Identifying and Graphing a Polar Equation

Identify and graph each equation:

(a) \(r=3\)

(b) \(\theta =\dfrac{\pi }{4}\)

Solution (a) If \(r\) is fixed at \(3\) and \(\theta\) is allowed to vary, the graph is a circle with its center at the pole and radius 3, as shown in Figure 39. To confirm this, we convert the polar equation \(r=3\) to a rectangular equation. \[ \begin{array}{rcl@{\qquad}l} r &=&3 \\ r^{2} &=&9\quad{\color{#0066A7}{{\hbox{Square both sides.}}}} \\ x^{2}+y^{2} &=&9\quad{\color{#0066A7}{{\hbox{\(r^{2} =x^{2} +y^{2}\)}}}} \end{array} \]

(b) If \(\theta\) is fixed at \(\dfrac{\pi }{4}\) and \(r\) is allowed to vary, the result is a line containing the pole, making an angle of \(\dfrac{\pi }{4}\) with the polar axis. That is, the graph of \(\theta =\dfrac{\pi }{4} \) is a line containing the pole with slope \(\tan \theta =\tan \dfrac{\pi }{4}=1\), as shown in Figure 40. To confirm this, we convert the polar equation to a rectangular equation. \[ \begin{array}{rcl} \theta &=&\dfrac{\pi }{4} \\ \tan \theta &=&\tan \dfrac{\pi }{4} \\ \dfrac{y}{x} &=&1 \\ y &=&x \end{array} \]