De Moivre's theorem — A/L Combined Mathematics

How do I practise De Moivre's theorem?

1 question with a worked solution

Check before you answer: Proof by Mathematical Induction: Presenting the complete three-step induction proof of De Moivre's theorem for positive integers.

1. According to De Moivre's Theorem, what is the value of (cos θ + i sin θ)⁴?
   1. A) cos(4θ) + i sin(4θ)
   2. B) cos⁴ θ + i sin⁴ θ
   3. C) 4 cos θ + 4i sin θ
   4. D) cos(θ/4) + i sin(θ/4)
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   Correct answer: A) cos(4θ) + i sin(4θ)

   De Moivre's Theorem states that for any real θ and integer n: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). Setting n = 4 yields cos(4θ) + i sin(4θ).

   Show worked solution

   Step 1: Apply De Moivre's theorem with n = 4. Step 2: (cos θ + i sin θ)⁴ = cos(4θ) + i sin(4θ). Answer: cos(4θ) + i sin(4θ).
   Modelled on: the A/L Combined Mathematics examination pattern

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Where does De Moivre's theorem fit in the syllabus?

Year 2 · Term 1