Hint for Problem 7: April 2024

In part (ii), solve for \(t\) in terms of \(x\) and then substitute into the equation involving \(y\).

At time \(t\), how far is the particle from the origin?

Starting with \(y^2=(\sin 2t)^2\), use trigonometric identities to eliminate all appearances of the variable \(t\). Remember that \(x=\cos t\).

As Circle 2 rolls around Circle 1, let \(t\) be the angle made by the positive \(x\)-axis and the line segment connecting the origin and the centre of Circle 2. It will help to draw a reasonably accurate diagram with Circle 2 rolled part of the way around Circle 1 (perhaps an angle of \(\dfrac{\pi}{3}\) or so). Once you have done this, mark the point on the circumference of Circle 2 that was originally at \((1,0)\) by \(P\) (or some other label). The objective is to find the coordinates of \(P\) in terms of \(t\). Since the circles roll without slipping, the arc length from the point of tangency along Circle 1 to \((1,0)\) should equal the arc length from the point of tangency along Circle 2 to \(P\).

As Circle 2 rolls along the inside of Circle 1, it (usually) intersects the \(x\)-axis at two points. Convince yourself that one of these two points must be the origin, then think about the other point.

Using a strategy similar to part (d), find a general formula for the coordinates of \(P\) in terms of the angle \(t\). Do this either in general for \(r\) or do it separately for the three relevant values of \(r\) in this question.

Find identities for \(\cos3t\) in terms of \(\cos t\) and \(\sin 3t\) in terms of \(\sin t\).

Find the parametric equations for the position of \(P\) when \(r=\dfrac{1}{3}\) if Circle 2 is rolled clockwise around Circle 1 instead of counterclockwise.