Question

For $x \in(0, \pi)$, the equation $\sin x+2 \sin 2 x-\sin 3 x=3$ has

• A. infinitely many solutions
• B. three solutions
• C. one solution
• D. no solution

Answer 1

Answer: (D)

$\sin x+2 \sin 2 x-\sin 3 x=3$
$\sin x+4 \sin x \cos x-3 \sin x+4 \sin ^{3} x=3$
$\sin x\left[-2+4 \cos x+4\left(1-\cos ^{2} x\right)\right]=3$
$\sin x\left[2-\left(4 \cos ^{2} x-4 \cos x+1\right)+1\right]=3$
$\sin x\left[3-(2 \cos x-1)^{2}\right]=3$
$\Rightarrow \sin x=1$ and $2 \cos x-1=0$
$\Rightarrow x=\frac{\pi}{2}$ and $x=\frac{\pi}{3}$