Stitz-Zeager_College_Algebra_e-book

# We would like t to begin at t 0 instead of t 2 the

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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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## 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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