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**Unformatted text preview: **Principle of Mathematical
Induction, z n = |z |n cis(nθ) for all natural numbers n.
The last property in Theorem 11.16 to prove is the quotient rule. Assuming |w| = 0 we have
z
w =
= |z |cis(α)
|w|cis(β )
|z | cos(α) + i sin(α)
|w| cos(β ) + i sin(β ) Next, we multiply both the numerator and denominator of the right hand side by (cos(β ) − i sin(β ))
which is the complex conjugate of (cos(β ) + i sin(β )) to get
z
w |z |
|w| = cos(α) + i sin(α) cos(β ) − i sin(β )
·
cos(β ) + i sin(β ) cos(β ) − i sin(β ) If we let N = [cos(α) + i sin(α)] [cos(β ) − i sin(β )] and simplify we get N = [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 ) Expand
= [cos(α) cos(β ) + sin(α) sin(β )] + i [sin(α) cos(β ) − cos(α) sin(β )] Rearrange and Factor = cos(α − β ) + i sin(α − β ) Diﬀerence Identities = cis(α − β ) Deﬁnition of ‘cis’ If we call the denominator...

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