Curve C is given in Polar Coordinates by the equation r=2+3sin(theta)

[xspook]

Homework Statement

Curve C is given in Polar Coordinates by the equation r=2+3sinθ.
Consider the usual Cartesian plane and take O as the pole and the positive x-axis as the polar axis.

Find points on the curve C where the tangent lines are horizontal or vertical and sketch the curve C.

Homework Equations

x^{2}+y^{2}=r^{2}
x=rcosθ
y=rsinθ
tanθ=\frac{y}{x}

The Attempt at a Solution

PART 1
For some reason I feel like the addition of 2 is throwing me off

r=2+3\frac{y}{r}
r^{2}=2+3y
x^{2}+y^{2}=2+3y
x^{2}+y^{2}-3y=2
x^{2}+y^{2}-3y+(\frac{-3}{2})^{2}=2+(\frac{-3}{2})^{2}
x^{2}+(y-\frac{3}{2})^{2}=\frac{17}{4}??

I don't know where to go from the last line above for the center, maybe (\frac{3}{2},0)??...

PART 2
I know also that I am supposed to take
\frac{∂r}{∂θ} which is 3cosθ

when I take
\frac{∂x}{∂θ} do I take the derivative of x=2+3(\frac{y}{r})(\frac{x}{r})?? And similarly for \frac{∂y}{∂θ}.

Lastly I know I have to take \frac{∂y}{∂x} which I hope I can easily do after I sort out the issue above.

Thank you

 

[Office_Shredder]

r = 2+3\frac{y}{r} should become r^2 = 2r+3y

You don't need to convert your curve to cartesian coordinates to sketch it though... you can plot them directly by finding the location of a bunch of points and drawing a curve through them

For \frac{\partial x}{\partial \theta} You should use x = r\cos(\theta) and do the product rule

 

[xspook]

What do I end up doing with the 2r now?

All of my examples from class always end up looking like
r^{2}=(some coefficient)(a variable)
we never have a term with r remaining

 

[xspook]

I guess I could divide by 2 and get r by itself

r=(\frac{x}{2})^{2}+(\frac{y}{2})^{2}-\frac{3y}{2}

but I don't know what I would do with that.