Consider the sum .

MATHEMATICS · Multi-part · A Level · Mathematics · Trigonometry · Trigonometry

Problem

Consider the sum cos⁡3θ+cos⁡5θ\cos3\theta+\cos5\theta. (i) Prove that cos⁡3θ+cos⁡5θ=2cos⁡4θ cos⁡θ\cos3\theta+\cos5\theta=2\cos4\theta\,\cos\theta. (ii) Hence express cos⁡3θ+cos⁡5θ\cos3\theta+\cos5\theta entirely in terms of sin⁡θ\sin\theta only. (iii) Evaluate the expression obtained in part (ii) when θ=18∘\theta=18^{\circ}. Give your answer correct to two decimal places.

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