91Ó°ÊÓ

A chord of a circle subtends an angle of \(\theta\) radians at the centre of the circle. If the area of the minor segment cut off by the chord is one sixth of the area of the circle prove that \(\sin \theta=\theta-\frac{\pi}{3}\).

If \(\theta=\frac{13 \pi}{6}, \cos \theta\) is: (a) \(\frac{1}{2}\) (b) \(-\frac{1}{2}\) (c) \(\frac{\sqrt{3}}{2}\) (d) \(-\frac{\sqrt{3}}{2}\) (e) \(\frac{\sqrt{2}}{2}\).

The associated acute angle for \(280^{\circ}\) is: (a) \(100^{\circ}\) (b) \(10^{\circ}\) (c) \(80^{\circ}\) (d) \(-80^{\circ}\) (e) \(190^{\circ}\).

\(\mathrm{P}\) and \(\mathrm{Q}\) are points on a circle of radius \(r\), and the chord PQ subtends an angle \(2 \theta\) radians at its centre 0 . If \(A\) is the area enclosed by the minor arc \(\mathrm{PQ}\) and the chord \(\mathrm{PQ}\), and if \(B\) is the area enclosed by the arc \(\mathrm{PQ}\) and the tangents to the circle at \(\mathrm{P}\) and \(\mathrm{Q}\), prove that $$ A-B \equiv r^{2}(2 \theta-\tan \theta-\sin \theta \cos \theta) $$

\(\sin \theta=0 \quad\) when \(\theta=n \pi\)