IIT JEE , AIEEE Forum - Mathematics , Trignometry

ankitagg

the minimum value of the expression sinA+sinB+sinC,where A,B,C are real positive angles satisfying A+B+C=180 is
a)positive
b)zero
c)negative
d)-3

rishipratimm

it shud b positive.

sin A +sin B + sin C

= 4 cos A/2 cos B/2 cos C/2

nw,as A/2,B/2,C/2 r always acute,their cosines r always +ve....

Werewolf

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

now a+b+c is always positive.

Conjurer

EDITED

Cut the chase,since angles are +ve they are all less than or equal to pi.So positive is the only answer satisfying.

ankitagg

can any one try it with A.M>G.M.i tried but i am not getting it.

anchitsaini

edit

ankit u have urself told the answer i think--

(sin A + sin B + sin C) / 3 >= ^{[ 3 ]}

sin A sin B sin C

(sin A + sin B + sin C) >= 3^{[ 3 ]}

sin A sin B sin C

now as A , B , C < = 180 hence

sin A , sin B , sin C is positive

thus 3^{[ 3 ]}

sin A sin B sin C is positive hence the answer is positive

nikhil_729

yes,sina+sinb+sinc/3>sin(a+b+c)/3

hence!

sina+sinb+sinc/3>sinpi/3

sina+sin+sinc/3>root3/2

hence,

sina+sin+sinc>3root3/2

hsbhatt

Related only to the post above:

When a function is concave, it has the property that

$f(w_1 x_1+w_2 x_2+w_3 x_3+...+w_n x_n) geq w_1f(x_1)+w_2f(x_2)+w_3f(x_3)+...+w_n f(x_n) \ \ ext{where} sum_{i=1}^n w_i = 1$

and all w_i are all positive

This is called Jensen's inequality. The inequality reverses if the function is convex.

In general, a function is concave if

$f(w_1 x_1+w_2 x_2) geq w_1 f(x_1)+w_2 f(x_2) ext {where} w_1+w_2 = 1$

If the function is differentiable, this condition translated to 'f"(x)

This is satisfied by sinx in the interval [0, rac{pi}{2}]

$ext{Hence} sin( rac{x_1+x_2+x_3}{3}) geq rac {sin(x_1)+sin(x_2)+sin(x_3)} {3}$

$x_1+x_2+x_3 = pi \ \ ext{Hence} sin(x_1) + sin(x_2) + sin(x_3) leq 3 sin ( rac{pi}{3}) = 3 rac{sqrt3}{2}$

ankur.kkhurana

edited......

hsbhatt

it is written in bold letters

computer001

ankur it says real positive angles

ankur.kkhurana

agree havent read that .......
answer is positive ......sry for mistake edited above

anchitsaini

sir did u put up your answer to increase our knowledge or did u give an answer to the above asked question.

hsbhatt

the just previous post was trying to prove that the min was 3 rac{sqrt3}{2}

I wanted to prove that this is the max.

If you set A=B =d where d is a small +ve angle, then C = 180-2d

sinA+sinB+sinC = 2sind+sin2d. You can make d as small as you please and so the min is zero.