Answer
$$\theta=\frac{\pi}{3}\pm2n\pi\hspace{0.5cm}and\hspace{0.5cm}\theta=\frac{2\pi}{3}\pm2n\pi\hspace{4mm} \forall n\in Z$$ are critical numbers of $g$.
Work Step by Step
How to find the critical numbers of a function $f$ according to definition
1) Find all numbers of $c$ satisfying whether $f'(c)=0$ or $f'(c)$ does not exist.
2) See that all the $c$ we get lie in the domain of $f$ $(D_f)$ or not.
- If $c$ lies in $D_f$, $c$ is a critical number of $f$.
- If $c$ does not lie in $D_f$, $c$ is not a critical number of $f$.
$$g(\theta)=4\theta-\tan\theta$$
$\tan\theta$ would be defined if $\cos\theta\ne0$ (for $\tan\theta=\frac{\sin\theta}{\cos\theta}$), which means $\theta\ne\frac{\pi}{2}\pm n\pi$ for all $n \in Z$. Therefore, $$D_g=\{\theta | \theta\ne\frac{\pi}{2}\pm n\pi \hspace{2mm} \forall n\in Z\}.$$
1) Now, we find $g'(\theta)$
$$g'(\theta)=4-\frac{1}{\cos^2\theta}$$
We find $g'(\theta)=0$, $$g'(\theta)=0$$
$$\frac{1}{\cos^2\theta}=4$$
$$\cos^2\theta=\frac{1}{4}$$
$$\cos\theta=\pm\frac{1}{2}$$
$$\theta=\frac{\pi}{3}\pm2n\pi\hspace{0.5cm}or\hspace{0.5cm}\theta=\frac{2\pi}{3}\pm2n\pi \hspace{4mm} \forall n\in Z$$
All of these values are in $D_g$, so they are critical numbers of $g$.
Also, looking at $$g'(\theta)=4-\frac{1}{\cos^2\theta}$$
reveals that for $\cos\theta=0$, $g'(\theta)$ would not exist. However, all values of $\theta$ satisfying $\cos\theta=0$ do not belong to $D_g$. Therefore, they are not critical numbers of $g$.
We conclude that only $$\theta=\frac{\pi}{3}\pm2n\pi\hspace{0.5cm}and\hspace{0.5cm}\theta=\frac{2\pi}{3}\pm2n\pi \hspace{4mm} \forall n\in Z$$ are critical numbers of $g$.
