Sum of Complex Numbers in Exponential Form

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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:

\(\displaystyle r_1 e^{i \theta_1} + r_2 e^{i \theta_2}\) \(=\) \(\displaystyle r_1 \left({\cos \theta_1 + i \sin \theta_1}\right) + r_2 \left({\cos \theta_2 + i \sin \theta_2}\right)\) $\quad$ Definition of Polar Form of Complex Number $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \left({r_1 \cos \theta_1 + r_2 \cos \theta_2}\right) + i \left({r_1 \sin \theta_1 + r_2 \sin \theta_2}\right)\) $\quad$ $\quad$

Then:

\(\displaystyle {r_3}^2\) \(=\) \(\displaystyle r_1^2 + r_2^2 + 2 r_1 r_2 \cos \left({\theta_1 - \theta_2}\right)\) $\quad$ Complex Modulus of Sum of Complex Numbers $\quad$
\(\displaystyle \leadsto \ \ \) \(\displaystyle r_3\) \(=\) \(\displaystyle \sqrt {r_1^2 + r_2^2 + 2 r_1 r_2 \cos \left({\theta_1 - \theta_2}\right)}\) $\quad$ $\quad$


and similarly:

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

$\blacksquare$


Sources