Our nal answer is x 2 cost y 3 sint for 2 t 2

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Unformatted text preview: he following theorem summarizes the advantages of working with complex numbers in polar form. Theorem 11.16. Products, Powers and Quotients Complex Numbers in Polar Form: Suppose z and w are complex numbers with polar forms z = |z |cis(α) and w = |w|cis(β ). Then • Product Rule: zw = |z ||w|cis(α + β ) • Power Rule:a z n = |z |n cis(nθ) for every natural number n • Quotient Rule: a z |z | = cis(α − β ), provided |w| = 0 w |w| This is DeMoivre’s Theorem The proof of Theorem 11.16 requires a healthy mix of definition, arithmetic and identities. We first start with the product rule. zw = [|z |cis(α)] [|w|cis(β )] = |z ||w| [cos(α) + i sin(α)] [cos(β ) + i sin(β )] We now focus on the quantity in brackets on the right hand side of the equation. [cos(α) + i sin(α)] [cos(β ) + i sin(β )] = cos(α) cos(β ) + i cos(α) sin(β ) + i sin(α) cos(β ) + i2 sin(α) sin(β ) = cos(α) cos(β ) + i2 sin(α) sin(β ) + i sin(α) cos(β ) + i cos(α) sin(β ) Rearranging terms = (cos(α) cos...
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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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