Rectangles and Poles: What's the Difference?

If we call the rectangular coordinates $(x, y)$ and the polar coordinates $(r, \theta)$ , we get that $x=r$ and $y=\theta$ . Then, let’s use some rectangular polar conversion formulas.

$x=r\cos(\theta)$ $x=x\cos(\theta)$ $\cos(\theta)=1$ $\theta=2\pi \times n$

Now, remember that an equation in the form $\theta=c$ is a line that goes through the origin, and an equation in the form $y=c$ is a line that has a slope of zero. Because both of these equations need to result in the same line in this case, the only line that works is $y=\theta=0$ , where the constant $c=0$ .

Wait, but what’s that you say? I made a mistake? Where? Aha! You have good eyes. In our work above, we divided by the variable $x$ . But what if $x=0$ ? Well, then, let’s use our other conversion formula.

$y=r\sin(\theta)$ $y=x\sin(\theta)$ $y=0 \times \sin(\theta)$ $y=0$

Again, we get to the same result, so we don’t have to change anything. We know that the equation of the line is $y=0$ , which can be written as $y=0x^{2}+0x+0$ . Thus, $a=b=c=0$ , and therefore, $a+b+c=\boxed{0}$ .