# Product Theorem

## Simplified way to multiply complex numbers.

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Product Theorem

What if you were given two complex numbers in polar form, such as and asked to multiply them? Would you be able to do this? How long would it take you?

### Product Theorem

Multiplication of complex numbers in polar form is similar to the multiplication of complex numbers in standard form. However, to determine a general rule for multiplication, the trigonometric functions will be simplified by applying the sum/difference identities for cosine and sine. To obtain a general rule for the multiplication 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 three facts: the definition , the sum identity , and the sum identity .

Now that the numbers have been designated, proceed with the multiplication of these binomials.

Therefore:

Use this general formula for the product of complex numbers to perform computations.

1. Find the product of complex numbers by using the Product Theorem

Find the product of the complex numbers and

Use the Product Theorem, .

Note: Angles are expressed unless otherwise stated.

2. Find the product of

First, calculate and

3. Find the product of the numbers and by first converting them to trigonometric form.

First, convert to polar form:

And now do the same with :

And now substituting these values into the product theorem:

### Examples

#### Example 1

Earlier, you were asked to multiply two complex numbers in polar form.

Since you want to multiply

where ,

you can use the equation

and calculate:

This simplifies to:

#### Example 2

Multiply together the following complex numbers. If they are not in polar form, change them before multiplying.

#### Example 3

Multiply together the following complex numbers. If they are not in polar form, change them before multiplying.

#### Example 4

Multiply together the following complex numbers. If they are not in polar form, change them before multiplying.

### Review

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

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

To see the Review answers, open this PDF file and look for section 6.10.

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