QCEVault

Mathematical induction and trigonometric proofs — Question 178

Original QCE Vault practice · 6 marks

Q178 · Practice questionTechnology-freeComplex familiar6 marks

QUESTION 178 (6 marks)

Consider (cos⁡θ+isin⁡θ)4(\cos\theta+i\sin\theta)^4.
a)
Use De Moivre's theorem to prove cos⁡4θ=8cos⁡4θ−8cos⁡2θ+1\cos4\theta=8\cos^4\theta-8\cos^2\theta+1.
[3 marks]
b)
Hence solve 8cos⁡4θ−8cos⁡2θ+1=−1/28\cos^4\theta-8\cos^2\theta+1=-1/2 for 0≤θ≤π/20\le\theta\le\pi/2.
[3 marks]
Question linkSyllabus coverage

Related questions

  1. 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
  2. Q180 · Original practice · 6 marks
    For positive integers n, consider 8n−18^n-1.
    Mathematical induction and trigonometric proofs
  3. 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
  4. Q182 · Original practice · 1 mark
    A student checks that n2+n+41n^2+n+41 is prime for n=0,…,39n=0,\ldots,39 and claims it is prime for every nonnegative integer. Which statement is correct?
    Mathematical induction and trigonometric proofs