# Determining 4 intersection points of two polar graphs?

1. Aug 26, 2007

### Tekee

1. The problem statement, all variables and given/known data
r = 2
r^2 = 9sin(2theta)
Find the 4 points of intersection

2. Relevant equations

3. The attempt at a solution
Since r = 2, 4 = 9sin(2theta)...
4/9 = sin(2theta)

Taking the inverse of 4/9 only gives me one answer on the calculator (obviously), and I do not know where to attain the 3 other points. (also, is there a way to do this without a calculator/calculus?) - IE 4/9 = 2sin(theta)cos(theta), although I do not see how this helps.

2. Aug 27, 2007

### red_dog

$$\sin 2\theta=\frac{4}{9}\Rightarrow 2\theta =(-1)^k\arcsin\frac{4}{9}+k\pi\Rightarrow\theta =(-1)^k\frac{\arcsin\frac{4}{9}}{2}+\frac{k\pi}{2}$$
$$k=0\Rightarrow\theta=\frac{1}{2}\arcsin\frac{4}{9}$$
$$k=1\Rightarrow\theta=\frac{\pi}{2}-\frac{1}{2}\arcsin\frac{4}{9}$$
$$k=2\Rightarrow\theta=\pi+\frac{1}{2}\arcsin\frac{4}{9}$$
$$k=3\Rightarrow\theta=\frac{3\pi}{2}-\frac{1}{2}\arcsin\frac{4}{9}$$