Why is the value of r in $r=\cos θ$ never negative? The graph of $r=\cos θ$ forms a circle with radius $.5$ and around $(1,0)$. However, in terms of Cartesian plane coordinates, why doesn't the $r$ value go into the second and third quadrants? Wouldn't $r$ be negative when $θ$ is in the second and third quadrants?
You are correct $r$ is negative when $\frac{\pi}{2} \leq \theta &lt; \frac{3\pi}{2}$, but since $r$ is negative, it points in the opposite direction of the $\theta$. Thus, the point $(r,\theta)=(-1,\pi)$ does not map to $(x,y)=(-1,0)$, but to $(x,y) = (1,0)$
I’d like you to first think of the equation $y=x^2$ in the Cartesian coordinate $(x,y)$-plane. That condition is a restriction over the plane that famously identifies those points whose $y$ value (height fron $x$ axis) is exactly the square of their $x$ value.

Now move back to polar coordinate $(r, \theta)$-plane. The equation $r = \cos\theta$ identifies those points whose $r$-value (distance from the origin) is exactly $\cos$ of their $\theta$ value (angle from $x$ axis). But $r$ is always greater than zero 
(and $\theta$ is always between $\pm\pi$), so the condition is not meaningless unless $\theta$ makes the cosine positive, that is $\theta$ is between $\pm\pi/2$.

You’d just gotten it the other way around.



ADDENDUM. (Technical) The reason why I require $r&gt;0$ and $-\pi&lt; \theta &lt; \pi$ is because I want polar coordinates to be consistent with the usual definitions of curvilinear coordinates in analysis, or coordinate charts in differential geometry. I want my coordinates to be a homeomorphism between an open subset of the plane (in this case the plane without the negative $x$ axis and the origin) and another open subset of the plane ($(0,+\infty)\times(-\pi,+\pi)$), and in particular a bijection. Otherwise it would be meaningless to even say that the polar coordinates of a point on the plane are such and such, because multiple answers would do. Notice that my approach provides a sensible answer to the OP’s doubts without recurring to weird ideas such as “negative distance from the origin”.