If a , b , c are a triangle angles, prove :
csc(a/2)+csc(b/2)+csc(c/2)≥6
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1 Answers
2305cosec(\theta ) = \frac{1}{sin(\theta )}
is a convex function.
Let :
f(\theta ) = cosec(\theta)
Therefore we can apply Jensen's Inequality,
f(\frac{a}{2})+f(\frac{b}{2})+f(\frac{c}{2})\geqslant 3f(\frac{{}\frac{a}{2}+\frac{b}{2}+\frac{c}{2}}{3}) = 3f(\frac{a+b+c}{6})=3f(30^{\circ}) = 6
[info on convex function and jensen inequality:
1)https://www.artofproblemsolving.com/Wiki/index.php/Convex_function
2)https://www.artofproblemsolving.com/Wiki/index.php/Jensen%27s_Inequality]
