ decrease as $x \to {270}^{o}$ from the right.

Since the limits from the left and right side are not the same, $\setminus m a t h b f \left(\tan x\right)$ of those angles is undefined.