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# Quotient Theorem

## Simplified way to divide complex numbers.

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Quotient Theorem

Suppose you are given two complex numbers in polar form, such as and and asked to divide them. Can you do this? How long will it take you?

### Quotient Theorem

Division of complex numbers in polar form is similar to the division of complex numbers in standard form. However, to determine a general rule for division, the denominator must be rationalized by multiplying the fraction by the complex conjugate of the denominator. In addition, the trigonometric functions must be simplified by applying the sum/difference identities for cosine and sine as well as one of the Pythagorean identities.

To obtain a general rule for the division of complex numbers in polar from, let the first number be and the second number be . The product can then be simplified by use of five facts: the definition , the difference identity , the difference identity , the Pythagorean identity, and the fact that the conjugate of is .

In general:

Use this rule for the computation of two complex numbers divided by one another in the following problems.

1. Find the quotient of

Express each number in polar form.

Now, plug in what we found to the Quotient Theorem.

2. Find the quotient of the two complex numbers and

3. Using the Quotient Theorem determine

Even though 1 is not a complex number, we can still change it to polar form.

### Examples

#### Example 1

Earlier, you were given two complex numbers in polar form and was asked to divide them.

You know that the 2 numbers to divide are and .

If you consider , you can use the formula:

Substituting values into this equation gives:

#### Example 2

Divide the following complex numbers. If they are not in polar form, change them before dividing.

#### Example 3

Divide the following complex numbers. If they are not in polar form, change them before dividing.

#### Example 4

Divide the following complex numbers. If they are not in polar form, change them before dividing.

### Review

Divide each pair of complex numbers. If they are not in trigonometric form, change them before dividing.

1. Can you divide a pair of complex numbers in standard form without converting to trigonometric form? How?

To see the Review answers, open this PDF file and look for section 6.11.

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