<s>[INST] To find maximum value If $A>0,B>0$ and $C>0$ and further it is known that $A+B+C=\frac{5\pi}{4}$,then find the maximum value of $\sin A+\sin B+\sin C$ 
 [/INST] $\sin{A}+\sin{B}+\sin{C}=2\sin{\frac{A+B}{2}}\cos{\frac{A-B}{2}}+\sin{C}\le 2\sin{\frac{A+B}{2}}+\sin{(\frac{5\pi}{4}-(A+B))}.$  Equality holds when $A=B$. Thus we consider the function $$f(x)=2\sin{x}+\sin{\left(\frac{5\pi}{4}-2x\right)}=2\sin{x}+\sin{\left(2x-\frac{\pi}{4}\right)}.$$
defined on $(0,\frac{5\pi}{8})$. Then $$f'(x)=2\cos{x}+2\cos{\left(2x-\frac{\pi}{4}\right)}$$
It achieves zero when $x=2x+\frac{3\pi}{4}+2k\pi$ or $x=2k\pi-2x+\frac{5\pi}{4}$.
That is, $0<x<\frac{5\pi}{12},f'>0$ and $5\pi/12<x<5\pi/8,f'<0$. Thus $f$ achieves maximum at $x=5\pi/12$. Namely, $A=B=C=5\pi/12$.
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