I'm extremely confused on what to do in part a of this question.
Consider the curve given by x=3sin(theta), y=1+2cos(theta), 0<=theta<=3pi/2
(a) Eliminate the parameter and find a Cartesian (Rectangular) equation for the curve.
Your basic replacement identities are:
r^2 = x^2 + y^2
x = rcosØ and y = rsinØ or cosØ = x/r and sinØ = y/r
so you have:
x =3sinØ ---> sinØ = x/3
y = 1+2cosØ
cosØ = (y-1)/2
but we know sin^2 Ø + cos^2 Ø = 1
x^2/9 + (y-1)^2 /4 = 1
looks like an ellipse to me
To eliminate the parameter and find a Cartesian equation for the curve, we need to express one variable in terms of the other.
In this case, we have x = 3sin(theta) and y = 1 + 2cos(theta).
To eliminate the parameter, theta, we can use trigonometric identities. Specifically, we can use the Pythagorean identity: sin^2(theta) + cos^2(theta) = 1.
Let's solve for sin(theta) in terms of cos(theta), which will allow us to eliminate theta from the equations.
Start by squaring both sides of the equation x = 3sin(theta):
x^2 = (3sin(theta))^2
x^2 = 9sin^2(theta)
Next, square both sides of the equation y = 1 + 2cos(theta):
y^2 = (1 + 2cos(theta))^2
y^2 = 1 + 4cos^2(theta) + 4cos(theta)
Now, using the Pythagorean identity, substitute 1 - cos^2(theta) for sin^2(theta) in the equation for x^2:
x^2 = 9(1 - cos^2(theta))
x^2 = 9 - 9cos^2(theta)
Now, substitute y^2 - 1 - 4cos(theta) for y^2 in the equation for x^2:
x^2 = 9 - 9cos^2(theta)
x^2 = 9 - 9cos^2(theta) + 4cos(theta) + 1 - 4cos(theta)
x^2 = 10 - 9cos^2(theta)
Finally, simplify the equation to get the Cartesian equation:
x^2 + 9cos^2(theta) = 10
So, the Cartesian equation for the curve is x^2 + 9cos^2(theta) = 10.