# Greatest value of sin*sin*sin

Trigonometry equalities, inequalities and expressions - sin, cos, tan, cot

### Greatest value of sin*sin*sin

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

MM

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### Re: Greatest value of sin*sin*sin

To find the greatest value of sin* sin*sin where $$\alpha, \beta, \gamma$$ are the angles of the triangle.

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

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### Re: Greatest value of sin*sin*sin

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

Math Tutor

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### Re: Greatest value of sin*sin*sin

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

R. Baber.
Guest

### Re: Greatest value of sin*sin*sin

The Baber's post is a real proof.

Math Tutor