Trouble solving a trigonometric equation I have this problem in the book and no matter how many times I try I cannot isolate θ:
$$2 = (θ - 2cosθ)(cosθ)\tag{displayed}$$

You can't isolate it.
Just like solving
$xe^x = a$
for $x$ in terms of $a$
required creating a new function
(Lambert W),
this can only be solved numerically.
I can see by looking at the equation
that there are an infinite number of solutions
(look near where
$\cos(\theta) = 0$).
Note that,
once
$\theta > 2$,
$\theta  -2\cos \theta> 0$,
so all the roots there
are supplied by the
$\cos \theta$ term.
Since $\cos \theta = 0$
implies that
$\theta = \pi(n+\frac12)$
for some integer $n$,
you can get an estimate for the root
near this
by letting
$r_n = x+\pi(n+\frac12)$
any applying one Newton's iteration
to get a better approximation.
For looking around $\theta = 0$,
let $\cos \theta \sim 1-\frac{\theta^2}{2}$
and try to get an approximation
for $\theta$.
So plot the function and see where the roots are.
