sin(A)+ sin(B )+ sin(C)≤ ((3(√3))/2) ≻ Solution ≺ ( A+B +C =π ) l.h.s = 2sin(((A+B)/2) )cos(((A−B)/2))+2sin((C/2))cos((C/2)) = 2cos ((C/2)){2cos(A/2) .cos((B/2) )} = 4cos((A/2)).cos((B/2)).cos((C/2)) l.h.s ≤_(post) ^(previoue) 4 (((3(√3))/8) )=((3(√3))/2) note: { (( I : cos(a)+cos(b)=2cos(((a+b)/2))cos(((a−b)/2)))),(( II: sin(a)+sin(b)=2sin(((a+b)/2))cos(((a−b)/2) ))),(( { (( cos((π/2) −α)=sin(α))),(( sin((π/2) −α)= cos(α))) :})) :}


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Question Number 174837 by mnjuly1970 last updated on 12/Aug/22
$sin(A)+ sin(B )+ sin(C)≤ ((3(√3))/2) ≻ Solution ≺ ( A+B +C =π ) l.h.s = 2sin(((A+B)/2) )cos(((A−B)/2))+2sin((C/2))cos((C/2)) = 2cos ((C/2)){2cos(A/2) .cos((B/2) )} = 4cos((A/2)).cos((B/2)).cos((C/2)) l.h.s ≤_(post) ^(previoue) 4 (((3(√3))/8) )=((3(√3))/2) note: { (( I : cos(a)+cos(b)=2cos(((a+b)/2))cos(((a−b)/2)))),(( II: sin(a)+sin(b)=2sin(((a+b)/2))cos(((a−b)/2) ))),(( { (( cos((π/2) −α)=sin(α))),(( sin((π/2) −α)= cos(α))) :})) :}$
$s i n (A) + s i n (B) + s i n (C) ⩽ \frac{3 \sqrt{3}}{2}$ $≻ S o l u t i o n ≺ (A + B + C = π)$ $l . h . s = 2 s i n (\frac{A + B}{2}) c o s (\frac{A - B}{2}) + 2 s i n (\frac{C}{2}) c o s (\frac{C}{2})$ $= 2 c o s (\frac{C}{2}) {2 c o s \frac{A}{2} . c o s (\frac{B}{2})}$ $= 4 c o s (\frac{A}{2}) . c o s (\frac{B}{2}) . c o s (\frac{C}{2})$ $l . h . s \underset{p o s t}{\overset{p r e v i o u e}{⩽}} 4 (\frac{3 \sqrt{3}}{8}) = \frac{3 \sqrt{3}}{2}$ $n o t e : {\begin{cases} I : c o s (a) + c o s (b) = 2 c o s (\frac{a + b}{2}) c o s (\frac{a - b}{2}) \\ I I : s i n (a) + s i n (b) = 2 s i n (\frac{a + b}{2}) c o s (\frac{a - b}{2}) \\ {\begin{cases} c o s (\frac{π}{2} - α) = s i n (α) \\ s i n (\frac{π}{2} - α) = c o s (α) \end{cases} \end{cases}$
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