OCR A Level: Further Pure 3 - Complex Numbers [January 2010 Q7]

$(\text{i})$ Solve the equation $\cos 6 \theta = 0$ , for $0<\theta<\pi$ .

$(\text{ii})$ By using de Moivre's theorem, show that $\cos 6 \theta \equiv (2 \cos^2 \theta -1)(16 \cos^4 \theta - 16\cos^2 \theta +1).$

$(\text{iii})$ Hence find the exact value of $\cos \left (\dfrac{\pi}{12} \right ) \cos \left (\dfrac{5 \pi}{12} \right ) \cos \left (\dfrac{7 \pi}{12} \right ) \cos \left (\dfrac{11 \pi}{12} \right )$ justifying your answer.

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If your answer to part $(\text{iii})$ is $\dfrac{a}{b}$ for co-prime integers $a$ and $b$ , input $a+b$ as your answer.

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There are 3 marks available for part (i), 5 marks for part (ii) and 5 marks for part (iii).

In total, this question is worth 18.1% of all available marks in the paper.

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This is part of the set OCR A Level Problems .