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Question Number 122636 by mnjuly1970 last updated on 18/Nov/20

$. . . n i c e c a l c u l u s . . .$ $I n A \overset{Δ}{B C} p r o v e ::$ $* : s i n (\frac{A}{2}) s i n (\frac{B}{2}) s i n (\frac{C}{2}) ⩽ \frac{1}{8}$ $. . . . . . . . . . . . . . . . . . . . . . . . .$ $* * :: m a x (c o s (\frac{A}{2}) c o s (\frac{B}{2}) c o s (\frac{C}{2})) = ?$
	Answered by TANMAY PANACEA last updated on 18/Nov/20
	$three points A,B and C lie on curve y=sin(x/2) (A,sin(A/2)) (B,sin(B/2))and (C,sin(C/2)) centroid P (((A+B+C)/3),((sin(A/2)+sin(B/2)+sin(C/2))/3)) ordinate of centroid P=(((sin(A/2)+sin(B/2)+sin(C/2))/3)) pointQ={ (((A+B+C)/3))sin((((A+B+C)/3)/2))}lies on y=sin((x/2)) sin(((A+B+C)/6))=ordinate of pointQ sin(((A+B+C)/6))>((sin(A/2)+sin(B/2)+sin(C/2))/3) using AM>GM sin(((180^o )/6))>((sin(A/2)+sin(B/2)+sin(C/2))/3)>(sin(A/2)sin(B/2)sin(C/2))^(1/3) ((1/2))^3 >(sin(A/2)sin(B/2)sin(C/2)) proved$
	$t h r e e p o i n t s A, B a n d C l i e o n c u r v e y = s i n \frac{x}{2}$ $(A, s i n \frac{A}{2}) (B, s i n \frac{B}{2}) a n d (C, s i n \frac{C}{2})$ $c e n t r o i d P (\frac{A + B + C}{3}, \frac{s i n \frac{A}{2} + s i n \frac{B}{2} + s i n \frac{C}{2}}{3})$ $o r d i n a t e o f c e n t r o i d P = (\frac{s i n \frac{A}{2} + s i n \frac{B}{2} + s i n \frac{C}{2}}{3})$ $p o i n t Q = {(\frac{A + B + C}{3}) s i n (\frac{\frac{A + B + C}{3}}{2})} l i e s$ $o n y = s i n (\frac{x}{2})$ $s i n (\frac{A + B + C}{6}) = o r d i n a t e o f p o i n t Q$ $s i n (\frac{A + B + C}{6}) > \frac{s i n \frac{A}{2} + s i n \frac{B}{2} + s i n \frac{C}{2}}{3}$ $u s i n g A M > G M$ $s i n (\frac{180^{o}}{6}) > \frac{s i n \frac{A}{2} + s i n \frac{B}{2} + s i n \frac{C}{2}}{3} > {(s i n \frac{A}{2} s i n \frac{B}{2} s i n \frac{C}{2})}^{\frac{1}{3}}$ ${(\frac{1}{2})}^{3} > (s i n \frac{A}{2} s i n \frac{B}{2} s i n \frac{C}{2})$ $p r o v e d$
	Commented by TANMAY PANACEA last updated on 18/Nov/20

	$s a m e m e t h o d f o r c o s θ c u r v e$ $f i n a l s t e p$ $c o s (\frac{180^{o}}{6}) > {(c o s \frac{A}{2} c o s \frac{B}{2} c o s \frac{C}{2})}^{\frac{1}{3}}$ ${(\frac{\sqrt{3}}{2})}^{3} > (c o s \frac{A}{2} c o s \frac{B}{2} c o s \frac{C}{2})$ $\frac{3 \sqrt{3}}{2} > (c o s \frac{A}{2} c o s \frac{B}{2} c o s \frac{C}{2})$
	Commented by mnjuly1970 last updated on 18/Nov/20

	$t h a n k y o u s i r t a n m a y . . .$ $e x c e l l e n t . . .$
	Commented by TANMAY PANACEA last updated on 18/Nov/20

	$m o s t w e l c o m e$
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