Answer
$(a)$
$135^{o}+180^{o}k$
$(b)$
$135^{o}$ and $315^{o}.$
Work Step by Step
The first task is to isolate the trigonometric function on one side:
Subtract 2 from both sides...
$2\tan\theta=-2\qquad $ ... divide with $2$
$\tan\theta=- 1\quad$
Now, we find a reference angle. From the table of characteristic angles, we know that $\tan 45^{o}=1.$
Next, we know that tangent is negative in quadrants II and IV,
so angles that satisfy the equation are
$180^{o}-45^{o}=135^{o}$
and
$360^{o}-45^{o}=315^{o}$
Finally, to each individual solution, add multiples of $360^{o}$ to cover all solutions:
$(a)$
$\theta=135^{o}+360^{o}k $ or
$\theta=315^{o}+360^{o}k $
In degrees, we have 135, 315, 495, 775...
which can be combined and written as $135^{o}+180^{o}k$
$(b)$
The solutions within the interval $ 0^{o}\leq\theta \lt 360^{o}:$
$135^{o}$ and $315^{o}.$
