Question

Sketch the curve with the given polar equation by first sketching the graph\({\rm{r}}\)as a function of\({\rm{\theta }}\)Cartesian coordinates.

Short answer

Equation is

To start sketching the curve, first need to find several points by calculating values offer various values\({\rm{\theta }}\). Then link the spots \({\rm{(r,\theta )}}\)to form the curve

\({\rm{\theta }}\)	\({\rm{r}}\)
\({\rm{1}}\)	\({\rm{0}}\)
\(\frac{{\rm{\pi }}}{{\rm{3}}}\)	\({\rm{ln}}\frac{{\rm{\pi }}}{{\rm{3}}}\)
\(\frac{{\rm{\pi }}}{{\rm{2}}}\)	\({\rm{ln}}\frac{{\rm{\pi }}}{{\rm{2}}}\)
\(\frac{{{\rm{2\pi }}}}{{\rm{3}}}\)	\({\rm{ln}}\frac{{{\rm{2\pi }}}}{{\rm{3}}}\)
\({\rm{\pi }}\)	\({\rm{ln\pi }}\)
\(\frac{{{\rm{4\pi }}}}{{\rm{3}}}\)	\({\rm{ln}}\frac{{{\rm{4\pi }}}}{{\rm{3}}}\)
\(\frac{{{\rm{3\pi }}}}{{\rm{2}}}\)	\({\rm{ln}}\frac{{{\rm{3\pi }}}}{{\rm{2}}}\)
\(\frac{{{\rm{5\pi }}}}{{\rm{3}}}\)	\({\rm{ln}}\frac{{{\rm{5\pi }}}}{{\rm{3}}}\)
\({\rm{2\pi }}\)	\({\rm{ln2\pi }}\)

Step-by-step solution

Step 1: Sketching the curve.

Equation is

To start sketching the curve, first need to find several points by calculating values offer various values\({\rm{\theta }}\). Then link the spots \({\rm{(r,\theta )}}\)to form the curve

\({\rm{\theta }}\)	\({\rm{r}}\)
\({\rm{1}}\)	\({\rm{0}}\)
\(\frac{{\rm{\pi }}}{{\rm{3}}}\)	\({\rm{ln}}\frac{{\rm{\pi }}}{{\rm{3}}}\)
\(\frac{{\rm{\pi }}}{{\rm{2}}}\)	\({\rm{ln}}\frac{{\rm{\pi }}}{{\rm{2}}}\)
\(\frac{{{\rm{2\pi }}}}{{\rm{3}}}\)	\({\rm{ln}}\frac{{{\rm{2\pi }}}}{{\rm{3}}}\)
\({\rm{\pi }}\)	\({\rm{ln\pi }}\)
\(\frac{{{\rm{4\pi }}}}{{\rm{3}}}\)	\({\rm{ln}}\frac{{{\rm{4\pi }}}}{{\rm{3}}}\)
\(\frac{{{\rm{3\pi }}}}{{\rm{2}}}\)	\({\rm{ln}}\frac{{{\rm{3\pi }}}}{{\rm{2}}}\)
\(\frac{{{\rm{5\pi }}}}{{\rm{3}}}\)	\({\rm{ln}}\frac{{{\rm{5\pi }}}}{{\rm{3}}}\)
\({\rm{2\pi }}\)	\({\rm{ln2\pi }}\)

Plotting the points.

Plot the points\({\rm{(0,1),}}\left( {{\rm{ln}}\frac{{\rm{\pi }}}{{\rm{3}}}{\rm{,}}\frac{{\rm{\pi }}}{{\rm{3}}}} \right){\rm{,}}\left( {{\rm{ln}}\frac{{\rm{\pi }}}{{\rm{2}}}{\rm{,}}\frac{{\rm{\pi }}}{{\rm{2}}}} \right){\rm{,}}\left( {{\rm{ln}}\frac{{{\rm{2\pi }}}}{{\rm{3}}}{\rm{,}}\frac{{{\rm{2\pi }}}}{{\rm{3}}}} \right){\rm{,(ln\pi ,\pi ),}}\left( {{\rm{ln}}\frac{{{\rm{4\pi }}}}{{\rm{3}}}{\rm{,}}\frac{{{\rm{4\pi }}}}{{\rm{3}}}} \right){\rm{,}}\left( {{\rm{ln}}\frac{{{\rm{3\pi }}}}{{\rm{2}}}{\rm{,}}\frac{{{\rm{3\pi }}}}{{\rm{2}}}} \right){\rm{,}}\left( {{\rm{ln}}\frac{{{\rm{5\pi }}}}{{\rm{3}}}{\rm{,}}\frac{{{\rm{5\pi }}}}{{\rm{3}}}} \right){\rm{,(ln2\pi ,2\pi )}}\)

Thus the curve is drawn with a polar equation.