cos(𝛂)+cos(𝛃)+cos(𝛄)≤(3/2) prove the inequality


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Question Number 88752 by wiWiw last updated on 12/Apr/20

$\cos (α) + \cos (β) + \cos (γ) ⩽ \frac{3}{2}$ $prove the inequality$
Commented by TANMAY PANACEA. last updated on 12/Apr/20

Commented by TANMAY PANACEA. last updated on 12/Apr/20

$a n o t h e r a p p r o a c h$ $p o i n t A, B a n d C l i e s o n c u r v e y = c o s x$ $p o i n t A = (A, c o s A)$ $p o i n t B = (B, c o s B)$ $p o i n t C = (C, c o s C)$ $c e n t r o i d o f t r i a n g l e A B C i s Q$ $Q = (\frac{A + B + C}{3}, \frac{c o s A + c o s B + c o s C}{3})$ $f r o m f i g u r e O P = \frac{A + B + C}{3}$ $P Q = \frac{c o s A + c o s B + c o s C}{3}$ $P o i n t R l i e s o n y = c o s x$ $c o o r d i n a t e o f p o i n t R {\frac{A + B + C}{3}, c o s (\frac{A + B + C}{3})}$ $f r o m f i g u r e i t i s c l e a r$ $P R > P Q$ $c o s (\frac{A + B + C}{3}) > \frac{c o s A + c o s B + c o s C}{3}$ $\frac{3}{2} > c o s A + c o s B + c o s C$
	Answered by mind is power last updated on 12/Apr/20

	$w e c a n a s s u m a + b + c = π, a, b, c \in [0, \frac{π}{2} [$ $c o s {(x)}^{″} = - c o s (x) < 0,$ $c o n c a v$ $\Rightarrow \frac{1}{3} [c o s (a) + c o s (b) + c o s (c)] ⩽ c o s (\frac{a + b + c}{3}) = c o s (\frac{π}{3})$ $\Rightarrow c o s (a) + c o z (b) + c o s (c) ⩽ \frac{3}{2}$
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