IIT JEE , AIEEE Forum - Mathematics , Algebra - prove:(1/sinA)+(1/sinB)+(1/sinC)>=2rt3 , if A,B,C are angles of triangle

savvej

prove:

(1/sinA)+(1/sinB)+(1/sinC) 2 , if A,B,C are angles of triangle.

ie.prove that 2rt3 is the minimum value of the above.

5 rates to one who answers.

note: if possible plz give an answer in which u prove the above ondependently.ie not using the "to prove" and utimately proving it something like > 0 ,thus true and hence above must be true.

roopa1991

(sinA)/a=(sinB)/b=(sinC)/c=1/2R

so 1/sinA=2R/a;1/sinB=2R/b;1/sinC=2R/c

=2R(1/a+1/b+1/c)

=2R(ab+bc+ac)/abc

change ab,bc,ca interms of cosa,cosb,cosc you will get.

savvej

no one has the gut s to solve this?

common u all r not dumb yaar!

atleast give a try?

savvej

some more answers?

savvej

the one who answers gets 10 rates.

common someone answer? TRIGONOMETRY isnt that pakao!

common newtons, eulid and arybhattas ,help me out..

hsbhatt

First you have to establish the inequality

$\sin A+ \sin B + \sin C \le \frac{3 \sqrt 3}{2}$

This easily done as sine is a concave function in the interval $[0, \pi]$ as f"(x) <0 in this interval

Hence, by Jensen's Inequality which states that for a concave function,

$\omega_1f(x_1)+\omega_2 f(x_2)+...+ \omega_n f(x_n) \le f(\omega_1 x_1 + \omega_2 x_2+...+\omega_n x_n) \\ \\<br/>\text{where} \ \sum_{i=1}^n \omega_i = 1$

So, $\frac{\sin A + \sin B + \sin C}{3} \le \sin {\left (\frac{A+B+C}{3} \right )}$

or $\sin A+ \sin B + \sin C \le \frac{3 \sqrt 3}{2}$ as A+B+C = $\pi$

Now, from AM-GM Inequality, $\left (\sin A + \sin B + \sin C \right ) \left ( \frac{1}{\sin A} +\frac{1}{\sin B}+\frac{1}{\sin C} \right) \ge 9$

$\Rightarrow \left ( \frac{1}{\sin A} +\frac{1}{\sin B}+\frac{1}{\sin C} \right) \ge \frac{9}{\sin A + \sin B + \sin C} \ge \frac{9} { \frac{3 \sqrt 3}{2}} = 2 \sqrt 3$

savvej

thank u very much hsbhatt sir.

shinee

can u plz tell me how u have taken that am gm relation
salute assured

allamraju

Ask & Discuss Questions with Community & Experts

(1/sinA)+(1/sinB)+(1/sinC) 2 , if A,B,C are angles of triangle.

@shinee,Let me explain u that am-gm.

We have, and

Multiplying these two inequalities will give u that

Hope u got it.