U6FM Hwk

Total: 27 marks. Show all working out. Those not showing mathematical rigour will be penalised.

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1. Express $\cos 6\theta$ in terms of $\cos^n \theta$

   [3 marks]

2. Express $\cos^6 \theta$ in terms of $\cos (n\theta)$

   [4 marks]

3. Solve, giving your answers in the form $re^{i\theta}$ with $-\pi < \theta \leq \pi$ , $$z^6=1-\sqrt3 i$$ Represent your solutions on an Argand diagram

   [4 marks]

4. Represent on an Argand diagram the locus of $z$ s.t.
   1. $\arg{(z-3-2i)} = -\frac{5\pi}{6}$

      [3 marks]

   2. $|z+1|=|z-2i|$ also give the cartesian equation

      [3 marks]

   3. $|z+1|=|4z-8i|$ also give the cartesian equation

      [4 marks]

   4. Give the cartesian equation and sketch the locus $$\arg (\frac{z+3}{z-3}) = 3\pi /4$$

      [6 marks]