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8 Applications of Trigonometry Copyright © 2009 Pearson Addison-Wesley 8.6-1 8 Applications of Trigonometry 8.1 The Law of Sines 8.2 The Law of Cosines 8.3 Vectors, Operations, and the Dot Product 8.4 Applications of Vectors 8.5 Trigonometric (Polar) Form of Complex Numbers; Products and Quotients 8.6 De Moivre’s Theorem; Powers and Roots of Complex Numbers 8.7 Polar Equations and Graphs 8.8 Parametric Equations, Graphs, and Applications Copyright © 2009 Pearson Addison-Wesley 8.6-2 8.6 De Moivre’s Theorem; Powers and Roots of Complex Numbers Powers of Complex Numbers (De Moivre’s Theorem) ▪ Roots of Complex Numbers Copyright © 2009 Pearson Addison-Wesley 1.1-3 8.6-3 De Moivre’s Theorem is a complex number, then In compact form, this is written Copyright © 2009 Pearson Addison-Wesley 1.1-4 8.6-4 Remember the following: r= Copyright © 2009 Pearson Addison-Wesley 2 + 2 , tan 1.1-5 θ= 8.6-5 Example 1 Find form. First write FINDING A POWER OF A COMPLEX NUMBER and express the result in rectangular in trigonometric form. Because x and y are both positive, θ is in quadrant I, so θ = 60°. Copyright © 2009 Pearson Addison-Wesley 1.1-6 8.6-6 Example 1 FINDING A POWER OF A COMPLEX NUMBER (continued) Now apply De Moivre’s theorem. 480° and 120° are coterminal. Rectangular form Copyright © 2009 Pearson Addison-Wesley 1.1-7 8.6-7 nth Root For a positive integer n, the complex number a + bi is an nth root of the complex number x + yi if Copyright © 2009 Pearson Addison-Wesley 1.1-8 8.6-8 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 Copyright © 2009 Pearson Addison-Wesley 1.1-9 8.6-9 Note In the statement of the nth root theorem, if θ is in radians, then Copyright © 2009 Pearson Addison-Wesley 1.1-10 8.6-10 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 Copyright © 2009 Pearson Addison-Wesley 1.1-11 8.6-11 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 Copyright © 2009 Pearson Addison-Wesley 1.1-12 8.6-12 Example 2 Copyright © 2009 Pearson Addison-Wesley FINDING COMPLEX ROOTS (continued) 1.1-13 8.6-13 Example 3 FINDING COMPLEX ROOTS Find all fourth roots of rectangular form. Write Write the roots in in trigonometric form: The fourth roots have absolute value and argument Copyright © 2009 Pearson Addison-Wesley 1.1-14 8.6-14 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°. Copyright © 2009 Pearson Addison-Wesley 1.1-15 8.6-15 Example 3 Copyright © 2009 Pearson Addison-Wesley FINDING COMPLEX ROOTS (continued) 1.1-16 8.6-16 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. Copyright © 2009 Pearson Addison-Wesley 1.1-17 8.6-17 Example 4 SOLVING AN EQUATION BY FINDING COMPLEX ROOTS Find all complex number solutions of x5 – 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: Copyright © 2009 Pearson Addison-Wesley 1.1-18 8.6-18 Example 4 SOLVING AN EQUATION BY FINDING COMPLEX ROOTS (continued) The fifth roots have absolute value argument and 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°} Copyright © 2009 Pearson Addison-Wesley 1.1-19 8.6-19 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. Copyright © 2009 Pearson Addison-Wesley 1.1-20 8.6-20 Copyright © 2009 Pearson Addison-Wesley 1.1-21 8.6-21