# Precalculus : Polar Coordinates and Complex Numbers

## Example Questions

### Example Question #13 : Find The Quotient Of Complex Numbers

Divide.

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be .

Now, distribute and simplify.

Recall that

### Example Question #21 : Products And Quotients Of Complex Numbers In Polar Form

Divide

Explanation:

Start by simplifying the fraction.

Recall that

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be .

Now, distribute and simplify.

Recall that

### Example Question #15 : Find The Quotient Of Complex Numbers

Simplify  into a number of the form .

Explanation:

We have

Multiply by the complex conjugate of the denominator.

The complex conjugate is the denominator with the sign changed:

Multiply fractions

FOIL the numerator and denominator

Apply the rule of :

Simplify.

Simplify further using the addition of fractions rule, then factor the i out of the 2nd fraction.

### Example Question #16 : Find The Quotient Of Complex Numbers

Divide:

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be .

Now, distribute and simplify.

Recall that

Then combine like terms:

Then since each term is a multiple of  you can simplify:

### Example Question #1 : Powers And Roots Of Complex Numbers

Find the magnitude of the complex number described by .

Explanation:

To find the magnitude of a complex number we use the formula:

,

where our complex number is in the form .

Therefore,

### Example Question #2 : Powers And Roots Of Complex Numbers

Find the magnitude of :

, where the complex number satisfies .

Explanation:

Note for any complex number z, we have:

.

Let .  Hence

Therefore:

This gives the result.

### Example Question #3 : Powers And Roots Of Complex Numbers

What is the magnitude of ?

Explanation:

To find the magnitude of a complex number we use the following formula:

, where .

Therefore we get,

.

Now to find

.

### Example Question #4 : Powers And Roots Of Complex Numbers

Simplify

Explanation:

We can use DeMoivre's formula which states:

Now plugging in our values of  and  we get the desired result.

### Example Question #5 : Powers And Roots Of Complex Numbers

Explanation:

First convert this point to polar form:

Since this number has a negative imaginary part and a positive real part, it is in quadrant IV, so the angle is

We are evaluating

Using DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get.

which is coterminal with since it is an odd multiplie

### Example Question #6 : Powers And Roots Of Complex Numbers

Evaluate

Explanation:

First, convert this complex number to polar form:

Since the real part is positive and the imaginary part is negative, this is in quadrant IV, so the angle is

So we are evaluating

Using DeMoivre's Theorem:

DeMoivre's Theorem is

We apply it to our situation to get.

is coterminal with  since it is an even multiple of