Answer
$(a)\quad t=\displaystyle \frac{3\pi}{2}+2k\pi$
$(b) \quad \displaystyle \frac{3\pi}{2}$
Work Step by Step
The first task is to isolate the trigonometric function on one side:
Add $(2\sin t-5)$ to both sides...
$3\sin t+2\sin t=-5\qquad $ ... simplify
$ 5\sin t=-5\quad$ ... divide with $5$
$\sin t=-1$
Now, we find a reference angle. From the table of characteristic angles, we know that $\displaystyle \sin\frac{\pi}{2}=1.$
Next, we know that within the interval $0\leq t \lt 2\pi $
the only radian angle that satisfies the equation is $\displaystyle \frac{3\pi}{2}$
Finally, to each individual solution, add multiples of $ 2\pi$ to cover all solutions:
$(a)$
$ t=\displaystyle \frac{3\pi}{2}+2k\pi$
$(b)$
The solutions within the interval $0\leq t \lt 2\pi:$
$\displaystyle \frac{3\pi}{2}.$
