identify the parametric equations that represent the same path as the following parametric equations. show your work

x=3+ cos theta
y= 2 sin theta

A. x=3+cos 2 theta
y=2sin 2theta

B.x=cos theta
y=3+2sin theta

C. x=3+2 cos theta
y= 4 sin theta

D. x=2 sin theta
y= 3+cos theta

I know it’s A but i’m not sure how to do the work :(

A parametric equation is an equation where the value of x variable given in terms of theta is also expressed in y variable in terms of the same parameter theta.

Given that:

x=3+ cos theta
y= 2 sin theta

where;

theta = parameter

The pair of (x and y) are called parametric equations.

At any given theta (θ), we can substitute it into the equation to determine the value of x and y.

Now, the cartesian form of the parametric equation will help us to determine if the parametric equation are represented in the same path.

The cartesian form is the equation with just x and y whereby the parameter is eliminated.

So, from the given equation;

x = 3 + cos θ

y = 2 sin θ

cos θ = x - 3

sin θ = y/2

We know from trigonometry identity that:

sin²θ + cos²θ = 1

we can eliminate the parameter by saying:

(y/2)² + (x-3)² = 1

From the given options, Option A represents the same path as the given parametric equation.

This is because:

x = 3 + cos 2 θ

y = 2sin 2 θ

cos 2θ = 3 - x

sin2θ = y/2

Eliminating the parameter, we have:

(y/2)² + (x-3)² = 1

Learn more about parametric equations here:

https://brainly.com/question/51019

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