Question

Sketch the curve with the given polar equation by

first sketching the graph of \({\rm{r}}\) as a function of \({\rm{\theta }}\) in Cartesian

coordinates.

Short answer

\(\left( {{\rm{r, \theta }}} \right)\) is plot points and connect the point finally get the graph.

Step-by-step solution

Finding r values.

The equation of r

\({\rm{r = 1 + 2cos(2\theta )}}\)

To draw the curve, we'll start by locating some points by determining the values of r for various values. Then plot points \(\left( {{\rm{r, \theta }}} \right)\)and connect them to get curve.

Curve values and points.

\({\rm{\theta }}\)	\({\rm{r}}\)
\({\rm{0}}\)	\({\rm{3}}\)
\(\frac{\pi }{{\rm{6}}}\)	\({\rm{2}}\)
\(\frac{\pi }{{\rm{3}}}\)	\({\rm{0}}\)
\(\frac{\pi }{{\rm{2}}}\)	\({\rm{ - 1}}\)
\(\frac{{{\rm{2}}\pi }}{{\rm{3}}}\)	\({\rm{0}}\)
\(\frac{{{\rm{5}}\pi }}{{\rm{6}}}\)	\({\rm{2}}\)
\(\pi \)	\({\rm{3}}\)
\(\frac{{{\rm{7}}\pi }}{{\rm{6}}}\)	\({\rm{2}}\)
\(\frac{{{\rm{4}}\pi }}{{\rm{3}}}\)	\({\rm{0}}\)
\(\frac{{{\rm{3}}\pi }}{{\rm{2}}}\)	\({\rm{ - 1}}\)
\(\frac{{{\rm{5}}\pi }}{{\rm{3}}}\)	\({\rm{0}}\)
\(\frac{{{\rm{11}}\pi }}{{\rm{6}}}\)	\({\rm{2}}\)
\({\rm{2}}\pi \)	\({\rm{3}}\)

Draw the graph of the table.

\(\left( {{\rm{r, \theta }}} \right)\) is plot points and connect the point and get the curve.

\(\left( {{\rm{0,}}\frac{\pi }{{\rm{3}}}} \right){\rm{,}}\left( {{\rm{0,}}\frac{{{\rm{2}}\pi }}{{\rm{3}}}} \right){\rm{,}}\left( {{\rm{0,}}\frac{{{\rm{4}}\pi }}{{\rm{3}}}} \right){\rm{,}}\left( {{\rm{0,}}\frac{{{\rm{5}}\pi }}{{\rm{3}}}} \right){\rm{.}}\)