Without using a calculator

MATHEMATICS · Multi-part · CBSE · Mathematics · Introduction to Trigonometry · Introduction to Trigonometry

Problem

Without using a calculator, show that the sums of the squares of the sines of 30∘,  45∘30^\circ,\;45^\circ and 60∘60^\circ is equal to the corresponding sum for the cosines. (i) Find the exact values of sin⁡30∘,  sin⁡45∘\sin 30^\circ,\; \sin 45^\circ and sin⁡60∘\sin 60^\circ, and hence compute sin⁡230∘,  sin⁡245∘\sin^2 30^\circ,\; \sin^2 45^\circ and sin⁡260∘\sin^2 60^\circ. (ii) Find the exact values of cos⁡30∘,  cos⁡45∘\cos 30^\circ,\; \cos 45^\circ and cos⁡60∘\cos 60^\circ, and hence compute cos⁡230∘,  cos⁡245∘\cos^2 30^\circ,\; \cos^2 45^\circ and cos⁡260∘\cos^2 60^\circ. (iii) Using the results of parts (i) and (ii), show that sin⁡230∘+sin⁡245∘+sin⁡260∘=cos⁡230∘+cos⁡245∘+cos⁡260∘.\sin^2 30^\circ + \sin^2 45^\circ + \sin^2 60^\circ = \cos^2 30^\circ + \cos^2 45^\circ + \cos^2 60^\circ.

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