Explanation:
x² + (y − 1)² = 9
This is a circle with center (0, 1) and radius 3. We can parameterize it using sine and cosine.
Use the starting point to determine which should be sine and which should be cosine.
Use the direction to determine the signs.
Use the number of revolutions and the interval to determine coefficient of t.
(A) Once around clockwise, starting at (3, 1). 0 ≤ t ≤ 2π.
The particle starts at (3, 1), which is 0 radians on a unit circle. It makes 1 revolution (2π radians). Therefore:
x = 3 cos t
y = 1 − 3 sin t
(B) Two times around counterclockwise, starting at (3, 1). 0 ≤ t ≤ 4π.
The particle starts at (3, 1), which is 0 radians on a unit circle. It makes 2 revolutions (4π radians). Therefore:
x = 3 cos t
y = 1 + 3 sin t
(C) Halfway around counterclockwise, starting at (0, 4). 0 ≤ t ≤ π.
The particle starts at (0, 4), which is π/2 radians on a unit circle. It makes 1/2 revolution (π radians). Therefore:
x = -3 sin t
y = 1 + 3 cos t