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.