Pre-CalculusChapter 6

Additional Topics in Trigonometry

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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.

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Graphical Representation of a Complex Number

Graph in coordinate plane called the complex plane

Horizontal axisis the real axis.

Vertical axis is the imaginaryaxis.

3 + 4i•-2 + 3i

•

• -5i

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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•

22 babiaz

5254343 22 i

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Examples Graph the complex number. Find the absolute value.

4 4z i

5z

5 6z i

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Trig Form of Complex Number Graph the complex number. Notice that a right triangle is formed.

2 2

cos sin

cos sin

where

a b

r ra r b r

r z a b

θ

a + bi•

b

a

r

How do we determine θ?

How do we determine θ?

1tanb

a

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Trig Form of Complex Number Substitute &

into z = a + bi. Result is

Sometimes abbreviated as

cos sina r b r

cos sinz r i r

cisz r sincos irz

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Examples Write the complex number –5 + 6i in trig

form. r = ? θ = ?

Write z = 3 cos 315° + 3i sin 315° in standard form. r = ? a = ? b = ?

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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.

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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.

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Examples Calculate using trig form and convert

answers to standard form.

35sin335cos3

240sin15240cos15.2

315sin6315cos6120sin4120cos4.1

i

i

ii

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Powers of Complex Numbers If z = r (cos θ + i sin θ), find z2.

What about z3?

2

2

cos sin cos sin

cos 2 sin 2

z r i r i

r i

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DeMoivre’s Theorem If z = r (cos θ + i sin θ) is a complex

number and n is a positive integer, then

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Examples Apply DeMoivre’s Theorem.

12

4

2

2

2

2.2

330sin330cos3.1

i

i

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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

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Solutions to Previous Example An equation such as x6 = 1 will have six

solutions. Each solution is a sixth root of 1.

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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.

n r

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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.

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Solution

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Homework 6.5 Worksheet

6.5