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De Moivre's theorem and exponential formEdexcel A-Level Further Maths: Flashcards

What these 13 flashcards ask

  • State de Moivre's theorem.
  • If z=r(\cos\theta+i\sin\theta), what is z^n?
  • What is z+\frac1z when z=\cos\theta+i\sin\theta?
  • What is z-\frac1z when z=\cos\theta+i\sin\theta?
  • What are z^n+z^{-n} and z^n-z^{-n}?
  • How do you find \cos3\theta in powers of \cos\theta?
  • Give \sin3\theta in terms of \sin\theta.
  • Give \tan3\theta in terms of \tan\theta.
  • How do you express \cos^4\theta in multiple angles?
  • Write \cos^4\theta in multiple angles.
  • What is the exponential form of a complex number?
  • Express \cos\theta and \sin\theta using exponentials.
  • Sum of 1+z+\dots+z^{n-1} when z=\cos\frac{\pi}{n}+i\sin\frac{\pi}{n}?

Exam questions on De Moivre's theorem and exponential form

  1. Let z=cos⁡π9+isin⁡π9z=\cos\frac{\pi}{9}+i\sin\frac{\pi}{9}.
    Find the exact value of z3+1z3z^3+\frac{1}{z^3}.2 marks
  2. The complex number z=2eiπ/6z=2e^{i\pi/6} is given.
    Find z4z^4 in the form x+iyx+iy, giving exact values.2 marks
  3. Let z=cos⁡θ+isin⁡θz=\cos\theta+i\sin\theta, where θ\theta is real.
    Use de Moivre's theorem to show that sin⁡3θ=3sin⁡θ−4sin⁡3θ\sin3\theta=3\sin\theta-4\sin^3\theta.3 marks
See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).