Graph the Cartesian equation: x = 2 sin t, y = 4 cos t

[Fatima Hasan]

Homework Statement

Parametric equations and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion.

x = 2 sin t, y = 4 cos t, 0 ≤ t ≤ 2π

Homework Equations

$cos^2t + sin^2t=1$

The Attempt at a Solution

$x= 2 sin t$
$y = 4 cos t$
Square both sides :
$\frac{x^2}{4} = sin t^2$
$\frac{y^2}{16} = cos^2 t$
Add the two equations :
$\frac{y^2}{16} + \frac{x^2}{4} = 1$
This equation forms an ellipse.
When t = 0 , x = 0 and y = 4
When t = $2\pi$ , x = 0 and y = 4
Counterclockwise from (0,4) to (0,4) , one rotation .

Could someone check my answer ?

 

[Mark44]

Homework Statement

Parametric equations and a parameter interval for the motion of a particle in the xy-plane are given. Identify the particle's path by finding a Cartesian equation for it. Graph the Cartesian equation. Indicate the portion of the graph traced by the particle and the direction of motion.

x = 2 sin t, y = 4 cos t, 0 ≤ t ≤ 2π

Homework Equations

$cos^2t + sin^2t=1$

The Attempt at a Solution

$x= 2 sin t$
$y = 4 cos t$
Square both sides :
$\frac{x^2}{4} = sin t^2$
$\frac{y^2}{16} = cos^2 t$
Add the two equations :
$\frac{y^2}{16} + \frac{x^2}{4} = 1$
This equation forms an ellipse.
When t = 0 , x = 0 and y = 4
When t = $2\pi$ , x = 0 and y = 4
Counterclockwise from (0,4) to (0,4) , one rotation .
View attachment 231780
Could someone check my answer ?

Looks good, except for a small typo. After you square both sides, it should be $\frac{x^2}{4} = sin^2 t$, not $sin t^2$