Question

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

\({\rm{r = 4sin3\theta }}\).

Short answer

In the Cartesian diagram,\({\rm{r = 4sin(3\theta )}}\) the value of the function increases from\({\rm{0}}\) to\({\rm{4}}\) in the interval\({\rm{0}} \le {\rm{\theta }} \le \frac{{\rm{\pi }}}{{\rm{6}}}\), and then decreases\({\rm{0}}\) as the angle travels\(\frac{{\rm{\pi }}}{{\rm{3}}}\). The first leaf of the rose on the right side represents the same curve in polar coordinates. Similarly, when the angle moves from \(\frac{{\rm{\pi }}}{{\rm{3}}}\)to\(\frac{{\rm{\pi }}}{{\rm{2}}}\), the function in the Cartesian coordinate decreases \({\rm{ - 4}}\)to\({\rm{0}}\) and subsequently raises to\({\rm{0}}\) when the angle \({\rm{\theta }}\)moves from \(\frac{{\rm{\pi }}}{{\rm{2}}}\)tos\(\frac{{{\rm{2\pi }}}}{{\rm{3}}}\). On the right, the polar curve's corresponding curve is marked (\({\rm{2}}\)).

Step-by-step solution

Step 1: Cartesian coordinate.

In the Cartesian diagram,\({\rm{r = 4sin(3\theta )}}\) the value of the function increases from\({\rm{0}}\) to\({\rm{4}}\) in the interval\({\rm{0}} \le {\rm{\theta }} \le \frac{{\rm{\pi }}}{{\rm{6}}}\), and then decreases\({\rm{0}}\) as the angle travels\(\frac{{\rm{\pi }}}{{\rm{3}}}\). The first leaf of the rose on the right side represents the same curve in polar coordinates. Similarly, when the angle moves from \(\frac{{\rm{\pi }}}{{\rm{3}}}\)to\(\frac{{\rm{\pi }}}{{\rm{2}}}\), the function in the Cartesian coordinate decreases \({\rm{ - 4}}\)to\({\rm{0}}\) and subsequently raises to\({\rm{0}}\) when the angle \({\rm{\theta }}\)moves from \(\frac{{\rm{\pi }}}{{\rm{2}}}\)tos\(\frac{{{\rm{2\pi }}}}{{\rm{3}}}\). On the right, the polar curve's corresponding curve is marked (\({\rm{2}}\)).

Step 2: Cartesian diagram.