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

In summary: Other than that, your summary is accurate and includes all the necessary steps to find the Cartesian equation, graph it, and identify the particle's path and direction of motion. Well done!
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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 .
image.png

Could someone check my answer ?
 

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  • #2
Fatima Hasan said:

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##
 
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1. What is the Cartesian equation for this graph?

The Cartesian equation for this graph is x = 2sin(t), y = 4cos(t).

2. What does the variable "t" represent in this equation?

The variable "t" represents the angle in radians.

3. What is the range of values for "t" in this equation?

The range of values for "t" is from 0 to 2π, or 0 to 360 degrees.

4. How do the values of "x" and "y" change as "t" increases?

As "t" increases, the values of "x" and "y" oscillate between -2 and 2 for "x" and between -4 and 4 for "y".

5. Is this graph a function?

No, this graph is not a function because for every value of "t", there are two values for both "x" and "y".

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