The particle starts at (3,5) and travels counter clockwise along a semi-circle with a radius of 2 and centered at (3,3) through the point (1,3) and stops at point (3,1).
First, let's look at the two equations:
x = 3 + 2 cos(t)
y = 3 + 2 sin(t)
If you ignore the limits specified for t, you'll realize that they describe a circle with a radius of 2, centered at (3, 3). Now that we know that the functions describe a circle, let's look at the limits specified for t and see what they imply.
Since the value is from π/2 to 3π/2, it will span a distance of π radians, so we know the particle will trace half of a full circle, or a semi-circle. Since the starting point is π/2, the starting coordinate of the particle will be:
(3 + 2 cos(Ď€/2), 3 + 2 sin(Ď€/2))
= (3 + 2*0, 3 + 2*1)
= (3+0, 3+2)
= (3,5)
And the ending point of the particle will obviously be (3,1). But to demonstrate:
(3 + 2 cos(3Ď€/2), 3 + 2 sin(3Ď€/2))
= (3 + 2*0, 3 + 2*(-1))
= (3+0, 3+ (-2))
= (3,1)
Now let's see where the particle is in the middle.
(3 + 2 cos(Ď€), 3 + 2 sin(Ď€))
= (3 + 2*(-1), 3 + 2*0)
= (3 + (-2), 3 + 0)
= (1, 3)
So the final description is:
The particle starts at (3,5) and travels counter clockwise along a semi-circle with a radius of 2 and centered at (3,3) through the point (1,3) and stops at point (3,1).
