# How do you divide  (-4-9i)/(6-2i)  in trigonometric form?

Apr 19, 2018

$\text{ }$
color(blue)(z=(sqrt(970))/20)color(blue)([cos 264^@ + i* sin 264^@]

#### Explanation:

$\text{ }$
Let color(red)(Z_1 = r_1(Cos theta_1 + i*sin theta_1)

Let color(red)(Z_2 = r_2(Cos theta_2 + i*sin theta_2)

color(blue)((Z_1/Z_2) = (r_1/r_2)[cos (theta_1 - theta_2)+i*sin(theta_1 - theta_2)]

Consider the problem given:

color(green)(Z_1 = (-4-9i) and

color(green)(Z_2 = (6-2i)

In complex numbers,

we know that $i = \sqrt{- 1} \mathmr{and} {i}^{2} = \left(- 1\right)$

${Z}_{1} / {Z}_{2} = \frac{- 4 - 9 i}{6 - 2 i}$

Multiply and divide by the Conjugate of the denominator to simplify.

Z_1/Z_2=(-4-9i)/(6-2i)*color(red)((6+2i)/(6+2i)

rArr [-24-8i-54i-18(i^2)]/(36-(2i)^2

$\Rightarrow \frac{- 24 - 62 i + 18}{\left(36 + 4\right)}$

$\Rightarrow \frac{- 62 i - 6}{40}$

$\Rightarrow \frac{2 \left(- 31 i - 3\right)}{2 \cdot 20}$

$\Rightarrow \frac{\cancel{2} \left(- 31 i - 3\right)}{\cancel{2} \cdot 20}$

$\Rightarrow \frac{- 3 - 31 i}{20}$

color(blue)( :. Z_1/Z_2= -(3/20)-(31/20)i

Express color(blue)(r in terms of color(blue)(a and b.

Using Pythagoras Theorem,

${r}^{2} = \left({a}^{2} + {b}^{2}\right)$

$r = \sqrt{{a}^{2} + {b}^{2}}$

$| z | = a + b i$

|z| = sqrt(a^2+b^2

$\sin \left(\theta\right) = \frac{b}{r}$

rArr color(red)( b=r*sin(theta)

$\cos \left(\theta\right) = \frac{a}{r}$

rArr color(red)( a=r*sin(theta)

color(blue)( Z_1/Z_2= -(3/20)-(31/20)i

|z|=sqrt((-3/20)^2+(-31/20)^2

Simplifying you get

rArr sqrt(970/400

$\Rightarrow \frac{\sqrt{970}}{20}$

To find color(red)(theta

$\theta = {\tan}^{-} 1 \left[\frac{- \frac{31}{20}}{- \frac{3}{20}}\right]$

Using the calculator to simplify, you get

$\theta \approx {84.47245985}^{\circ}$

$\therefore \theta \approx {84.5}^{\circ}$

You have to add 180 to the angle when you recognize that the angle lies in the third quadrant.

Hence $\theta \approx {84.5}^{\circ} + {180}^{\circ}$

$\Rightarrow \theta \approx {264}^{\circ}$

Hence, the final representation of $Z$ will be

color(red)(Z=sqrt(970)/20[cos(264^@)+i*sin(264^@)]

Hope it helps.