Dear Uncle Colin,

I have the polar equation $\theta = k$, and I want to turn it into rectangular coordinates. I worked it out to be the line $y = x \tan(k)$, but the solution says it’s only a half-line. Why is that?

Perplexed Over Line And Radius

Hi, POLAR, and thanks for your message!

This one has a fairly simple answer, once you think about the argument of points on your line.

Suppose you had $k = \piby 4$, for the sake of argument ((No pun intended. Honest.)). Then the answer you’ve given would work out to be $y=x$. A point like $(1,1)$ does indeed satisfy your equation and the polar equation.

However, a point like $(-1,-1)$, while it lies on your line, has an argument of $\frac{5}{4}\pi$, so it doesn’t satisfy the polar equation.

The line has to be restricted to a half-line so that only points with the correct argument are included.

Hope that helps!

- Uncle Colin