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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:
cis(α − β ), provided |w| = 0
|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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