To solve part (b), let's examine the requirements based on part (a).

Problem Breakdown

1. The equation of the circle given is: $x^2 + (y - 1)^2 = 16$ This represents a circle centered at $(0, 1)$ with a radius of $4$.

2. In part (a), the parametric equations provided are: $x = 4 \cos(t), \quad y = 1 - 4 \sin(t)$ for $0 \leq t \leq 2\pi$, which describes a clockwise movement around the circle starting at $(4, 1)$.

Part (b): Three Times Counterclockwise

For three counterclockwise rotations:

• We need the parameter $t$ to vary from $0$ to $6\pi$, as each full rotation is $2\pi$, so three rotations would be $3 \times 2\pi = 6\pi$.
• To change the direction from clockwise to counterclockwise, we should modify the sign in the sine term.

Thus, the parametric equations for part (b) are: $x = 4 \cos(t), \quad y = 1 + 4 \sin(t), \quad 0 \leq t \leq 6\pi.$

These equations describe three counterclockwise rotations around the circle starting at $(4, 1)$.

Would you like a more detailed breakdown of the steps?

Here are some related questions:

1. How do we determine the starting point of parametric equations on a circle?
2. What changes if we need a counterclockwise rotation versus a clockwise one?
3. Why do we use the range $0 \leq t \leq 6\pi$ for three rotations?
4. How can we determine the radius and center of a circle from its equation?
5. What modifications would be necessary if the circle's radius or center changed?

Tip: Remember that changing the sign of the sine or cosine term in parametric equations can alter the direction of traversal around the circle.