how do we know if the curve theta = pi/2 is a circle, or semicircle, or spiral?

I've tried to change the curve to x and y, but both sides of the equation are the same, and tan(pi/2)=infinity.

The curve is a straight line, at angle theta = pi/2. It does not matter what r is, it will be a point on that vertical line.

Similarly, the graph of r=3 is a circle. It does not matter what theta is, the point will lie on a circle of radius 3.