So here's my function: $g(θ) = 20θ − 5 tan θ$
The instructions are: Find the critical numbers of the function. (Enter your answers as a comma-separated list. Use $n$ to denote any arbitrary integer values. If an answer does not exist, enter DNE.)
I managed to get the derivative to be: $20 – 5sec^2\theta$ and I managed to get to $sec \theta = 2$ but I'm stuck here. Can someone please help me?
Best Answer
Note that $sec^2\theta = 4$ means that $sec\theta$ can be $2$ or $-2$.
$sec\theta = \pm2 \implies cos\theta = \pm\tfrac{1}{2}$.
For what values of $\theta$ does $cos\theta = \tfrac{1}{2}$ or $\tfrac{-1}{2}$?
Well, there are an infinite number of them, but we can reduce all of them to four common forms:
This is all for some integer $n$. We get this because $2\pi n$ for some $n \in \mathbb{Z}$ is a full rotation around the circle, so the angle is the same. And the four points $\theta = \dfrac{\pi}{3}, \dfrac{2\pi}{3}, \dfrac{4\pi}{3}, \dfrac{5\pi}{3}$ are the four points in the range $[0 ,2\pi )$ where $cos\theta = \pm\frac{1}{2}$.