Solve trigonometric equation for $y$

Question

I have the following equation: $$y(\cos\theta) - x(\sin\theta) = sin(10* \big( y\big(\sin\theta\big)+(x(\cos\theta\big)))$$ I know that:

$sin(a+b) = sin(a) cos(b) + cos(a) sin(b)$, but i do not know how to solve?

Answer 1

Here's a sketch that sort of gets the answer. Let $z = y\sin\theta +x \cos \theta.$ Then your equation is $z'= \sin 10 z.$ This differential equation is solvable, but ugly.

But now you have $ y\sin\theta +x \cos \theta = \mbox{ugly}(\theta),$ which you can solve for $y$.