In a `Delta ABC` `sin Asin B sin C <= (3sqrt3)/8`

Explanation

(a) We have already prove that in `triangle ABC`,
`sin^(2)A+sin^(2)B+sin^(2)Cle(9)/(4)`
Now, using `AMgeGM`, we have
`therefore (sin^(2)A+sin^(2)B+sin^(2)C)/(3)ge3sqrt(sin^(2)Asin^(2)Bsin^(2)C)`
Therefore, from Eq. (1) .
`(((9)/(4)))/(3)ge(sin^(2)A+sin^(2)B+sin^(2)C)/(3)`
`therefore (3)/(4)gt(sin A sin B sinC)^((2)/(3))`
or `sin A sin B sin C le(3sqrt(3))/(8)`
(b) We have already proved that in `triangle ABC`,
`sinA sin B sinCle3sqrt(3)/(8)`
Now, `sin2A+sin2B+sin2C=4sinA sinB sinC`
`le4(3sqrt(3)/(8))`
`therefore sin 2A+sin2B+sin2Cle(3sqrt(3))/(2)`