# 6.10: Product Theorem

**At Grade**Created by: CK-12

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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?

After completing this Concept, you'll know the Product Theorem, which will make it easier to multiply complex numbers.

### Watch This

In the first part of this video you'll learn about the product of complex numbers in trigonometric form.

James Sousa: The Product and Quotient of Complex Numbers in Trigonometric Form

### Guidance

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:

We can use this general formula for the product of complex numbers to perform computations.

#### Example A

Find the product of the complex numbers and

**Solution:** Use the Product Theorem, .

Note: Angles are expressed unless otherwise stated.

#### Example B

Find the product of

**Solution:** First, calculate and

#### Example C

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

**Solution:**

First, convert to polar form:

And now do the same with :

And now substituting these values into the product theorem:

### Vocabulary

**Product Theorem:** The ** product theorem** is a theorem showing a simplified way to multiply complex numbers.

### Guided Practice

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

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

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

**Solutions:**

1.

2.

3.

### Concept Problem Solution

Since you want to multiply

where ,

you can use the equation

and calculate:

This simplifies to:

### Practice

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

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

### Image Attributions

Here you'll learn to derive and apply the Product Theorem, which simplifies the multiplication of complex numbers.