An isosceles triangle of vertical angle `2theta` is inscribed in a circle of radius `a` . Show that the area of the triangle is maximum when `theta=pi/6` .

Explanation

To solve the problem of maximizing the area of an isosceles triangle inscribed in a circle of radius \( a \) with a vertical angle \( 2\theta \), we will follow these steps:

### Step 1: Understand the Geometry
We have a circle with radius \( a \) and an isosceles triangle \( ABC \) inscribed in it. The angle at vertex \( A \) is \( 2\theta \), and the angles at vertices \( B \) and \( C \) are each \( \theta \).

### Step 2: Determine the Area of the Triangle
The area \( A \) of triangle \( ABC \) can be expressed in terms of the base \( BC \) and height \( AD \) (where \( D \) is the foot of the altitude from \( A \) to \( BC \)):
\[
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