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

**Geometry**Level 4

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

**Input the largest possible value of \( \beta\) as your answer.**

###### 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.

###### This is part of the set OCR A Level Problems.

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