a 12.6 problem

Hi Professor,

I am currently working on webwork right now and I am having trouble with one of the problems. The question is on converting from rectangular to cylindrical coordinates and I have plugged in all the numbers correctly so far but it keeps telling me that my theta is incorrect. I attached a screenshot of my answers with the problem and am going to move on from it for now but I was just wondering if you would be able to point out what I am doing wrong.

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Yeah, you'll have the problem with polar coordinates too. The situation is that the graph of the tangent function looks like this:

You should recall that a function is invertible only if it satisfies the horizontal line test, which this tan(x) doesn't, for example x satisfies the equation tan(x) = 1 if x = π/4, or 5π/4 or -3π/4 etc. We deal with this situation by restricting the domain of tan(x) to the range (-π/2, π/2), and then all of the values of tan^{-1}(x) are in this range too. This corresponds to (x,y) in the first and fourth quadrants. BUT <-4,1> is in the second quadrant, and this vector is exactly opposite of the vector <4,-1> that corresponds to the angle tan^{-1}(-1/4).  To get the right angle just add π to the the answer you are getting.
PS, I get the same question every semester, but it's a lot easier to explain once people have gotten frustrated trying to do it.

PPS. btw, the moral of this whole story is that you have to think about what things mean.

PPPS, if you click the "email instructor" button at the bottom of the page it will send the problem to me and you don't have to copy/paste.