sin-A-sin-B-sin-C-3-3-2-Solution-A-B-C-pi-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-

Question Number 174837 by mnjuly1970 last updated on 12/Aug/22

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}