Answer: (a) \(x=0.983,\frac{\pi}{2}\). (b) \(\theta=-0.865,-0.374,0.944,1.44\).

(a) From

\((2-3\cot x)\cos x=0,\)

either \(\cos x=0\), or \(2-3\cot x=0\).

So

\(x=\frac{\pi}{2}\)

or

\(\cot x=\frac23,\quad\text{so}\quad \tan x=\frac32.\)

In the given interval, this gives

\(x=0.982793\ldots.\)

Thus \(x=0.983,\frac{\pi}{2}\).

(b) Let \(u=2\theta+1\). Then

\(2\operatorname{cosec}u-12\sin u=5.\)

Writing this in terms of \(\sin u\),

\(\frac{2}{\sin u}-12\sin u=5.\)

Multiplying by \(\sin u\),

\(12\sin^2u+5\sin u-2=0.\)

Factorising,

\((4\sin u-1)(3\sin u+2)=0.\)

So

\(\sin u=\frac14\quad\text{or}\quad \sin u=-\frac23.\)

Since \(-1\leqslant\theta\leqslant2\), we have \(-1\leqslant u\leqslant5\).

The valid values give

\(\theta=-0.373659\ldots,\quad0.944456\ldots,\quad-0.864863\ldots,\quad1.43566\ldots.\)

So

\(\theta=-0.865,-0.374,0.944,1.44.\)