Writing a Cartesian Equation in Polar Form

The goal is to eliminate $x$ and $y$ from the equation and introduce $r$ and $\theta .$ Ideally, we would write the equation $r$ as a function of $\theta .$ To obtain the polar form, we will use the relationships between $\left(x,y\right)$ and $\left(r,\theta \right).$ Since $x=r\cos \theta$ and $y=r\sin \theta ,$ we can substitute and solve for $r.$

Thus, $x^2+y^2=9,r=3,$ and $r=-3$ should generate the same graph. See Figure 10.

Figure 10 (a) Cartesian form $x^2+y^2=9$ (b) Polar form $r=3$

To graph a circle in rectangular form, we must first solve for $y.$

Note that this is two separate functions, since a circle fails the vertical line test. Therefore, we need to enter the positive and negative square roots into the calculator separately, as two equations in the form $Y_1=\sqrt{9-x^2}$ and $Y_2=-\sqrt{9-x^2}.$ Press GRAPH.