Converting from Polar Coordinates to Rectangular - Problem 3

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.