Question

What is the domain of the secant function?

Short answer

Answer: The domain of the secant function is all real numbers x, except x ≠ (2n+1) * (π/2), where n is an integer. In interval notation, the domain can be written as: ((-\infty, π/2)∪(π/2, 3π/2)∪(3π/2, 5π/2)∪ ... ∪(-π/2,π/2)∪(-3π/2,-π/2)∪...)

Step-by-step solution

Define the secant function

The secant function, denoted as sec(x), is the reciprocal of the cosine function, cos(x). So, sec(x) = 1/cos(x).

Identify the points where the cosine function is equal to zero

Cosine function values range between -1 and 1. Cosine function is equal to zero for all values of x such that x = (2n+1) * \frac{\pi}{2}, where n is an integer.

Determine the domain of the secant function

Since the secant function is undefined where the cosine function is equal to zero, we should exclude these points from the domain of the secant function. Therefore, the domain of the secant function is all real numbers x, except x ≠ (2n+1) * \frac{\pi}{2}, where n is an integer. In interval notation, the domain can be written as: \[((-\infty, \frac{\pi}{2})\bigcup(\frac{\pi}{2}, \frac{3\pi}{2})\bigcup(\frac{3\pi}{2}, \frac{5\pi}{2})\bigcup ... \bigcup(-\frac{\pi}{2},\frac{\pi}{2})\bigcup(-\frac{3\pi}{2},-\frac{\pi}{2})\bigcup...)\]