Question

Sketch the strophoid \(r=\sec \theta-2 \cos \theta,\) \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\). Convert this equation to rectangular coordinates. Find the area enclosed by the loop.

Short answer

The rectangular equation is \(x=y\tan(\theta)-2y\). The area enclosed by a loop can be calculated by determining the bounds for \(\theta\) from the plot and substituting them into the equation for the area in polar coordinates.

Step-by-step solution

Sketching the Polar Curve

Plot the polar function \(r=\sec \theta-2 \cos \theta,\) for \(-\frac{\pi}{2}<\theta<\frac{\pi}{2}\). This can be done by making a table and calculating \(r\) for different \(\theta\) values and then plotting them on polar graph paper.

Convert to Rectangular Coordinates

Using the transformation \(r = x\cos(\theta) = y\sin(\theta)\), the equation becomes \(x = y\tan(\theta)- 2y\). This can be done by plugging in the identity \(sec(\theta) = 1/cos(\theta)\) and rearranging the equation.

Calculate the area of the Loop

Once we have the plot, we can determine the bounds for \(\theta\) which form a loop and Integrate using the formula for the area in polar coordinates \(A = 1/2 \int_{a}^{b} r^2 d\theta\), where \(a\) and \(b\) are the bounds determined from the plot.