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

Algebra Level 4

\((\text{i})\) 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.

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

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.

This is part of the set OCR A Level Problems.

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