Complex Numbers
De Moivre's Theorem
Premium Question
Grade 11
Question:
If $z + \frac{1}{z} = 2 \cos \theta$, then $z^n + \frac{1}{z^n}$ equals:
(1) $\cos n\theta$
(2) $2 \cos n\theta$
(3) $2 \sin n\theta$
(4) $2^n \cos \theta$
Step-by-Step Solution
Key Concept: The condition forces $z = e^{\pm i\theta}$. For $z = e^{i\theta}$: $z^n + 1/z^n = e^{in\theta} + e^{-in\theta} = 2 \cos n\theta$.
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Correct Answer: (2)