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Pre-Calculus Chapter 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 axis is the real axis. Vertical axis is the imaginary axis. 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 •

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Examples Graph the complex number. Find the absolute value.

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**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?

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**Trig Form of Complex Number**

Substitute & into z = a + bi. Result is Sometimes abbreviated as

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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. 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.

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**Powers of Complex Numbers**

If z = r (cos θ + i sin θ), find z2. What about z3?

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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.

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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.

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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

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