# Trigonometry : Complex Numbers

## Example Questions

### Example Question #1 : Complex Numbers

Name the real part of this expression and the imaginary part of this expression: .

All parts are real; there are no imaginary parts

Real:

Imaginary:

Real:

Imaginary:

All parts are imaginary; there are no real parts

Real:

Imaginary:

Real:

Imaginary:

Explanation:

The real part of this expression includes any terms that do not have  attached to them. Therefore the real part of this expression is 3. The imaginary part of this expression includes any terms with  that cannot be further reduced; the imaginary part of this expression is .

### Example Question #2 : Complex Numbers

Find the product of the complex number and its conjugate:

Explanation:

To solve this problem, we must first identify the conjugate of this complex number. The conjugate keeps the real portion of the number the same, but changes the sign of the imaginary part of the number. Therefore the conjugate of  is . Now, we need to multiply these together using distribution, combining like terms, and substituting .

### Example Question #3 : Complex Numbers

What is the complex conjugate of ?

Explanation:

To solve this problem, we must understand what a complex conjugate is and how it relates to a complex number. The conjugate of a number  is . Therefore the conjugate of  is .

### Example Question #11 : Complex Numbers/Polar Form

Simplify .

Explanation:

To add complex numbers, we must combine like terms: real with real, and imaginary with imaginary.

### Example Question #682 : Trigonometry

Simplify  .

Explanation:

In order to solve this problem, we must combine real numbers with real numbers and imaginary numbers with imaginary numbers. Be careful to distribute the subtraction sign to all terms in the second set of parentheses.

### Example Question #6 : Complex Numbers

Simplify .

Explanation:

To solve this problem, make sure you set it up to multiply the entire parentheses by itself (a common mistake it to try to simply distribute the exponent 2 to each of the terms in the parentheses.)

(recall that )

Please note that while the answer choice  is not incorrect, it is not fully simplified and therefore not the correct choice.

### Example Question #7 : Complex Numbers

What is the complex conjugate of 5? What is the complex conjugate of 3i?

Complex conjugates do not exist for these terms

Explanation:

While these terms may not look like they follow the typical format of , don't let them fool you! We can read 5 as  and we can read 3i as . Now recalling that the complex conjugate of  is , we can see that the complex conjugate of  is just  and the complex conjugate of  is

### Example Question #8 : Complex Numbers

Perform division on the following expression by utilizing a complex conjugate:

Explanation:

To perform division on complex numbers, multiple both the numerator and the denominator of the fraction by the complex conjugate of the denominator. This looks like:

### Example Question #9 : Complex Numbers

Which of the following represents  graphically?

Explanation:

To represent complex numbers graphically, we treat the x-axis as the "axis of reals" and the y-axis as the "axis of imaginaries." To plot , we want to move 6 units on the x-axis and -3 units on the y-axis. We can plot the point P to represent , but we can also represent it by drawing a vector from the origin to point P. Both representations are in the diagram below.

### Example Question #10 : Complex Numbers

The following graph represents which one of the following?