630 CHAPTER 9 Polar Coordinates; Vectors 3 Find Products and Quotients of Complex Numbers The exponential form of a complex number is particularly useful for finding products and quotients of complex numbers. The following theorem states that the laws of exponents can be used. Solution To plot the complex number π π ( ) = + = π⋅ z i e 2 cos 6 sin 6 2 , i 6 plot the point whose polar coordinates are θ π ( ) ( ) = r, 2, 6 as shown in Figure 42. To express z in rectangular form, expand π π ( ) = + z i 2 cos 6 sin 6 . π π ( ) = + = + ⎛ ⎝ ⎜⎜⎜ ⎞ ⎠ ⎟⎟ ⎟ = + z i i i 2 cos 6 sin 6 2 3 2 1 2 3 Now Work PROBLEM 25 THEOREM Suppose θ θ ( ) = = + θ z r e r i cos sin i 1 1 1 1 1 1 and θ θ ( ) = = + θ z r e r i cos sin i 2 2 2 2 2 2 are two complex numbers. Then θ θ θ θ [ ] ( ) ( ) = = + + + θ θ ( ) + z z r r e r r i cos sin i 1 2 1 2 1 2 1 2 1 2 1 2 (5) If ≠ z 0, 2 then θ θ θ θ [ ] ( ) ( ) = = − + − θ θ ( ) − z z r r e r r i cos sin i 1 2 1 2 1 2 1 2 1 2 1 2 (6) Proof We prove formula (5). The proof of formula (6) is left as an exercise (see Problem 70). = ⋅ = θ θ θ θ ( ) + z z r e r e r r e i i i 1 2 1 2 1 2 1 2 1 2 and θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ θ ( ) ( ) ( )( ) [ ] ( ) ( ) [ ] ( ) ( ) = + ⋅ + = + + = − + + = + + + z z r i r i r r i i r r i r r i cos sin cos sin cos sin cos sin cos cos sin sin sin cos cos sin cos sin 1 2 1 1 1 2 2 2 1 2 1 1 2 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 ■ Figure 42 = π⋅ z e2 i /6 Imaginary axis Real axis 22 2 2 2 z 5 2(cos 1 i sin ) 5 2e O p– 6 p– 6 p– 6 p/6 i . Need to Review? The Laws of Exponents are discussed in Section A.1, pp. A7–A9. In Words The magnitude of a complex number z is r, and its argument is θ, so when z r i re cos sin i θ θ ( ) = + = θ the magnitude of the product (quotient) of two complex numbers equals the product (quotient) of their magnitudes; the argument of the product (quotient) of two complex numbers is determined by the sum (difference) of their arguments. Finding Products and Quotients of Complex Numbers If π π ( ) = + z i 3 cos 9 sin 9 and π π ( ) = + w i 5 cos 5 9 sin 5 9 , find (a) zw (b) z w Express the answers in polar form and in exponential form. Solution EXAMPLE 4 (a) π π π π π π ( ) ( ) ( ) = + ⋅ + = ⋅ = ⋅ ⋅ = = + π π π π π ( ) ⋅ ⋅ ⋅ + ⋅ zw i i e e e e i 3 cos 9 sin 9 5 cos 5 9 sin 5 9 3 5 3 5 15 15 cos 2 3 sin 2 3 i i i i 9 5 9 9 5 9 2 3 Write the complex numbers in exponential form. Use formula (5). Exponential form of the product zw Polar form of the product zw
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