Problem of the Month
Hint for Problem 7: April 2024

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

2. At time $t$, how far is the particle from the origin?

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

4. 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 ${\displaystyle \frac{\pi }{3}}$ 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 $P$.

5. 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.

6. 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.

   1. Find identities for $\cos 3t$ in terms of $\cos t$ and $\sin 3t$ in terms of $\sin t$.

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