# How do you add (-7-5i) and (-2+2i) in trigonometric form?

May 31, 2017

$z = 9.49 \left(\cos \left({18.43}^{\circ}\right) + i \sin \left(18.43\right)\right)$ or simply $\left(9.49 , {18.43}^{\circ}\right)$

#### Explanation:

Strategy. First add them up, while they are still in rectangular form. Then convert the single term rectangular number into trigonometric form. Choose degrees or radians for the angle. I choose degrees.

Step 1. Add the two rectangular complex numbers. The result will be in standard rectangular form $a + b i$ or $\left(a , b\right)$

$\left(- 7 - 5 i\right) + \left(- 2 + 2 i\right) = \left(- 7 - 2 - 5 i + 2 i\right) = - 9 - 3 i$

Here, $a = - 9$ and $b = - 3$

Step 2. Given the conversion formulas, translate to trig form, which is of the form $z = r \left(\cos \left(\theta\right) + i \sin \left(\theta\right)\right)$ or in polar form $\left(r , \theta\right)$

$\theta = {\tan}^{-} 1 \left(\frac{b}{a}\right) = {\tan}^{-} 1 \left(\frac{- 3}{-} 9\right) = {\tan}^{-} 1 \left(\frac{1}{3}\right) \approx {18.43}^{\circ}$

$r = \sqrt{{a}^{2} + {b}^{2}} = \sqrt{{\left(- 9\right)}^{2} + {\left(- 3\right)}^{2}} = \sqrt{90} \approx 9.49$

$z = 9.49 \left(\cos \left({18.43}^{\circ}\right) + i \sin \left(18.43\right)\right)$ or simply $\left(9.49 , {18.43}^{\circ}\right)$