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JEE Main 2020 · Mathematics · Trigonometry and Inverse Trigonometric Functions · medium

The maximum value of 3sin x+5sin (x+(π)/(3)) over all real x is

Single correct option

The maximum value of 3sin⁡x+5sin⁡ ⁣(x+π3)3\sin x+5\sin\!\left(x+\dfrac{\pi}{3}\right) over all real xx is
  1. A. 88
  2. B. 34\sqrt{34}
  3. C. 535\sqrt3
  4. D. 77
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Answer: (D) 77

Expand the second term: 5sin⁡ ⁣(x+π3)=52sin⁡x+532cos⁡x5\sin\!\left(x+\tfrac\pi3\right)=\tfrac52\sin x+\tfrac{5\sqrt3}{2}\cos x. The expression becomes (3+52)sin⁡x+532cos⁡x=112sin⁡x+532cos⁡x.\left(3+\tfrac52\right)\sin x+\tfrac{5\sqrt3}{2}\cos x=\tfrac{11}{2}\sin x+\tfrac{5\sqrt3}{2}\cos x. For Asin⁡x+Bcos⁡xA\sin x+B\cos x the maximum is A2+B2\sqrt{A^2+B^2}. Here A2+B2=1214+754=1964=49A^2+B^2=\dfrac{121}{4}+\dfrac{75}{4}=\dfrac{196}{4}=49, so the maximum is 77. The tempting value 8=3+58=3+5 would need both sines to peak together, which cannot happen since they are out of phase by 60∘60^\circ.
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