To determine the domain of the function [tex]\( y = \sec(x) \)[/tex], we first need to understand the properties of the secant function. The secant function is defined as the reciprocal of the cosine function:
[tex]\[ \sec(x) = \frac{1}{\cos(x)} \][/tex]
For [tex]\(\sec(x)\)[/tex] to be defined, [tex]\(\cos(x)\)[/tex] must be non-zero since division by zero is undefined. Therefore, we need to find the values of [tex]\(x\)[/tex] for which [tex]\(\cos(x) = 0\)[/tex] and exclude these from the domain.
The cosine function [tex]\(\cos(x)\)[/tex] is zero at the following values:
[tex]\[ x = \frac{\pi}{2} + n\pi \quad \text{where} \quad n \text{ is any integer} \][/tex]
These are the points where the cosine function crosses the x-axis, which corresponds to [tex]\(90^\circ, 270^\circ,\)[/tex] and so on for all integers [tex]\(n\)[/tex].
Therefore, the secant function [tex]\(\sec(x)\)[/tex] will be undefined at:
[tex]\[ x = n\pi + \frac{\pi}{2} \quad \text{where} \quad n \text{is any integer} \][/tex]
Because [tex]\(\sec(x)\)[/tex] is defined for all other values of [tex]\(x\)[/tex], we conclude that the domain of [tex]\( y = \sec(x) \)[/tex] is all real numbers [tex]\(x\)[/tex] except those on the form [tex]\( n\pi + \frac{\pi}{2} \)[/tex], where [tex]\( n \)[/tex] is any integer.
Hence, the detailed description of the domain of [tex]\(y = \sec(x)\)[/tex] is:
[tex]\[ \text{all real numbers except } n \pi + \frac{\pi}{2}, \text{ where } n \text{ is any integer} \][/tex]
