ta có $\sin A+\sin B+\sin C+\sin \frac{\pi }{3}=2\sin \frac{A+B}{2}\cos \frac{A-B}{2}+2\sin (\frac{C}{2}+\frac{\pi }{6})\cos (\frac{C}{2}-\frac{\pi }{6})$$\leq 2\sin \frac{A+B}{2}+2\sin (\frac{C}{2}+\frac{\pi }{6})=4\sin (\frac{A+B+C}{4}+\frac{\pi }{12})\cos (\frac{A+B-C}{4}-\frac{\pi }{12})\leq 4\sin \frac{\pi }{3}$$\Rightarrow \sum sin\leq 3\sin \frac{\pi }{3}=\frac{3\sqrt{3}}{2}$dấu "=' $\Leftrightarrow \Delta ACB đều$
ta có $\sin A+\sin B+\sin C+\sin \frac{\pi }{3}=2\sin \frac{A+B}{2}\cos \frac{A-B}{2}+2\sin( \frac{C}{2}+\frac{\pi }{6})\cos (\frac{C}{2}-\frac{\pi }{6})$$\leq 2\sin \frac{A+B}{2}+2\sin (\frac{C}{2}+\frac{\pi }{6})=4\sin (\frac{A+B+C}{4}+\frac{\pi }{12})\cos (\frac{A+B-C}{4}-\frac{\pi }{12})\leq 4\sin \frac{\pi }{3}$$\Rightarrow \sum sin\leq 3\sin \frac{\pi }{3}=\frac{3\sqrt{3}}{2}$dấu "=' $\Leftrightarrow \Delta ACB đều$
ta có $\sin A+\sin B+\sin C+\sin \frac{\pi }{3}=2\sin \frac{A+B}{2}\cos \frac{A-B}{2}+2\sin
(\frac{C}{2}+\frac{\pi }{6})\cos (\frac{C}{2}-\frac{\pi }{6})$$\leq 2\sin \frac{A+B}{2}+2\sin (\frac{C}{2}+\frac{\pi }{6})=4\sin (\frac{A+B+C}{4}+\frac{\pi }{12})\cos (\frac{A+B-C}{4}-\frac{\pi }{12})\leq 4\sin \frac{\pi }{3}$$\Rightarrow \sum sin\leq 3\sin \frac{\pi }{3}=\frac{3\sqrt{3}}{2}$dấu "=' $\Leftrightarrow \Delta ACB đều$