Question

what is the domain of $y = \sec(x)$?
\bigcirc all real numbers
\bigcirc all real numbers except $n\pi+ \frac{\pi}{2}$, where $n$ is any integer
\bigcirc all real numbers except $n\pi$, where $n$ is any integer
\bigcirc all real numbers except $n\pi$ and $n\pi+ \frac{\pi}{2}$, where $n$ is any integer

Explanation:

Step1: Recall secant function definition

The secant function is defined as \(\sec(x)=\frac{1}{\cos(x)}\). So, we need to find where \(\cos(x)
eq0\) because division by zero is undefined.

Step2: Solve \(\cos(x) = 0\)

We know that \(\cos(x)=0\) when \(x = \frac{\pi}{2}+n\pi\), where \(n\) is any integer (since the cosine function has a period of \(2\pi\) and is zero at odd multiples of \(\frac{\pi}{2}\)). So, the domain of \(\sec(x)\) is all real numbers except \(x=n\pi+\frac{\pi}{2}\), \(n\in\mathbb{Z}\).

Answer:

all real numbers except \(n\pi+\frac{\pi}{2}\), where \(n\) is any integer (the second option)