Using the fact that (45^ = 3 15^ )

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

Problem

Using the fact that 45∘=3×15∘45^\circ = 3 \times 15^\circ, obtain exact values of certain trigonometric ratios. (i) Express sin⁡45∘\sin 45^\circ in terms of sin⁡15∘\sin 15^\circ only and show that sin⁡15∘\sin 15^\circ satisfies a cubic equation. (ii) Solve the cubic equation obtained in part (i) to determine the exact value of sin⁡15∘\sin 15^\circ. (iii) Hence, find the exact value of cos⁡15∘\cos 15^\circ.

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