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?

By the end of this Concept, you'll know how to divide complex numbers using the Quotient Theorem.

### Watch This

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

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

### Guidance

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:

We can use this rule for the computation of two complex numbers divided by one another.

#### Example A

Find the quotient of

**
Solution:
**
Express each number in polar form.

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

#### Example B

Find the quotient of the two complex numbers and

**
Solution:
**

#### Example C

Using the Quotient Theorem determine

**
Solution:
**

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

### Vocabulary

**
Quotient Theorem:
**
The
**
quotient theorem
**
is a theorem showing a simplified way to divide complex numbers.

### Guided Practice

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

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

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

**
Solutions:
**

1.

2.

3.

### Concept Problem Solution

You know that the 2 numbers to divide are and .

If you consider , you can use the formula:

Substituting values into this equation gives:

### Practice

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

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