Polar Coordinates and Complex Numbers
PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
We're talking about lines and polar coordinates. Remember, this is the equation for our line and polar coordinates. R equals d over cosine theta minus beta where d, beta, are the polar coordinates of the point on the line closest to the origin. So let's do some examples.
Here, I've got a vertical line that passes through 3,pi. According to the the formula, the equation for this would be r equals, and d, this is the closest point to the origin. The distance, d would be 3, and the cosine of theta minus beta. Beta is the angle of this point, so it would be pi and that's your equation.
Of course another way to get the equation for line, is to make the observation that its equation in rectangular is x equals -3. So you can use the conversion; r cosine theta equals -3. R equals -3 secant theta, so that's another equation for the same line.
Here, we can use the equation r equals d over cosine theta minus beta. The close point here d, beta, is this point. This is the closest point to the origin. D is 8, beta is pi over 2. So we get r equals 8 over the cosine of theta minus pi over 2. So that's one equation. But if you recognize that this is y equals 8, then you can use the conversion y is r sine theta. So r sine theta equals 8, r equals 8 cosecant theta. These are equivalent equation.