Answer
$(a)$
$\theta=30^{o}+360^{o}k $ or
$\theta=330^{o}+360^{o}k $
$(b)$
$30^{o}$ and $330^{o}.$
Work Step by Step
The first task is to isolate the trigonometric function on one side:
Add $\sqrt{3}$ to both sides...
$2\cos\theta=\sqrt{3}\qquad $ ... divide with 2
$\displaystyle \cos\theta=\frac{\sqrt{3}}{2}$
Now, we find a reference angle. From the table of characteristic angles, we know that $\displaystyle \cos 30^{o}=\frac{\sqrt{3}}{2}.$
Next, we know that cosine is positive in quadrant IV as well,
so another angle that satisfies the equation is
$360^{o}-30^{o}=330^{o}$
Finally, to each individual solution, add multiples of $360^{o}$ to cover all solutions:
$(a)$
$\theta=30^{o}+360^{o}k $ or
$\theta=330^{o}+360^{o}k $
$(b)$
The solutions within the interval $ 0^{o}\leq\theta \lt 360^{o}:$
$30^{o}$ and $330^{o}.$
