4. $y=\sin \left(\frac{\pi}{3}+x\right)-\sin 3 x$ 的最大值为 $\qquad$

The maximum value of $y=\sin \left(\frac{\pi}{3}+x\right)-\sin 3 x$ is $\qquad$
4. $\frac{8 \sqrt{3}}{9}$ Analysis: Let $\frac{\pi}{3}+x=t$, then $x=t-\frac{\pi}{3}, \sin 3 x=\sin (3 t-\pi)=-\sin 3 t, y=\sin t+\sin 3 t=$ $4 \sin t\left(1-\sin ^{2} t\right)$, hence $y^{2}=16 \sin ^{2} t\left(1-\sin ^{2} t\right)^{2}=8\left(2 \sin ^{2} t\right)\left(1-\sin ^{2} t\right)\left(1-\sin ^{2} t\right) \leqslant 8 \times\left(\frac{2}{3}\right)^{3}=\frac{64}{27}$, therefore, $y \leqslant \frac{8 \sqrt{3}}{9}$, equality holds when $\sin ^{2} t=\frac{1}{3}$.