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The number of solutions of the equation sin x cos 3x = sin 3x cos 5x in [ 0, π/2 ] is(a) 3 (b) 4 (c) 5 (d) 6 |
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Answer» Answer : (c) 5 sin x cos 3x = sin 3x cos 5x ⇒ 2 sin x cos 3x – 2 sin 3x cos 5x = 0 ⇒ [sin (3x + x) – sin (3x – x)] – [sin (3x + 5x) – sin (5x – 3x)] = 0 ⇒ sin 4x – sin 2x – sin 8x + sin 2x = 0 ⇒ sin 4x – sin 8x = 0 ⇒ 2 cos \(\big( \frac{4x+8x}{2}\big)\) sin \(\big( \frac{8x-4x}{2}\big)\) = 0 ⇒ 2 cos 6x sin 2x = 0 ⇒ cos 6x = 0 or sin 2x = 0 ⇒ 6x = (2n + 1) π/2 or 2x = nπ ⇒ x = (2n + 1) π/2 or x = \(\frac{n\pi}{2}\) ⇒ x = 0, \(\frac{\pi}{2}\), \(\frac{\pi}{12}\), \(\frac{3\pi}{12}\),\(\frac{5\pi}{12}\) in [0. \(\frac{\pi}{2}\)] = 5 |
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