632 CHAPTER 9 Polar Coordinates; Vectors The proof of De Moivre’s Theorem requires mathematical induction (which is not discussed until Section 12.4), so it is omitted here. The theorem is actually true for all integers n. You are asked to prove this in Problem 73. Using De Moivre’s Theorem Express π π ( ) ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ i 2 cos 9 sin 9 3 in (a) exponential form, (b) polar form, and (c) rectangular form. Solution EXAMPLE 5 (a) π π ( ) ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ = = π π ( ) ⋅ ⋅ ⋅ i e e 2 cos 9 sin 9 2 8 i i 3 3 3 9 3 (b) π π π π π π ( ) ( ) ( ) ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ = + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ i i i 2 cos 9 sin 9 2 cos 3 9 sin 3 9 8 cos 3 sin 3 3 3 (c) π π π π ( ) ( ) ( ) ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ + ⎣ ⎢ ⎤ ⎦ ⎥ = + ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = + i i i i 2 cos 9 sin 9 8 cos 3 sin 3 8 1 2 3 2 4 4 3 3 Now Work PROBLEM 45 Using De Moivre’s Theorem Express ( ) +i 1 5 in (a) exponential form, (b) polar form, and (c) rectangular form. Algebraic Solution To use De Moivre’s Theorem, first convert the complex number to polar form and exponential form. Since the magnitude of +i 1 is + = 1 1 2, 2 2 we write π π ( ) ( ) + = + = + = π⋅ i i i e 1 2 1 2 1 2 2 cos 4 sin 4 2 i 4 Polar form Exponential form (a) ( ) ( ) ( ) + = = = π π π ( ) ⋅ ⋅ ⋅ ⋅ i e e e 1 2 2 4 2 i i i 5 4 5 5 5 4 5 4 (b) π π π π π π ( ) ( ) ( ) ( ) ( ) ( ) ( ) + = + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ i i i i 1 2 cos 4 sin 4 2 cos 5 4 sin 5 4 4 2 cos 5 4 sin 5 4 5 3 5 (c) Using the polar form from part (b), we have π π ( ) ( ) + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − − ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = − − i i i 4 2 cos 5 4 sin 5 4 4 2 2 2 2 2 4 4 EXAMPLE 6 Figure 43 Graphing Solution A TI-84 Plus CE graphing calculator provides the solution shown in Figure 43. NOTE In the algebraic solution of Example 6, the approach used in Example 2 could also be used to write +i 1 in polar form. j 5 Find Complex Roots Suppose w is a complex number, and ≥ n 2 is a positive integer.Any complex number z that satisfies the equation = z w n is a complex nth root of w. In keeping with previous usage, if = n 2, the solutions of the equation = z w 2 are called complex square roots of w, and if = n 3, the solutions of the equation = z w 3 are called complex cube roots of w.
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