Question

To sketch the polar curve from the given Cartesian curve as shown in Figure.

Short answer

It can be observed that the curve is symmetric with respect to both the axis.

Step-by-step solution

Given data

The given data is the Cartesian curve.

Concept of Polar curve

Polar curves are defined by points that are a variable distance from the origin (the pole) which depends on the angle measured off the positive\(x\)-axis.

Calculation for the polar curve

It can be observed that the maximum value obtained by the given graph is \(2\) whereas the minimum value obtained by the graph is \(0.5\).

From the given graph, when \(\theta \) increases from 0 to \(\frac{\pi }{2}\), polar curve \(r\) increases from \(0.5\) to \(2\) in polar graph.

When \(\theta \) increases from \(\frac{\pi }{2}\) to \(\pi \), polar curve \(r\) decreases from \(2\) to \(0.5\).

When \(\theta \) increases from \(\pi \) to \(\frac{{3\pi }}{2}\), polar curve \(r\) increases from \(0.5\) to \(2\).

When \(\theta \) increases from \(\frac{{3\pi }}{2}\) to \(2\pi \), polar curve \(r\) decreases from \(2\) to \(0.5\).

The procedure is same for next values of \(\theta \).

Plot the curve form the Cartesian curve

The curve from the given Cartesian curve is plotted and shown below.

From Figure, it can be observed that the curve is symmetric with respect to both the axis.