Points of intersection

Find the points of intersection (r,theta) of the curves. Points intersect at the pole and one other point.

r=2(1-cos(theta))
r=4sin(theta)

How do you find theta? I know the one of the points is zero but i'm looking for the other point of intersection. I also know what r and theta is if i graph these equations but how do you work it out without a calculator? Details would be appreciated.

One way is to solve for r, rather than $\theta$ .

It's not necessarily obvious. You must think about it. Sometimes you must try new things.

From the cosine equation: $\theta\;=\;acos\left(\frac{r-2}{2}\right)$

Substitute into the sine equation - and simplify: $r\;=\;2\sqrt{4r-r^{2}}$

You should be able to get that far. Only a little algebra remains.

Quote:

Originally Posted by TKHunny

One way is to solve for r, rather than $\theta$ .

It's not necessarily obvious. You must think about it. Sometimes you must try new things.

From the cosine equation: $\theta\;=\;acos\left(\frac{r-2}{2}\right)$

Substitute into the sine equation - and simplify: $r\;=\;2\sqrt{4r-r^{2}}$

You should be able to get that far. Only a little algebra remains.

general Warning: When finding the intersection points of two curves defined in polar coordinates, graphs should ALWAYS be drawn because the solution to the simultaneous equations does not always give ALL intersection points. This is because there is NOT a one-to-one correspondence between points and coordinates in the polar coordinate system.

For example, the intersection point at the pole of $r = \sqrt{3} \cos \theta$ and $r = \sin \theta$ does NOT show as a solution to the simultaneous equations. This is because the pole is not on the two curves ‘simultaneously’ in the sense of being reached at the same value of $\theta$ : on the curve $r = \sin \theta$ the pole is reached when $\theta = 0$ , whereas on the curve $r = \sqrt{3} \cos \theta$ the pole is reached when $\theta = \frac{\pi}{2}$ . (This can be compared with parametric equations - although the paths followed by two particles may intersect, the particles themselves do not necessarily meet).

Example: The curves $r^2 = 4 a^2 \cos \theta$ and $r = a(1 - \cos \theta )$ intersect at FOUR points:

1. The two points with polar coordinates $[r_1, \theta_1]$ where $r_1 = 2a(\sqrt{2} - 1)$ and $\cos \theta_1 = 3 - 2\sqrt{2}$ , are found from the simultaneous solution.

2. The two points with Cartesian coordinates (0, 0) and (-2a, 0) are disclosed only by graphing the curves.

The Cartesian point (0, 0) has polar coordinates $[0, \frac{\pi}{2}]$ on the curve $r^2 = 4 a^2 \cos \theta$ and polar coordinates [0, 0] on the curve $r = a(1 - \cos \theta )$ . Clearly it is not a simultaneous solution in polar coordinates.

Similarly, the Cartesian point (-2a, 0) has polar coordinates $[-2a, 0]$ on the curve $r^2 = 4 a^2 \cos \theta$ and polar coordinates $[0, \pi]$ on the curve $r = a(1 - \cos \theta )$ .

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It is the intersectin of a cardioid and a circle.
Use the fact that 1-cos(t)=2sin(t/2)^2 and sin(t)=2sin(t/2)cos(t/2).