<p>The figure below shows a pair of coaxial gears with a common centre \(O\). Gear I has a radius of \(16\) cm and Gear II has a radius of \(4\) cm. At the moment \(t = 0\) (in seconds), the pair of gears starts rotating anti-clockwise. Let&nbsp;\(A\), \(B\) be the points on the edges of Gear I and Gear II such that&nbsp;\(O\), \(A\) and \(B\) are collinear and form a horizontal line at \(t = 0\).</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1770034682377-uq4yfc5tlci.png" style="max-width: min(220px, 100%); height: auto; display: block;" alt="A pair of coaxial gears with centre O and points A and B on Gear I and Gear II respectively." class="aligncenter custom-width" /></figure></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Given that Gear I complete one full rotation every \(6\) seconds, find the time for Gear II to complete one full rotation.</span></div></li><li><div class="list-item-paragraph">Let&nbsp;\(y_1\) cm (respectively, \(y_2\) cm) be the horizontal distance of point \(A\) (respectively, point \(B\)) from centre \(O\). Based on your answer in <strong>(i)</strong>, Express \(y_1\), \(y_2\) as a function of the time \(t\), respectively.</div></li><li><div class="list-item-content"><span class="list-item-text">On the same diagram, sketch the graphs of \(y_1\) and \(y_2\) for&nbsp;\(0 \leq t \leq 12\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the distance between point \(A\) and point \(B\) at \(t = 6\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Let&nbsp;\(T\) be the moment when the points \(O\), \(A\) and \(B\) form a vertical line for the \(n^{\text{th}}\) time. Express \(T\) in terms of \(n\), stating your workings clearly.</span></div></li></ol>

The figure below shows a pair of coaxial gears with a common centre \(O\). Gear I has a radius of \(16\) cm and Gear II has a radius of \(4\) cm. At the moment \(t = 0\) (in seconds), the pair of gears starts rotating anti-clockwise. Let \(A\), \(B\) be the points on the edges of Gear I and Gear II such that \(O\), \(A\) and \(B\) are collinear and form a horizontal line at \(t = 0\).

Given that Gear I complete one full rotation every \(6\) seconds, find the time for Gear II to complete one full rotation.

Let \(y_1\) cm (respectively, \(y_2\) cm) be the horizontal distance of point \(A\) (respectively, point \(B\)) from centre \(O\). Based on your answer in (i), Express \(y_1\), \(y_2\) as a function of the time \(t\), respectively.

Let \(y_1\) cm (respectively, \(y_2\) cm) be the horizontal distance of point \(A\) (respectively, point \(B\)) from centre \(O\). Based on your answer in **(i)**, Express \(y_1\), \(y_2\) as a function of the time \(t\), respectively.

On the same diagram, sketch the graphs of \(y_1\) and \(y_2\) for \(0 \leq t \leq 12\).

Find the distance between point \(A\) and point \(B\) at \(t = 6\).

Let \(T\) be the moment when the points \(O\), \(A\) and \(B\) form a vertical line for the \(n^{\text{th}}\) time. Express \(T\) in terms of \(n\), stating your workings clearly.

Trigonometric Functions and Graphs

Trigonometric Functions and Graphs