So I've gone back to school after a few years of being away. Just got a calculus assignment and I'm blanking completely. The problem I'm struggling with is:
Find the domain of $S(\theta)=\frac{1}{\cos^{2} \theta + \cos \theta}$
I know the answer is $(\frac{\pi}{2} < \theta-2 \pi n < \pi, n \in \mathbb{Z})$, but I don't remember how to get there and Google is not being helpful. I'm not looking for a full rundown, but maybe a hint or two to jog my memory would be good.
Let $f(\theta) = \cos(\theta)^2 + \cos(\theta)$.
Since we are considering $1/f(\theta)$ and we don't want this to be infinity, we require that $f(\theta) \neq 0$.
We know that $\cos(\theta) = 0$ whenever $\theta = \frac{\pi}{2} + n \pi$.
So $\cos(\theta)^2 + \cos(\theta) = 0 + 0 = 0$
We also know that $\cos(\theta) = -1$ whenever $\theta = \pi + 2n\pi$.
So $\cos(\theta)^2 + \cos(\theta) = (-1)^2 - 1 = 1 - 1 = 0$
These are the values at which $f(\theta) = 0$. At all other values the function is well defined.