Question

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

\({\rm{r = 2 + sin3\theta }}\)

Short answer

The graph shows a transformation \({\rm{y = sinx}}\) with amplitude 1 and shifted up by 2 units.

Step-by-step solution

Step 1: Draw a Graph\({\rm{y = 2 + sin3\theta }}\).

First, make a Cartesian graph \({\rm{\theta }}\)at the horizontal axis and \({\rm{r}}\)the vertical axis. This is the transformation of \({\rm{y = sinx}}\)with amplitude 1, shifted up via 2 units. So,\({\rm{1}} \le {\rm{r}} \le {\rm{3}}\).

Then, the function has a period of\(\frac{{{\rm{2\pi }}}}{{\rm{3}}}\).

Draw a Graph\({\rm{r = 2 + sin3\theta }}\).

To Plot the polar graph considers an increment of \({\rm{\pi /6}}\)approximately equal to\({\rm{0}}{\rm{.5236}}\).

\({\rm{\theta }}\)	\({\rm{r = 2 + sin3\theta }}\)
0.00000	2.000
0.52360	3.000
1.04720	2.000
1.57080	1.000
2.09440	2.000
2.61799	3.000
3.14159	2.000
3.66519	1.000
4.18879	2.000
4.71239	3.000
5.23599	2.000
5.75959	1.000
6.28319	2.000

A petal starts at \({\rm{ - }}\frac{{\rm{\pi }}}{{\rm{6}}}\) the ends \({\rm{ - }}\frac{{\rm{\pi }}}{{\rm{6}}}{\rm{ + }}\frac{{{\rm{2\pi }}}}{{\rm{3}}}\) and then repeats the same pattern 2 more times. The curve never gets closer than a distance \({\rm{r = 2}}\)from the pole.

Thus, the graph shows a transformation \({\rm{y = sinx}}\) with amplitude 1 and shifted up by 2 units.