Answer: (a) \(\theta=26.6^\circ,108.4^\circ\); (b) \(\phi=-\frac{\pi}{2},\frac{5\pi}{6}\).

(a) Use

\(\operatorname{cosec}^2\theta=1+\operatorname{cot}^2\theta.\)

Then

\(3(1+\operatorname{cot}^2\theta)-5=5\operatorname{cot}\theta.\)

So

\(3\operatorname{cot}^2\theta-5\operatorname{cot}\theta-2=0.\)

Factorising,

\((3\operatorname{cot}\theta+1)(\operatorname{cot}\theta-2)=0.\)

Thus

Equivalently,

For \(0^\circ\leq\theta\leq180^\circ\), this gives

(b) Let

\(u=\phi+\frac{\pi}{3}.\)

Since \(-\pi\leq\phi\leq\pi\),

\(-\frac{2\pi}{3}\leq u\leq \frac{4\pi}{3}.\)

In this interval,

\(\sin u=-\frac12\)

when

Therefore

or