a circle of radius 1 rolls around the outside of a circle of radius 2 without slipping. the curve traced by a point on the circumfarence of the smaller circle is callled an epicycloid. use the angle theta to find a set of parametric equations for this curve.

To find the parametric equations for the epicycloid, we'll use the concept of parametric equations in polar coordinates.

Let's start by considering the bigger circle of radius 2. The equation for the polar coordinates of a point on its circumference can be expressed as:

x = 2 * cos(theta)
y = 2 * sin(theta)

Now, let's focus on the smaller circle of radius 1 that is rolling around the outside of the bigger circle. The path traced by a point on its circumference can be represented by an epicycloid.

To find the coordinates of this point on the epicycloid, we need to consider two things: the rotation of the smaller circle itself and the rolling motion around the bigger circle.

First, the rotation of the smaller circle is defined by the angle theta. As it rolls, the distance traveled on the smaller circle can be expressed as the product of the radius of the smaller circle (1) and the angle theta.

Next, to account for the rolling motion around the bigger circle, we need to find the point of contact between the two circles. This occurs when the angle between the vertical line passing through the center of the smaller circle and the radius connecting the center of the smaller circle to the point of contact is equal to theta.

Using this information, we can find the coordinates of the point on the epicycloid:

x = (2 * cos(theta)) + (1 * cos(theta))
y = (2 * sin(theta)) + (1 * sin(theta))

Therefore, the set of parametric equations for the epicycloid are:

x = 3 * cos(theta)
y = 3 * sin(theta)

These equations describe the curve traced by a point on the circumference of the smaller circle as it rolls around the outside of the bigger circle without slipping.