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JEE Advanced 2016 Paper 2 · Mathematics · Trigonometry and Inverse Trigonometric Functions · medium

Consider the equation sin x+sin 2x+sin 3x=0.

One or more options correct

Consider the equation sin⁡x+sin⁡2x+sin⁡3x=0\sin x+\sin 2x+\sin 3x=0. Which of the following statements is/are correct?
  1. A. It has exactly 77 solutions in [0,2π][0,2\pi]
  2. B. The sum of all its solutions in [0,2π][0,2\pi] is 6π6\pi
  3. C. It has exactly 77 solutions in [−π,π][-\pi,\pi]
  4. D. The largest solution in [0,2π)[0,2\pi) is 4π3\dfrac{4\pi}{3}
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Answer: A, C

Group the terms: sin⁡x+sin⁡3x=2sin⁡2xcos⁡x\sin x+\sin3x=2\sin2x\cos x, so the equation is sin⁡2x (2cos⁡x+1)=0\sin2x\,(2\cos x+1)=0. Either sin⁡2x=0\sin2x=0, giving x=kπ2x=\frac{k\pi}{2}, or cos⁡x=−12\cos x=-\frac12, giving x=±2π3+2kπx=\pm\frac{2\pi}{3}+2k\pi. In [0,2π][0,2\pi] the solutions are 0,π2,π,3π2,2π,2π3,4π30,\frac\pi2,\pi,\frac{3\pi}2,2\pi,\frac{2\pi}3,\frac{4\pi}3: seven values, so (A) is true. Their sum is 5π+2π=7π≠6π5\pi+2\pi=7\pi\ne6\pi, so (B) is false. In [−π,π][-\pi,\pi] we get −π,−2π3,−π2,0,π2,2π3,π-\pi,-\frac{2\pi}3,-\frac\pi2,0,\frac\pi2,\frac{2\pi}3,\pi, also seven, so (C) is true. In [0,2π)[0,2\pi) the largest is 3π2>4π3\frac{3\pi}{2}>\frac{4\pi}3, so (D) is false.
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