Question

Describe the motion of a particle with position (x, y) as t varies in the given interval. (For each answer, enter an ordered pair of the form x, y.) x = 1 + sin(t), y = 3 + 2 cos(t), π/2 ≤ t ≤ 2π

Answer 1

Answer:

The motion of the particle describes an ellipse.

Step-by-step explanation:

The characteristics of the motion of the particle is derived by eliminating $t$ in the parametric expressions. Since both expressions are based on trigonometric functions, we proceed to use the following trigonometric identity:

$\cos^{2} t + \sin^{2} t = 1$ (1)

Where:

$\cos t = \frac{y-3}{2}$ (2)

$\sin t = x - 1$ (3)

By (2) and (3) in (1):

$\left(\frac{y-3}{2} \right)^{2} + (x-1)^{2} = 1$

$\frac{(x-1)^{2}}{1}+\frac{(y-3)^{2}}{4} = 1$ (4)

The motion of the particle describes an ellipse.