which of the following is an asymptote of y = sec(x)?\no x=-2π\no x = -\\frac{π}{6}\no x = π\no x =…
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}
Answer
Explanation:
Step1: Recall secant - cosine relation
We know that $\sec(x)=\frac{1}{\cos(x)}$. The vertical asymptotes of $y = \sec(x)$ occur where $\cos(x)=0$.
Step2: Find cosine - zero values
The general solution of the equation $\cos(x)=0$ is $x=(2n + 1)\frac{\pi}{2}$, where $n\in\mathbb{Z}$ (the set of all 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:
$x=\frac{3\pi}{2}$