1. Pre-Calculus
Chapter 6
Additional Topics in Trigonometry

2. 6.5 Trig Form of a Complex Number
Objectives:
Find absolute values of complex numbers.
Write trig forms of complex numbers.
Multiply and divide complex numbers written in trig form.
Use DeMoivre’s Theorem to find powers of complex numbers.
Find nth roots of complex numbers.

3. Graphical Representation of a Complex Number
Graph in coordinate plane called the complex plane
Horizontal axis is the real axis.
Vertical axis is the imaginary axis.
3 + 4i •
-2 + 3i •
• -5i

4. Absolute Value of a Complex Number
Defined as the length of the line segment from the origin (0, 0) to the point.
Calculate using the Distance Formula.
3 + 4i •

5. Examples
Graph the complex number.
Find the absolute value.

6. Trig Form of Complex Number
Graph the complex number.
Notice that a right triangle is formed. θ
a + bi • b a r
How do we determine θ?

7. Trig Form of Complex Number
Substitute & into z = a + bi.
Result is
Sometimes abbreviated as

8. Examples Write the complex number –5 + 6i in trig form. r = ? θ = ?
Write z = 3 cos 315° + 3i sin 315° in standard form.
a = ?
b = ?

9. Product of Trig Form of Complex Numbers
Given
and
It can be shown that the product is
That is,
Multiply the absolute values.
Add the angles.

10. Quotient of Trig Form of Complex Numbers
Given
and
It can be shown that the quotient is
That is,
Divide the absolute values.
Subtract the angles.

11. Examples
Calculate using trig form and convert answers to standard form.

12. Powers of Complex Numbers
If z = r (cos θ + i sin θ), find z2.
What about z3?

13. DeMoivre’s Theorem
If z = r (cos θ + i sin θ) is a complex number and n is a positive integer, then

14. Examples
Apply DeMoivre’s Theorem.

15. Roots of Complex Numbers
Recall the Fundamental Theorem of Algebra in which a polynomial equation of degree n has exactly n complex solutions.
An equation such as x6 = 1 will have six solutions. Each solution is a sixth root of 1.
In general, the complex number u = a + bi is an nth root of the complex number z if

16. Solutions to Previous Example
An equation such as x6 = 1 will have six solutions. Each solution is a sixth root of 1.

17. nth Roots of a Complex Number
For a positive integer n, the complex number z = r (cos θ + i sin θ) has exactly n distinct nth roots given by
Note: The roots are equally spaced around a circle of radius centered at the origin.

18. Example Find the three cube roots of z = –2 + 2i.
Write complex number in trig form.
Find r.
Find θ.
Use the formula with k = 0, 1, and 2.

19. Solution

20. Homework 6.5
Worksheet 6.5