Question

Sketch the polar curve.

\({\rm{r = 1 + cos2\theta }}\)

Short answer

The polar curve \({\rm{r = 1 + cos2\theta }}\) is symmetric about both the x and y axes, as can be seen in Figure 1.

Step-by-step solution

Definition of Concept

Polar curve: A polar curve is a shape made with the polar coordinate system. Polar curves are defined by points that vary in distance from the origin (the pole) based on the angle measured off the positive x-axis.

Sketch the polar curve

Considering the given information:

Polar equation for the variable

\({\rm{r = 1 + cos2\theta }}\)

Puttingthe value\({{\rm{0}}^{\rm{^\circ }}}\)for\({\rm{\theta }}\) and obtain the value of r as,

\(\begin{aligned}{c}{\rm{r = }}\left( {{\rm{1 + }}\left( {{\rm{cos}}\left( {{\rm{2 \times }}{{\rm{0}}^{\rm{^\circ }}}} \right){\rm{ \times }}\frac{{\rm{\pi }}}{{{\rm{180}}}}} \right)} \right)\\{\rm{ = 2}}\end{aligned}\)

Similarly, obtain the values of\({\rm{\theta }}\)and r .

Draw the polar curve for the equation \({\rm{r = 1 + cos2\theta }}\)using the points as shown in Figure 1.

Therefore, the polar curve \({\rm{r = 1 + cos2\theta }}\) is symmetric about both the x and y axes, as can be seen in Figure 1.