InterviewSolution
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InterviewSolution
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Express each of the following as the product of sine and cosine(i) sin A + sin 2A (ii) cos 2A + cos 4A (iii) sin 6θ – sin 2θ (iv) cos 2θ – cos θ |
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Answer» (i) sin A + sin 2A = 2 sin(\(\frac{A+2A}{2}\)) cos(\(\frac{A-2A}{2}\)) [∵ sin C + sin D = sin(\(\frac{C+D}{2}\)) cos(\(\frac{C-D}{2}\))] = 2 sin\(\frac{3A}{2}\) cos\(\frac{A}{2}\) [∵ cos(-θ) = cos θ] (ii) cos 2A + cos 4A = 2 cos(\(\frac{2A+4A}{2}\)) cos(\(\frac{2A-4A}{2}\)) [∵ cos C + cos D = 2 cos(\(\frac{C+D}{2}\)) cos(\(\frac{C-D}{2}\))] = 2 cos(\(\frac{6A}{2}\)) cos(\(\frac{6-2A}{2}\)) = 2 cos(3A) cos (-A) [∵ cos(-θ) = cos θ] = 2 cos 3A cos A (iii) sin 6θ – sin 2θ = 2 cos(\(\frac{6\theta+2\theta}{2}\)) cos(\(\frac{6\theta-2\theta}{2}\)) [∵ sin C – sin D = 2 cos(\(\frac{C+D}{2}\)) sin(\(\frac{C-D}{2}\))] = 2 cos(\(\frac{8\theta}{2}\)) sin(\(\frac{4\theta}{2}\)) = 2 cos 4θ sin 2θ (iv) cos 2θ – cos θ = -2 sin(\(\frac{2\theta+\theta}{2}\)) sin(\(\frac{2\theta-\theta}{2}\)) [∵ cos C – cos D = -2 sin(\(\frac{C+D}{2}\)) sin(\(\frac{C-D}{2}\))] = -2 sin(\(\frac{3\theta}{2}\)) sin(\(\frac{\theta}{2}\)) |
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