Describe the motion of a particle with position (x, y) as t varies in the given interval. x = 2 sin(t), y = 3 + cos(t), 0 ≤ t ≤ 3π/2

Question

Describe the motion of a particle with position (x, y) as t varies in the given interval.

x = 2 sin(t),    y = 3 + cos(t),    0 ≤ t ≤ 3π/2

Answer 1

Note the 3 and 2 in the equations for x and y and that sin and cos are reversed.

Also, note that we go only from 0 to 3 π/2, and thta the angular velocity ω is 1.

This is an ellipse centered at (0, 3); x is the major axis of length 2*2 = 4, and y is the minor axis of length 1 * 2 = 2.

Because sin and cos are reversed, we start at due North for the ellipse and go clockwise 270° until we reach due West. Due to the translation and scaling, we start at (0, 3) + (0, 1) = (0, 4) , passing through (0, 3) + (2, 0) = (2, 3) and (0, 3) + (0, -1) = (0, 2), until we reach (0, 3) + (-2, 0) = (-2, 3)

Answer 2

Briana:

The trick is to use identity sin^2(t)+cos^2(t) = 1

We know x=2*sin(t) so x/2 = sin(t)

We also know y = 3 + cos(t) so y-3 = cos(t)

Then (x/2)^2+(y-3)^2 = sin^2(t)+cos^2(t) =1

We obtain x²/4 + (y-3)²/1 = x²/2² +(y-3)²/1² = 1

This is the equation of an ellipse. Try to figure what the bounds should be on 0<=t<=3pi/2