sin ³x cos3x + cos ³x sin3x= 3/8 với x ∈ (0, π/2)

Ptrinh tương đương vs

$4\sin^3x \cos(3x) + 4\cos^3x \sin(3x) = \dfrac{3}{2}$

Ta có

$\sin(3x) = 3\sin x - 4\sin^3x$

$<-> \sin^3x = \dfrac{3\sin x - \sin(3x)}{4}$

và

$\cos(3x) = 4\cos^3x - 3\cos x$

$<-> \cos^3x = \dfrac{\cos(3x) + 3\cos x}{4}$

THay vào ptrinh ta có

$\cos(3x) (3\sin x - \sin(3x)) + \sin(3x) (\cos(3x) + 3\cos x) = \dfrac{3}{2}$

$<-> 3\cos(3x) \sin x + 3\sin(3x) \cos x = \dfrac{3}{2}$

$<-> \cos(3x) \sin x + \sin(3x) \cos x = \dfrac{1}{2}$

$<-> \sin(4x) = \dfrac{1}{2}$

Vậy $4x = \dfrac{\pi}{6} + 2k\pi$ hoặc $4x = \dfrac{5\pi}{6} + 2k\pi$.

Do đó $x = \dfrac{\pi}{24} + \dfrac{k\pi}{2}$ hoặc $x = \dfrac{5\pi}{24} + \dfrac{k\pi}{2}$