Answer
1. Evaluate several points in rectangular coordinates corresponding to $\theta = 0,\frac{\pi }{4},\frac{\pi }{2},...,2\pi $
2. Then plot the points in rectangular coordinates and sketch the curve by connecting these points.
Work Step by Step
The rectangular coordinates of a point on the curve corresponding to $\theta$ is given by
$\left( {x,y} \right) = \left( {r\cos \theta ,r\sin \theta } \right)$,
$\left( {x,y} \right) = \left( {1 + \cos \theta } \right)\left( {\cos \theta ,\sin \theta } \right)$.
For the interval $0 \le \theta \le 2\pi $, we evaluate several points in rectangular coordinates corresponding to $\theta = 0,\frac{\pi }{4},\frac{\pi }{2},...,2\pi $ and list them on a table:
$\begin{array}{l}
\begin{array}{*{20}{c}}
\theta &{\left( {x,y} \right)}
\end{array}\\
\begin{array}{*{20}{c}}
0&{\left( {2,0} \right)}
\end{array}\\
\begin{array}{*{20}{c}}
{\frac{\pi }{4}}&{\left( {\frac{1}{2}\left( {1 + \sqrt 2 } \right),\frac{1}{2}\left( {1 + \sqrt 2 } \right)} \right)}
\end{array}\\
\begin{array}{*{20}{c}}
{\frac{\pi }{2}}&{\left( {0,1} \right)}
\end{array}\\
\begin{array}{*{20}{c}}
{\frac{{3\pi }}{4}}&{\left( {\frac{1}{2}\left( {1 - \sqrt 2 } \right),\frac{1}{2}\left( { - 1 + \sqrt 2 } \right)} \right)}
\end{array}\\
\begin{array}{*{20}{c}}
\pi &{\left( {0,0} \right)}
\end{array}\\
\begin{array}{*{20}{c}}
{\frac{{5\pi }}{4}}&{\left( {\frac{1}{2}\left( {1 - \sqrt 2 } \right),\frac{1}{2}\left( {1 - \sqrt 2 } \right)} \right)}
\end{array}\\
\begin{array}{*{20}{c}}
{\frac{{3\pi }}{2}}&{\left( {0, - 1} \right)}
\end{array}\\
\begin{array}{*{20}{c}}
{\frac{{7\pi }}{4}}&{\left( {\frac{1}{2}\left( {1 + \sqrt 2 } \right),\frac{1}{2}\left( { - 1 - \sqrt 2 } \right)} \right)}
\end{array}\\
\begin{array}{*{20}{c}}
{2\pi }&{\left( {2,0} \right)}
\end{array}
\end{array}$
Then we plot the points in rectangular coordinates and sketch the curve by connecting these points.
