Chapter 10.6, Problem 73PE

If a circle of radius $r$ rolls around the exterior of a circle of radius $R\ge r,$ a fixed point on the outer circle traces out a curve called an epicycloid. For Exercises 73-74, use the following parametric equations for an epicycloid.

$x=\left(R+r\right)\cos t-r\cos \left[\left(\frac{R+r}{r}\right)t\right]$ and $y=\left(R+r\right)\sin t-r\sin \left[\left(\frac{R+r}{r}\right)t\right]$

a. Write parametric equations for a epicycloid with $R=1$ and $r=1$ . The curve defined by these equations is called a cardioid.

b. Graph the circle given by $x=\cos t$ and $y=\sin t$ and the cardioid from part (a). Use $0\le t\le 2\pi$ and a viewing window of $\left[-6.4,6.4,1\right]$ by $\left[-4,4,1\right]$ .