Question

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

Answer 1

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 "i"`
  where `a` is the real part and `b` the imaginary part.
• Cartesian form: `z = (x, y)`
  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(y/x)`. It must be always between `-pi/2` and `pi/2`
• Cartesian form: `z = r cdot (cos phi+ "i" cdot sin phi)`
• 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 `{8"i"+2}/{-"i"+2}` 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 "i" + 2 = 8.2462_1.33`
• `-"i" + 2 = 2.2361_{-0.46}`

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

`{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 (cos 1.79 + "i" sin 1.79)`