FP2 June 2014 Q4
4.
(a) Use de Moivre’s theorem to show that \[\cos 6\theta = 32\cos^6\theta - 48\cos^4\theta + 18\cos^2\theta - 1\] (5)
(b) Hence solve for \(0 \leqslant \theta \leqslant \dfrac{\pi}{2}\) \[64\cos^6\theta - 96\cos^4\theta + 36\cos^2\theta - 3 = 0\] giving your answers as exact multiples of \(\pi\). (5)
| Scheme | Marks |
|---|---|
| \(\cos 6\theta = \mathrm{Re}[(\cos\theta + \mathrm{i}\sin\theta)^6]\) Ignore any imaginary parts included in their expansion | |
| \((\cos\theta + \mathrm{i}\sin\theta)^6 = c^6 + 6c^5\mathrm{i}s + 15c^4\mathrm{i}^2s^2 + 20c^3\mathrm{i}^3s^3 + 15c^2\mathrm{i}^4s^4 + 6c\mathrm{i}^5s^5 + \mathrm{i}^6s^6\) Attempt to expand correctly or only show real terms (May be implied) Often seen with powers of i simplified. If \(\mathrm{i}s^n\) seen, but becomes \(\mathrm{i}^ns^n\) (oe) later, deduct the final A mark of (a) even if no further errors. | M1 |
| \(\cos 6\theta = c^6 - 15c^4s^2 + 15c^2s^4 - s^6\) M1: Attempt to identify real parts. These 2 M marks may be awarded together A1: Correct expression | M1A1 |
| \(= c^6 - 15c^4(1 - c^2) + 15c^2(1 - c^2)^2 - (1 - c^2)^3\) Correct use of \(s^2 = 1 - c^2\) in all their sine terms | M1 |
| \(\cos 6\theta = c^6 - 15c^4 + 15c^6 + 15c^2(1 - 2c^2 + c^4) - (1 - 3c^2 + 3c^4 - c^6)\) | |
| \(\cos 6\theta = 32\cos^6\theta - 48\cos^4\theta + 18\cos^2\theta - 1\) * (\(\cos 6\theta\) must be seen somewhere) | A1cso |
| (5) |
| Scheme | Marks |
|---|---|
| \(64\cos^6\theta - 96\cos^4\theta + 36\cos^2\theta - 3 = 0\) \(\Rightarrow 2\cos 6\theta - 1 = 0 \therefore \underline{\cos 6\theta = \tfrac{1}{2}}\) or 0.5 M1: Uses part (a) to obtain an equation in \(\cos 6\theta\) A1: Correct underlined equation | M1A1 |
| \(\cos 6\theta = \tfrac{1}{2} \Rightarrow (6\theta =)\ \dfrac{\pi}{3}, \dfrac{5\pi}{3}, \dfrac{7\pi}{3}\) | |
| \(\theta = \dfrac{\pi}{18}, \dfrac{5\pi}{18}, \dfrac{7\pi}{18}\) M1: Valid attempt to solve \(\cos 6\theta = k,\ -1 \leqslant k \leqslant 1\) leading to \(\theta = \ldots\) Can be degrees A1 2 correct answers A1 3rd correct answer with no extras within the range, ignore extras outside the range. Must be radians Answers in degrees or decimal answers score A0A0 | M1 A1A1 |
| (5) | |
| (10 marks) |