# How to find the asymptotes of y = 1/2 sec ( x - pi/6)?

Aug 12, 2017

$x = 2 k \pi + \frac{3 \pi}{2} \mathmr{and} x = 2 k \pi - \frac{\pi}{3} , k \in \mathbb{Z}$

#### Explanation:

$y = \frac{1}{2} \sec \left(x - \frac{\pi}{6}\right) = \frac{1}{2} \cdot \frac{1}{\cos} \left(x - \frac{\pi}{6}\right)$

We know that when $\cos \left(x - \frac{\pi}{6}\right) \rightarrow 0$ then $y \rightarrow \pm \infty$
so everytime that $\cos \left(x - \frac{\pi}{6}\right) = 0$ the function has a vertical asymptode.

Let's solve the equation $\cos \left(x - \frac{\pi}{6}\right) = 0$ :

$\cos \left(x - \frac{\pi}{6}\right) = 0 \iff \cos \left(x - \frac{\pi}{6}\right) = \cos \left(\frac{\pi}{2}\right) \iff$

$x - \frac{\pi}{6} = 2 k \pi \pm \frac{\pi}{2} \iff x = 2 k \pi + \frac{2 \pi}{3} \mathmr{and} x = 2 k \pi - \frac{\pi}{3}$

So the function has the folowing vertical asyptodes :

$x = 2 k \pi + \frac{3 \pi}{2} \mathmr{and} x = 2 k \pi - \frac{\pi}{3} , k \in \mathbb{Z}$