Trigonometry - Trigonometric Graphs and Parameters

The figure below shows a pair of interlocked gears, where Gear I has a radius of \(12\) cm and Gear II has a radius of \(8\) cm. At the moment \(t = 0\) (in seconds), the pair of gears starts rotating, where Gear I rotates clockwise and Gear II rotates anti-clockwise. Let \(A\), \(B\) be the points on the edges of Gear I and Gear II where the two gears meet at \(t = 0\).

1. Given that Gear I completes one full rotation every \(6\) seconds, find the time for Gear II to complete one full rotation.
2. Let \(y_1\) cm (respectively, \(y_2\) cm) be the vertical distance of point \(A\) (respectively, point \(B\)) from the centre of Gear I (respectively, Gear II). Based on your answer in (i), Express \(y_1\), \(y_2\) as a function of the time \(t\), respectively.
3. On the same diagram, sketch the graphs of \(y_1\) and \(y_2\) for \(0 \le t \le 12\).
4. Find the distance between point \(A\) and point \(B\) at \(t = 6\).

5. Let \(T\) be the moment when the two gears meet each other at points \(A\) and \(B\) for the \(n^{\mathrm{th}}\) time. Express \(T\) in terms of \(n\), stating your workings clearly.