# Addition operations in polar form? sqrt(18)(cos175°+isin175°)+(cos40°+isin40°)

## What would be the best way to perform addition operations in the polar form? Convert to cartesian form then add Re/Im components or is there another way? I only know how to multiply/divide in polar form.

Jun 25, 2018

color(brown)(=> -3.4605 + 1.0126 i

#### Explanation:

$z = a + b i = r \left(\cos \theta + i \sin \theta\right)$

$r = \sqrt{{a}^{2} + {b}^{2}} , \text{ } \theta = {\tan}^{-} 1 \left(\frac{b}{a}\right)$

${r}_{1} \left(\cos \left({\theta}_{1}\right) + i \sin \left({\theta}_{2}\right)\right) + {r}_{2} \left(\cos \left({\theta}_{2}\right) + i \sin \left({\theta}_{2}\right)\right) = {r}_{1} \cos \left({\theta}_{1}\right) + {r}_{2} \cos \left({\theta}_{2}\right) + i \left({r}_{1} \sin \left({\theta}_{1}\right) + {r}_{2} \sin \left({\theta}_{2}\right)\right)$
sqrt18(cos 175 + i sin 175) + 9cos 40 + i sin 40)

$\implies \left(\sqrt{18} \cos 175 + \cos 40\right) + i \left(\sqrt{18} \sin 175 + \sin 40\right)$

$\implies \left(- 4.2265 + 0.7660\right) + i \left(0.3698 + 0.6428\right)$

color(brown)(=> -3.4605 + 1.0126 i