Find the minimum value of the following expression sin ⁡ α sin ⁡ β sin 2 ⁡ γ 2 + sin ⁡ β sin ⁡ γ sin 2 ⁡ α 2 + sin ⁡ γ sin ⁡ α sin 2 ⁡ β 2 ,where α , β and γ −− angles of some triangle?

Question

Find the minimum value of the following expression            sin      ⁡      α      sin      ⁡      β                      sin        2            ⁡              γ        2              +            sin      ⁡      β      sin      ⁡      γ                      sin        2            ⁡              α        2              +            sin      ⁡      γ      sin      ⁡      α                      sin        2            ⁡              β        2              ,where   α  ,  β and   γ  −− angles of some triangle?

relientaaho2 · Accepted Answer

HINT: use  γ  =  π  −  α  −  βor  α  +  β  +  γ  =  πand the Lagrange Multiplier Method now i have another idea: with  2  A  =  b  c  sin  ⁡  (  α  )  ,  2  A  =  a  c  sin  ⁡  (  β  )  ,  sin  ⁡  (  γ      /    2  )  =                    (        s        −        a        )        (        s        −        b        )                    a        b            we get for the first summand (after some Algebra)            4              A        2                    (      s      −      a      )      (      s      −      b      )              c        2            etc and our sum is given by  4  s  (            s      −      a              a      2        +            s      −      b              b      2        +            s      −      c              c      2        )

Find the minimum value of the following expression sin ⁡

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