Dabour Hub EGEgyptian Curriculum 2026/2027
2-2 (a) De Moivre's theorem for a positive integer power

2-2 (a) De Moivre's theorem for a positive integer power

Point 6 of 8 — Lesson 1: Complex Numbers in Trigonometric and Exponential Form and De Moivres TheoremQuestion bank: 7 questions (7 with pictures)

06 2-2 (a) De Moivre's theorem for a positive integer power

Statement

  • For every positive integer n: (cos⁡θ+isin⁡θ)n=cos⁡nθ+isin⁡nθ
  • Consequently [r(cos⁡θ+isin⁡θ)]n=rn(cos⁡nθ+isin⁡nθ).
  • In exponential form: (reiθ)n=rneinθ.

Using the theorem to prove trigonometric identities

  • Expand (cos⁡θ+isin⁡θ)n with the binomial theorem.
  • Equate it with cos⁡nθ+isin⁡nθ.
  • Equate the real parts to get cos⁡nθ and the imaginary parts to get sin⁡nθ.
  • Use sin⁡2θ=1−cos⁡2θ (or the reverse) to write the result in a single function.
✍️ Worked examples

Example 13: Express cos⁡3θ in terms of cos⁡θ.

Solution:

  • By De Moivre: (cos⁡θ+isin⁡θ)3=cos⁡3θ+isin⁡3θ.
  • By the binomial theorem: (cos⁡θ+isin⁡θ)3=cos⁡3θ+3icos⁡2θsin⁡θ−3cos⁡θsin⁡2θ−isin⁡3θ.
  • Equate the real parts: cos⁡3θ=cos⁡3θ−3cos⁡θsin⁡2θ.
  • Replace sin⁡2θ=1−cos⁡2θ: cos⁡3θ=cos⁡3θ−3cos⁡θ+3cos⁡3θ.
  • cos⁡3θ=4cos⁡3θ−3cos⁡θ.
  • In the same way, equating the imaginary parts gives sin⁡3θ=3sin⁡θ−4sin⁡3θ.

📝 Question bank for this point

7 questions linked to this point only, 7 of them with pictures.