How do you add (2-3i) and (12-2i) in trigonometric form?

May 17, 2016

$\sqrt{221} \left(\cos \left(0.343\right) - i \sin \left(0.343\right)\right)$

Explanation:

A complex number z = x +iy can be expressed in trig. form as shown.

$z = x + i y = r \left(\cos \theta + i \sin \theta\right) \text{ where}$

•r=sqrt(x^2+y^2)" and " theta=tan^-1(y/x)

Now to get this sum in trig form we have to add the numbers together and then convert to trig.

$\Rightarrow \left(2 - 3 i\right) + \left(12 - 2 i\right) = 14 - 5 i$

Using x = 14 and y = -5 , convert to trig form.

$\Rightarrow r = \sqrt{{14}^{2} + {\left(- 5\right)}^{2}} = \sqrt{221} \text{ does not simplify further}$

and theta=tan^-1(-5/14)≈-0.343" radians"

$\Rightarrow 14 - 5 i = \sqrt{221} \left(\cos \left(- 0.343\right) + i \sin \left(- 0.343\right)\right)$

using $\cos \left(- \theta\right) = \cos \theta \text{ and } \sin \left(- \theta\right) = - \sin \theta$

we can also express in trig form as

$14 - 5 i = \sqrt{221} \left(\cos \left(0.343\right) - i \sin \left(0.343\right)\right)$