# How do you divide ( 8i+2) / (-i +2) in trigonometric form?

Dec 25, 2015

The key to solve this problem is to know trigonometric form of a complex number.

#### Explanation:

Tip: this is just a theoretical introductory; you can jump to the text below the image to see the problem solution.

A complex number $z$ can be written in many ways:

• Binomial form: $z = x + y \cdot \text{i}$
where $a$ is the real part and $b$ the imaginary part.
• Cartesian form: $z = \left(x , y\right)$
just as the binomial form, but written as an ordered pair.
• Polar form: $z = {r}_{\phi}$
where $r$ is the modulus (or absolute value) of the number and $\theta$ is the argument.
-- The modulus is obtained by: $r = \sqrt{{x}^{2} + {y}^{2}}$
-- The argument is obtained by: $\phi = \arctan \left(\frac{y}{x}\right)$. It must be always between $- \frac{\pi}{2}$ and $\frac{\pi}{2}$
• Cartesian form: $z = r \cdot \left(\cos \phi + \text{i} \cdot \sin \phi\right)$
• Exponential form: $z = r \cdot {e}^{\phi}$
where $e$ is the exponential.

To sum up, there are two ways to represent a complex number:
- Depending on its coordinates, $x$ and $y$.
- Depending on its vectorial magnitudes, $r$ and $\phi$. If we want to add and substract complex numbers, we should use cartesian or binomial forms; however, if we want to solve a product or a fraction, we should use polar form, and then transform into the one which interests us.

Let us divide $\left\{8 \text{i"+2}/{-"i} + 2\right\}$ in trigonometric form.
First of all, we must transform both numbers (numerator and denominator) from binomial to polar form, and then we will transform the result into trigonometric form.

• $8 \text{i} + 2 = {8.2462}_{1.33}$
• $- \text{i} + 2 = {2.2361}_{- 0.46}$

And now, we divide both modulus, and we substract both arguments:

$\frac{{8.2462}_{1.33}}{{2.2361}_{- 0.46}} = {3.6878}_{1.79}$

Finally, we transform it into trigonometric form:
${3.6878}_{1.79} \equiv 3.6878 \cdot \left(\cos 1.79 + \text{i} \sin 1.79\right)$