29
RCMultiple Choice

Applying Vectors in the Plane — Cumulative exam

Question 29 • Precalculus

Review the proof of de Moivre’s theorem. Proof of de Moivre's Theorem [cos(θ) + i sin(θ)]k + 1A= [cos(θ) + i sin(θ)]k ∙ [cos(θ) + i sin(θ)]1B= [cos(kθ) + i sin(kθ)] ∙ [cos(θ) + i sin(θ)]C= cos(kθ)cos(θ) − sin(kθ)sin(θ) + i [sin(kθ)cos(θ) + cos(kθ) sin(θ)]D= cos(kθ + θ) + i sin(kθ + θ)E= cos[(k + 1)θ] + i sin[(k + 1)θ]

Answer
A
distributive property
B
factoring
C
multiplication rule
D
assumption (for n = k step)

Explanation

Step B replaces [cos(θ) + i sin(θ)]^k with cos(kθ) + i sin(kθ). That statement is the induction assumption, which says de Moivre’s theorem is true for n = k so it can be used to prove the case n = k + 1. Thus the correct choice is D, assumption (for n = k step).

AI-written and checked against the verified answer.

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