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

Explanation:

Step1: Recall secant - cosine relationship

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

Step2: Find cosine - zero values

The general solution for $\cos(x)=0$ is $x=\frac{\pi}{2}+n\pi$, where $n\in\mathbb{Z}$ (the set of 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:

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