<p>An isosceles triangle \(ABC\), with \(AB = AC\), is inscribed in a fixed circle of radius 1 unit and centre \(O\). It is given that angle \(BOC = 2\theta\), where&nbsp;\(\theta\) is acute. Using calculus, show that the area of the triangle \(ABC\) is a maximum when it is equilateral. State the maximum value of this area exactly.<span class="inline-marks">[8]</span></p>

An isosceles triangle \(ABC\), with \(AB = AC\), is inscribed in a fixed circle of radius 1 unit and centre \(O\). It is given that angle \(BOC = 2\theta\), where \(\theta\) is acute. Using calculus, show that the area of the triangle \(ABC\) is a maximum when it is equilateral. State the maximum value of this area exactly.