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Copyright © 2009 Pearson Addison-Wesley 8.6-1 8 Applications of Trigonometry

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Copyright © 2009 Pearson Addison-Wesley 8.6-2 8.1 The Law of Sines 8.2 The Law of Cosines 8.3 Vectors, Operations, and the Dot Product 8.4Applications of Vectors 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients 8.6 De Moivres Theorem; Powers and Roots of Complex Numbers 8.7Polar Equations and Graphs 8.8Parametric Equations, Graphs, and Applications 8 Applications of Trigonometry

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Copyright © 2009 Pearson Addison-Wesley1.1-3 8.6-3 De Moivres Theorem; Powers and Roots of Complex Numbers 8.6 Powers of Complex Numbers (De Moivres Theorem) Roots of Complex Numbers

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Copyright © 2009 Pearson Addison-Wesley1.1-4 8.6-4 De Moivres Theorem is a complex number, then In compact form, this is written

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Copyright © 2009 Pearson Addison-Wesley1.1-5 8.6-5 Remember the following:

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Copyright © 2009 Pearson Addison-Wesley1.1-6 8.6-6 Example 1 FINDING A POWER OF A COMPLEX NUMBER Find and express the result in rectangular form. First write in trigonometric form. Because x and y are both positive, θ is in quadrant I, so θ = 60°.

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Copyright © 2009 Pearson Addison-Wesley1.1-7 8.6-7 Example 1 FINDING A POWER OF A COMPLEX NUMBER (continued) Now apply De Moivres theorem. 480° and 120° are coterminal. Rectangular form

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Copyright © 2009 Pearson Addison-Wesley1.1-8 8.6-8 nth Root For a positive integer n, the complex number a + bi is an nth root of the complex number x + yi if

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Copyright © 2009 Pearson Addison-Wesley1.1-9 8.6-9 nth Root Theorem If n is any positive integer, r is a positive real number, and θ is in degrees, then the nonzero complex number r(cos θ + i sin θ) has exactly n distinct nth roots, given by where

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Copyright © 2009 Pearson Addison-Wesley1.1-10 8.6-10 Note In the statement of the nth root theorem, if θ is in radians, then

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Copyright © 2009 Pearson Addison-Wesley1.1-11 8.6-11 Example 2 FINDING COMPLEX ROOTS Find the two square roots of 4i. Write the roots in rectangular form. Write 4i in trigonometric form: The square roots have absolute value and argument

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Copyright © 2009 Pearson Addison-Wesley1.1-12 8.6-12 Example 2 FINDING COMPLEX ROOTS (continued) Since there are two square roots, let k = 0 and 1. Using these values for, the square roots are

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Copyright © 2009 Pearson Addison-Wesley1.1-13 8.6-13 Example 2 FINDING COMPLEX ROOTS (continued)

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Copyright © 2009 Pearson Addison-Wesley1.1-14 8.6-14 Example 3 FINDING COMPLEX ROOTS Find all fourth roots of Write the roots in rectangular form. Write in trigonometric form: The fourth roots have absolute value and argument

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Copyright © 2009 Pearson Addison-Wesley1.1-15 8.6-15 Example 3 FINDING COMPLEX ROOTS (continued) Since there are four roots, let k = 0, 1, 2, and 3. Using these values for α, the fourth roots are 2 cis 30°, 2 cis 120°, 2 cis 210°, and 2 cis 300°.

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Copyright © 2009 Pearson Addison-Wesley1.1-16 8.6-16 Example 3 FINDING COMPLEX ROOTS (continued)

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Copyright © 2009 Pearson Addison-Wesley1.1-17 8.6-17 Example 3 FINDING COMPLEX ROOTS (continued) The graphs of the roots lie on a circle with center at the origin and radius 2. The roots are equally spaced about the circle, 90° apart.

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Copyright © 2009 Pearson Addison-Wesley1.1-18 8.6-18 Example 4 SOLVING AN EQUATION BY FINDING COMPLEX ROOTS Find all complex number solutions of x 5 – i = 0. Graph them as vectors in the complex plane. There is one real solution, 1, while there are five complex solutions. Write 1 in trigonometric form:

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Copyright © 2009 Pearson Addison-Wesley1.1-19 8.6-19 Example 4 SOLVING AN EQUATION BY FINDING COMPLEX ROOTS (continued) The fifth roots have absolute value and argument Since there are five roots, let k = 0, 1, 2, 3, and 4. Solution set: {cis 0°, cis 72°, cis 144°, cis 216°, cis 288°}

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Copyright © 2009 Pearson Addison-Wesley1.1-20 8.6-20 Example 4 SOLVING AN EQUATION BY FINDING COMPLEX ROOTS (continued) The graphs of the roots lie on a unit circle. The roots are equally spaced about the circle, 72° apart.

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Copyright © 2009 Pearson Addison-Wesley1.1-21 8.6-21

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