Greatest value of sin*sin*sin

[MM]

Find the greatest value of [tex]\sin \left(\alpha\right)\sin \left(\beta\right)\sin \left(\gamma\right)[/tex] where [tex]\alpha[/tex], [tex]\beta[/tex] and [tex]\gamma[/tex] are angles of triangle.

[adamsmith]

To find the greatest value of sin* sin*sin where [tex]\alpha, \beta, \gamma[/tex] are the angles of the triangle.

Let the sides of the triangle be 5,3,4
[tex]sin \alpha * sin \beta *sin \gamma[/tex]
= (1)* (4/5)*(3/5)
=(4/5)*(3/5)
=12/25
The greatest value is 12/25.

[Math Tutor]

You do not prove anything with that. It just a special case.

[Guest]

Spoiler: show

[tex]\sin(x)\sin(y) = (\cos(x-y)-\cos(x+y))/2[/tex], so if any pair of the angles are not equal, say [tex]\alpha\neq\beta[/tex] we can increase the product by replacing [tex]\alpha,\beta[/tex] with [tex](\alpha+\beta)/2,(\alpha+\beta)/2[/tex]. The maximum occurs when [tex]\alpha=\beta=\gamma=60^\circ[/tex].

R. Baber.

[Math Tutor]

The Baber's post is a real proof.