OCR A Level: Core 3 - Trigonomoetry [January 2006 Q9]

$(\text{i})$ By first writing $\sin 3 \theta$ as $\sin (2 \theta + \theta)$ , show that $\sin 3 \theta = 3 \sin \theta - 4 \sin^3 \theta .$

$(\text{ii})$ Determine the greatest possible value of $9 \sin \left (\dfrac{10}{3} \alpha \right )- 12 \sin^3 \left (\dfrac{10}{3} \alpha \right )$ and find the smallest positive value of $\alpha$ (in degrees) for which that value occurs.

$(\text{iii})$ Solve, for $0°< \beta < 90°$ , the equation $3 \sin 6 \beta \, \text{csc} \, 2 \beta = 4$ . Give your answer(s) to 3 significant figures.

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Input the largest possible value of $\beta$ as your answer.

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There are 4 marks available for part (i), 3 marks for part (ii) and 6 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 .