How to solve $\,A\sin(\theta_2-\theta_1) - B\sin(\theta_1) = 0$

Question

I want to find the solutions of the following equation (In order to find the singular points of a robot).

$A,B$ are positive numbers, and actually :
$A = 0.2531,$
$B = 0.2455.$

$$ A\sin(\theta_2 - \theta_1) - B\sin(\theta_1) = 0 $$

I plotted the result in matlab, and it seems there are a lot of solutions. I have no idea how to find an analytic result, and even if such a result is possible.

Answer 1

For any $\theta_1\in\mathbb{R}$, there is a solution (in fact, infinitely many), where the corresponding $\theta_2$ satisfies $$\sin(\theta_2-\theta_1)={B\over A}\sin(\theta_1)\implies\theta_2=\theta_1+ (-1)^n \sin^{-1}\left({B\over A}\sin(\theta_1) \right)+n\pi, \quad n\in \mathbb{Z}.$$

Answer 2

You're interested in the intersection curves between surfaces:

$$ z = a\sin(y-x) - b\sin(x), \quad \quad \quad z = 0 $$

$\quad\quad\quad\quad\quad\quad$

which correspond to plane curves:

$$ y = x + (-1)^k\arcsin\left(\frac{b}{a}\,\sin(x)\right) + k\,\pi \quad \quad \quad \text{with} \; k \in \mathbb{Z} $$

$\quad\quad\quad\quad\quad\quad$