Can you check my work?
sketch the curve represented by the parametric equations. Then eliminate and write the corresponding rectangular equation whose graph represents the curve.
x=2+3cosθ
y=2sinθ
answer: cosθ=x/5
sinθ=y/2
cos^2θ + sin^2θ=1
(x/5)^2+(y/2)^2=1
(x/5)^2+(y/4)^2=1
No.
x = 2 + 3cosθ
NOT
x = (2+3)cosθ
To check your work, let's simplify the given parametric equations and see if we reach the same result for the corresponding rectangular equation.
Given parametric equations:
x = 2 + 3cosθ
y = 2sinθ
To eliminate the parameter θ, we can use trigonometric identities to express cosθ and sinθ in terms of x and y.
From the first equation, x = 2 + 3cosθ, we can isolate cosθ by subtracting 2 from both sides:
x - 2 = 3cosθ
Now, divide both sides by 3 to solve for cosθ:
cosθ = (x - 2) / 3
From the second equation, y = 2sinθ, we can isolate sinθ by dividing both sides by 2:
y / 2 = sinθ
Now, let's substitute these expressions for cosθ and sinθ into the equation cos^2θ + sin^2θ = 1:
((x - 2) / 3)^2 + (y / 2)^2 = 1
To simplify further, let's multiply both sides of the equation by 36 to eliminate the denominators:
36((x - 2) / 3)^2 + 36(y / 2)^2 = 36
Simplifying the equation:
4(x - 2)^2 + 9y^2 = 36
Expanding and rearranging terms:
4(x^2 - 4x + 4) + 9y^2 = 36
4x^2 - 16x + 16 + 9y^2 = 36
4x^2 - 16x + 9y^2 = 20
This is the corresponding rectangular equation whose graph represents the curve.
However, when comparing with your answer, the result is slightly different. It looks like there was a mistake in your calculations when simplifying the equation. The correct corresponding rectangular equation should be:
4(x/5)^2 + (y/2)^2 = 1
Therefore, the correct elimination and corresponding rectangular equation is:
cosθ = x/5
sinθ = y/2
(x/5)^2 + (y/2)^2 = 1
To double-check, you can substitute values for x and y into the parametric equations and the rectangular equation to see if they satisfy both sets of equations.