The number of solutions of the equation
$$8000\sin^3 x + 3375\cos^3 x + 1728 = 5400\sin 2x$$
in $(0,\, 2\pi)$ is/are
Step-by-Step Solution
Key Concept: Note $8000 = 20^3$, $3375 = 15^3$, $1728 = 12^3$, and $5400\sin 2x = 3\cdot 20\sin x\cdot 15\cos x\cdot 12$. The equation becomes $(20\sin x)^3 + (15\cos x)^3 + 12^3 = 3(20\sin x)(15\cos x)(12)$. Apply: $a^3+b^3+c^3 = 3abc \iff a+b+c=0$ or $a=b=c$.
Case I: $20\sin x + 15\cos x + 12 = 0$ → Use $t = \tan(x/2)$: $3t^2 - 40t - 27 = 0$. Discriminant positive, both roots in $(0,\pi)$ giving 2 solutions in $(0,2\pi)$. Case II: $20\sin x = 15\cos x = 12$ → $\sin x = 3/5$, $\cos x = 4/5$ → 1 solution. Total: $\mathbf{3}$.
Correct Answer: 3