How do I solve two equations for two unknown angles?

Question

I need to solve these two equations for two unknown angles for a robot arm I am trying to program.

1. $\sin\theta_{1} + \sin\theta_{2} = c_{1}$
2. $\cos\theta_1 + \cos\theta_{2} = c_{2}$

I multiplied equation 1 with $\cos \theta_{1}$, and equation 2 with $\sin \theta _{1}$, and subtracted both equations and used a trig rule to arrive at this but I am stuck. How do I go further to solve one of the variables?

$\sin({\theta_{2}-\theta_{1}}) = c_{1}\cos{\theta_{1}} - c_{2}\sin{\theta_{1}} $

Alternatively for sum to product I get

1. $2\sin({(\theta_{1}+\theta_{2})/2})\cos({(\theta_{1}-\theta_{2})/2}) = c_{1}$
2. $2\cos({(\theta_{1}+\theta_{2})/2})\cos({(\theta_{1}-\theta_{2})/2}) = c_{2}$

Adding them up, simplifying, squaring gives me

$\cos^2 (\theta_{1}-\theta_{2})/2) = (c_{1}/2)^2 + (c_{2}/2)^2$

How can I go further this route to solve for the angles?

Answer 1

Hint: Use the formulas for sum to product.
This will give you two equations which you can easily manipulate to receive a solvable equation for the sum of the angles.