SECTION 9.3 The Complex Plane; De Moivre’s Theorem 633 Notice that each complex root of − + i 1 3 has the same magnitude 2. 3 This means that each point corresponding to a cube root is the same distance from the origin on a circle with center at the origin and radius 2. 3 Furthermore, the arguments of these cube roots are π π 2 9 , 8 9 , and π 14 9 . The difference between consecutive pairs THEOREM Finding Complex Roots Exponential Form Suppose = θ w rei is a complex number and ≥ n 2 is an integer. If ≠ w 0, there are n distinct complex roots of w, given by the formula = θ π ( )( ) + z r e k n i n k 1 2 (7a) where = − k n 0, 1, 2, ..., 1. Polar Form Suppose θ θ ( ) = + w r i cos sin is a complex number and ≥ n 2 is an integer. If ≠ w 0, there are n distinct complex roots of w given by the formula θ π θ π ( ) ( ) ( ) ( ) = + + + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ z r n k i n k cos 1 2 sin 1 2 k n (7b) where = − k n 0, 1, 2, ..., 1. Finding Complex Cube Roots Find the complex cube roots of − + i 1 3. Express the answers in exponential form and in polar form. Solution EXAMPLE 7 First, express − + i 1 3 in exponential and polar form. Since the magnitude of − + i 1 3 is 2, we write π π ( ) − + = − + ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = + = π ⋅ i i i e 1 3 2 1 2 3 2 2 cos 2 3 sin 2 3 2 i 2 3 Polar form Exponential form Then, using Formula (7a), we find that the exponential form of the complex cube roots of − + = π ⋅ i e 1 3 2 i 2 /3 is = = π π ( )( ) + z e k 2 0, 1, 2 k i k 3 1 3 2 3 2 The three roots, in exponential form, are = = = = = = π π π π π π π π π ( )( ) ( )( ) ( )( ) + ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ z e e z e e z e e 2 2 2 2 2 2 i i i i i i 0 3 1 3 2 3 2 0 3 2 9 1 3 1 3 2 3 2 1 3 8 9 2 3 1 3 2 3 2 2 3 14 9 Next, from Formula (7b), the polar form of the complex cube roots of π π ( ) − + = + i i 1 3 2 cos 2 3 sin 2 3 is π π π π ( ) ( ) ( ) ( ) = + + + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = z k i k k 2 cos 1 3 2 3 2 sin 1 3 2 3 2 0, 1, 2 k n The three roots, in polar form, are π π π π π π π π π π π π π π π π π π ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = + ⋅ ⋅ + + ⋅ ⋅ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + = + ⋅ ⋅ + + ⋅ ⋅ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + = + ⋅ ⋅ + + ⋅ ⋅ ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = + z i i z i i z i i 2 cos 1 3 2 3 2 0 sin 1 3 2 3 2 0 2 cos 2 9 sin 2 9 2 cos 1 3 2 3 2 1 sin 1 3 2 3 2 1 2 cos 8 9 sin 8 9 2 cos 1 3 2 3 2 2 sin 1 3 2 3 2 2 2 cos 14 9 sin 14 9 0 3 3 1 3 3 2 3 3
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