# How do you add (-6+i)+(9+3i) in trigonometric form?

Aug 6, 2018

color(maroon)((-6 + i) + (9 + 3i) = 3 + 4 i, " I Quadrant"

#### Explanation:

$\left(- 6 + i\right) + \left(9 + 3 i\right)$

$z = x + i y$

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

$r = | z | = | \sqrt{{x}^{2} + {y}^{2}} |$

${r}_{1} = \sqrt{- {6}^{2} + {1}^{2}} = \sqrt{37}$

${\theta}_{1} = \arctan \left(\frac{y}{x}\right) = {\tan}^{-} 1 \left(\frac{1}{-} 6\right) = {170.5377}^{\circ} , \text{ II Quadrant}$

${r}_{2} = \sqrt{{9}^{2} + {3}^{2}} = \sqrt{90}$

${\theta}_{2} = \arctan \left(\frac{y}{x}\right) = {\tan}^{-} 1 \left(\frac{3}{9}\right) = {18.4349}^{\circ} , \text{ I Quadrant}$

$\left(- 6 + i\right) + \left(9 + 3 i\right) = \sqrt{37} \left(\cos 170.5377 + \sin 170.5377\right) + \sqrt{90} \left(\cos 18.4349 + \sin 18.4349\right)$

$\implies \sqrt{37} \cos 170.5377 + \sqrt{90} \cos 18.4349 + i \left(\sqrt{37} \sin 170.5377 + \sqrt{90} \sin 18.4349\right)$

$\implies - 6 + 9 + 1 i + 3 i$

color(maroon)((-6 + i) + (9 + 3i) = 3 + 4 i, " I Quadrant"