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Converting from Polar Coordinates to Rectangular - Problem 3

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

I’m converting polar equations to rectangular. I have the equation r equals 2 over cosine theta minus pi over 6. I want to convert this to rectangular and then identify the resulting shape.

First thing I want to do is multiply the cosine out of the denominator. So let me do that. I get r cosine theta minus pi over 6, equals 2. Now I really would like an r cosine theta or something like that on this left side, but I’ve got theta minus pi over 6. How can I deal with that? Well there is a way. We can use the cosine of the different formula to break that cosine of theta minus pi over 6 apart. So let’s use that.

I get r times, then I’ll use some brackets here. Cosine theta, cosine pi over 6 plus, remember the cosine of a difference has a plus in it, sine theta, sine pi over 6. All that equals 2. So I have an r cosine theta when I distribute. Times and the cosine of pi over 6 root 3 over 2 plus.

R times sine theta, times the sine of pi over 6 which is 1/2, and all that equals 2. Let me multiply through by 2. I get root 3 r cosine theta plus r sine theta equals 4. And now I’m going to convert. R cosine theta is x, root 3 times x plus r sine theta is y, equals 4.

So let me just move this term over the left side, y equals negative root 3x plus 4. This is a line; the slope negative root 3 and y intercept 0,4. So this equation, the strange looking equation is actually the equation of a line.

Don’t be afraid to use the cosine of a difference formula any identity that you have from the past, when you are converting from polar to rectangular.

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