I need help with finding the domain of this trigonometric function?

Question

I'm confused and I need to understand this better, how do I find the domain of:

f(t) = sec [(pi*t)/4]

Could you please explain in detail, step by step? I'm asking this because I don't actually have the textbook associated with these kinds of problems.

Answer 1

f(x) = cos(x) has domain (- infty, infty).

sec(x) = 1 / cos(x), and therefore has the same domain excluding all points where cos(x) = 0.

Those points are where:

x = (2n + 1)pi / 2 and n is any integer.

Replacing x by pi * t / 4 means the points satisfying

pi * t / 4 = (2n + 1)pi / 2 must be excluded.

That is values of t satisfying:

t = 2(2n + 1).

The domain of f(t) is (- infty, infty) \ { 2(2n + 1) } where n is any integer.