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.

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