Question

Sketch the polar curve.

\({\rm{r = 1 - cos\theta }}\)

Short answer

As,\(\theta \)increases from\(\frac{{3\pi }}{2}\)to values of r decreases from 1 to 2.

\({\rm{Now sketch that part of the curve}}{\rm{.}}\)

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

\(r = 1 - \cos \theta \)

To draw this curve, first draw the function \({\rm{r = 1 - cos\theta }}\) in Cartesian coordinates. This will allow us to see which values of r correspond to which values of \({\rm{\theta }}\). This will show us where the polar curve will begin and where it will go. So, on the y-axis, we'll see values for r, and on the x-axis, we'll see values for \({\rm{\theta }}\).

We can see from the previous graph that as theta rises from 0 to \(\frac{{\rm{\pi }}}{{\rm{2}}}\;\), the value of r rises from 0 to 1. Now we'll draw that section of the curve.

As \({\rm{\theta }}\) rises from \(\frac{{\rm{\pi }}}{{\rm{2}}}\) to \({\rm{\pi }}\), the value of r rises from one to two. However, because \(\frac{\pi }{2} \le \theta \le \pi \), this portion of the curve must be in the second quadrant. Now draw that section of the curve.

As, \({\rm{\theta }}\) rises from \({\rm{\pi }}\) to \(\frac{{{\rm{3\pi }}}}{{\rm{2}}}\) values of r fall from 2 to 1. However, since \(\pi \le \theta \le \frac{{3\pi }}{2}\) and \({\rm{r > 0}}\), \({\rm{\theta }}\)and r are in the same quadrant, the third quadrant. Now draw that portion of the curve.

Therefore, as, \(\theta \) increases from \(\frac{{3\pi }}{2}\) to values of r decreases from 1to 2.