# Problem of the MonthHint 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 $$\dfrac{\pi}{3}$$ 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 $$P$$.

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 $$\cos3t$$ in terms of $$\cos t$$ and $$\sin 3t$$ in terms of $$\sin t$$.

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