which of the following is an asymptote of y = sec(x)?\no x=-2π\no x = - \frac{π}{6}\no x = π\no x=\frac{3π}{2}

Explanation:

Step1: Recall secant function definition

$\sec(x)=\frac{1}{\cos(x)}$. Asymptotes occur where $\cos(x) = 0$.

Step2: Find cosine - zero values

The general form for where $\cos(x)=0$ is $x=\frac{\pi}{2}+n\pi$, where $n\in\mathbb{Z}$ (integers).

Step3: Check each option

For $x = - 2\pi$, $\cos(-2\pi)=1\neq0$. For $x=-\frac{\pi}{6}$, $\cos(-\frac{\pi}{6})=\frac{\sqrt{3}}{2}\neq0$. For $x = \pi$, $\cos(\pi)=-1\neq0$. For $x=\frac{3\pi}{2}$, $\cos(\frac{3\pi}{2}) = 0$.

Answer:

$x=\frac{3\pi}{2}$