# How do you write the following in trigonometric form and perform the operation given (4i)/(-4+2i)?

Mar 23, 2017

From Cartesian, $a + b i$, to trig $r \left(\cos \left(\theta\right) + i \sin \left(\theta\right)\right)$:

$r = \sqrt{{a}^{2} + {b}^{2}}$
theta = {(tan^-1(b/a);a>0,b>0),(pi/2; a=0,b>0),(pi+tan^-1(b/a),a<0),((3pi)/2;a=0,b<0), (2pi+tan^-1(b/a);a>0,b<0):}

#### Explanation:

For the numerator, a = 0, b = 4:

$r = \sqrt{{0}^{2} + {4}^{2}}$

$r = 4$
$\theta = \frac{\pi}{2}$

The trig form of the numerator: $4 \left(\cos \left(\frac{\pi}{2}\right) + i \sin \left(\frac{\pi}{2}\right)\right)$

For the denominator, a = -4, b = 2:

$r = \sqrt{{\left(- 4\right)}^{2} + {2}^{2}}$

$r = \sqrt{16 + 4}$

$r = \sqrt{20}$

$r = 2 \sqrt{5}$

$\theta = \pi + \tan \left(\frac{2}{-} 4\right)$

$\theta \approx 2.68$

The trig form of the numerator: $2 \sqrt{5} \left(\cos \left(2.68\right) + i \sin \left(2.68\right)\right)$

To divide you subtract the angle of the denominator from the angle of the numerator:

$\theta = \frac{\pi}{2} - 2.68$

$\theta \approx - 1.1$

And divide the magnitudes:

$r = \frac{4}{2 \sqrt{5}}$

$r = 2 \frac{\sqrt{5}}{5}$

The trig form is:

$2 \frac{\sqrt{5}}{5} \left(\cos \left(- 1.1\right) + i \sin \left(- 1.1\right)\right)$