Sum of Complex Numbers in Exponential Form

Theorem
Let $z_1 = r_1 e^{i \theta_1}$ and $z_2 = r_2 e^{i \theta_2}$ be complex numbers expressed in exponential form.

Let $z_3 = r_3 e^{i \theta_3} = z_1 + z_2$.

Then:
 * $r_3 = \sqrt {r_1^2 + r_2^2 + 2 r_1 r_2 \cos \left({\theta_1 - \theta_2}\right)}$
 * $\theta_3 = \arctan \left({\dfrac {r_1 \sin \theta_1 + r_2 \sin \theta_2} {r_1 \cos \theta_1 + r_2 \cos \theta_2}}\right)$

Proof
We have:

Then:

Thus:
 * $r_3 = \sqrt {r_1^2 + r_2^2 + 2 r_1 r_2 \cos \left({\theta_1 - \theta_2}\right)}$

and similarly:
 * $\theta_3 = \arctan \left({\dfrac {r_1 \sin \theta_1 + r_2 \sin \theta_2} {r_1 \cos \theta_1 + r_2 \cos \theta_2} }\right)$