1. ## Confusing polar coordinates

Find Cartesian coordinates for all points of intersection of the curves given below:
r=1-sin(theta) and r^2=4sin(theta)
Sketch these.

I am not sure how to do this. Do you square the r=1-sin(theta) and set it equal to 4 sin (theta) and solve for theta? That seems complicated. What is the best approach?

Find Cartesian coordinates for all points of intersection of the curves given below:
r=1-sin(theta) and r^2=4sin(theta)
Sketch these.

I am not sure how to do this. Do you square the r=1-sin(theta) and set it equal to 4 sin (theta) and solve for theta? That seems complicated. What is the best approach?

$\displaystyle (1-sin\theta)^2=r^2=4sin\theta$

$\displaystyle 1-6sin\theta+sin^{2}\theta=0$

$\displaystyle (sin\theta-3)^{2}-9+1=0$

$\displaystyle (sin\theta-3)^{2}=8$

$\displaystyle sin\theta-3=\pm{\sqrt{8}}$

$\displaystyle sin\theta=3\pm{2\sqrt{2}}$

Since, $\displaystyle sin\theta\leq{1}\Rightarrow{sin\theta=3-2\sqrt{2}}$

$\displaystyle sin\theta=3-2\sqrt{2}$

This value satisfies both the given equations.

Therefore, $\displaystyle \theta=n\pi+(-1)^{n}sin^{-1}(3-2\sqrt{2})$ ; where $\displaystyle n\in{Z}$

Hope this helps.

3. thank you