Determining 4 intersection points of two polar graphs?

[Tekee]

Homework Statement

r = 2
r^2 = 9sin(2theta)
Find the 4 points of intersection

Homework Equations

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.

 

[red_dog]

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

 

[Architeuthis Dux]

I would approach this problem by first understanding the polar coordinate system and its relationship to the Cartesian coordinate system. I would also review the properties of polar graphs and how they are affected by changes in the equations.

To find the intersection points of two polar graphs, I would start by setting the two equations equal to each other and solving for theta. This will give me the values of theta at which the two graphs intersect. Then, I would substitute these values of theta back into one of the original equations to find the corresponding values of r. This will give me the coordinates of the intersection points.

In this case, the two equations are r = 2 and r^2 = 9sin(2theta). Setting them equal to each other, we get:

2 = 9sin(2theta)

Dividing both sides by 9, we get:

2/9 = sin(2theta)

Taking the inverse sine of both sides, we get:

sin^-1(2/9) = 2theta

Dividing both sides by 2, we get:

sin^-1(2/9)/2 = theta

Using a calculator, we can find the value of theta to be approximately 0.482. Substituting this back into the original equation r = 2, we get r = 2. Therefore, the first intersection point is (2, 0.482).

To find the other three intersection points, we can use the symmetry properties of polar graphs. Since the equations are symmetric about the x-axis, we can add or subtract π to the value of theta to get the other three points. Therefore, the other three points are (2, -0.482), (-2, 2.66), and (-2, -2.66).

In summary, the four intersection points of the two polar graphs are (2, 0.482), (2, -0.482), (-2, 2.66), and (-2, -2.66). These can also be verified by graphing the two equations on a polar coordinate system.