How to solve for $\theta$ in an expression involving linear and $\sin$ terms While trying to solve a spatial geometry problem, I came across the expression:
$$156θ-36\sin\theta=554.8$$
And I have no idea where to even begin. 
 A: There isn't a nice way to find an exact expression to solve this (possibly if you get really fancy with the Lambert W function, which still isn't really much of a solution). However, you can find increasingly accurate solutions via numerical methods such as Newton-Raphson.
A: Equations in which are present polynomial terms and trigonometric functions do not shaow analytical solutions and only numerical methods could be used.
For your specific case, since $-1 \leq \sin(\theta) \leq 1$, you can easily detect that the solution is between the solutions of $$156 \theta-36=554.8$$ $$156\theta+36=554.8$$  that is to say $$3.32564 \lt \theta \lt 3.78718$$
If you just need an approximate solution, you could develop the equation as a Taylor expansion built at $\theta =\pi$ and get $$156\theta-36\sin(\theta)-554.8 \approx \left(156 \pi -\frac{2774}{5}\right)+192 (\theta -\pi )+O\left((\theta -\pi
   )^2\right)$$ and get $$\theta =\frac{1387+90 \pi}{480} \approx 3.47863$$
If you need an accurate solution, use Newton method which, starting from a reasonable guess $\theta_0$, will update it according to $$\theta_{n+1}=\theta_{n}-\frac{f(\theta_{n})}{f'(\theta_{n})}$$ For your case $$f(\theta)=156\theta-36\sin(\theta)-554.8$$ $$f'(\theta)=156-36\cos(\theta)$$ Let us be lazy and start using $\theta_0=\pi$; so, the following iterates are generated : $3.47863$, $3.47983$ which is the solution for six significant figures.
Just to show how fast this scheme converges, let me give you the iterates with $25$ decimal figures up to solution

$3.478631955881419565545079$
$3.479834304497599014491005$
$3.479834349849334038884468$
$3.479834349849334103555807$

A: You could express $\sin \theta$ as its Taylor series and then get an approximate answer (you will have a 5th/7th power to solve depending on how accurate you go.)
