Multiple choice

$AB$ is the diameter of a circle, centre $O$. $C$ is a point on the circumference such that $\angle COB=\theta $. The area of the minor segment cut off by $AC$ is equal to twice, the area of the sector $BOC$. Find whether the statement $\sin { \cfrac { \theta }{ 2 } } \cos { \cfrac { \theta }{ 2 } } =\pi \left( \cfrac { 1 }{ 2 } -\cfrac { \theta }{ 120 } \right) $ is True/False

  1. True

  2. False
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A Correct answer
Explanation

This is a geometric identity verification. The area of the minor segment is (1/2)R^2(theta - sin(theta)). The area of the sector is (1/2)R^2(theta). The condition leads to the stated trigonometric identity.

AI explanation

The area of sector BOC with angle theta is (1/2)r^2 theta. The minor segment cut off by AC has an area equal to the area of sector AOC minus the area of triangle AOC, which is (1/2)r^2(pi - theta) - (1/2)r^2 sin(pi - theta). Setting this equal to twice the area of sector BOC gives (1/2)r^2(pi - theta - sin theta) = r^2 theta. Dividing by r^2 and rearranging yields pi - sin theta = 3 theta. Using the double angle identity, sin theta is 2 sin(theta/2) cos(theta/2), resulting in 2 sin(theta/2) cos(theta/2) = pi - 3 theta. Dividing by 2 confirms that sin(theta/2) cos(theta/2) = pi(1/2 - theta/120), making the statement True.

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