QCEVault

Mathematical induction and trigonometric proofs — Question 267

Original QCE Vault practice · 6 marks

Q267 · Practice questionTechnology-freeComplex familiar6 marks

QUESTION 267 (6 marks)

Consider (cos⁡θ+isin⁡θ)4(\cos\theta+i\sin\theta)^4.
a)
Use the binomial expansion and De Moivre's theorem to prove sin⁡4θ=4sin⁡θcos⁡θ(1−2sin⁡2θ)\sin4\theta=4\sin\theta\cos\theta(1-2\sin^2\theta).
[4 marks]
b)
Hence solve 4sin⁡θcos⁡θ(1−2sin⁡2θ)=04\sin\theta\cos\theta(1-2\sin^2\theta)=0 for 0≤θ<π0\le\theta<\pi.
[2 marks]
Question linkSyllabus coverage

Related questions

  1. Q178 · Original practice · 6 marks
    Consider (cos⁡θ+isin⁡θ)4(\cos\theta+i\sin\theta)^4.
    Mathematical induction and trigonometric proofs
  2. Q179 · Original practice · 6 marks
    For positive integers n, let Sn=∑k=1n(2k−1)3S_n=\sum_{k=1}^n(2k-1)^3.
    Mathematical induction and trigonometric proofs
  3. Q180 · Original practice · 6 marks
    For positive integers n, consider 8n−18^n-1.
    Mathematical induction and trigonometric proofs
  4. Q181 · Original practice · 6 marks
    A programmable light display contains k2kk2^k lights in row k, for k=1,…,nk=1,\ldots,n.
    Mathematical induction and trigonometric proofs