OCR A Level: Core 3 - Trigonometry [January 2013 Q9]

Geometry Level 4

\((\text{i})\) Prove that \[\cos^2{(\theta + 45°)}-\dfrac{1}{2}(\cos 2 \theta - \sin 2 \theta) \equiv \sin^2 \theta . \]

\((\text{ii})\) Hence solve \[6\cos^2{ \left ( \dfrac{\theta}{2} + 45° \right )}-3(\cos \theta - \sin \theta) = 2\] for \(-90°< \theta < 90°\).

\((\text{iii})\) It is given that there are two values of \(\theta\), where \(-90°< \theta < 90°\), satisfying \[6\cos^2{\left (\frac{\theta}{3} + 45°\right )}-3\left (\cos \frac{2 \theta}{3} - \sin \dfrac{2 \theta}{3} \right ) = k.\]

Find the set of possible values for \(k\).

If \(a<k<\dfrac{b}{c}\) where \(a\), \(b\) and \(c\) are integers with \(b\) and \(c\) coprime, input \(a+b+c\) as your answer.

There are 4 marks available for part (i), 3 marks for part (ii) and 3 marks for part (iii).
In total, this question is worth 13.9% of all available marks in the paper.

This is part of the set OCR A Level Problems.

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