Q. 21 - Calculus

Question

Graph the equation in the θr-plane. Label each arc of your curve with the quadrant in which the corresponding polar graph will occur.   

$r={\cos }^2\theta$

Step-by-Step Solution

Answer

The graph of the equation:

Step 1. Given information

The equation of the curve:

$r={\cos }^2\theta$  

Step 2. Simplify the equation:

The function can be simplified as:

$$\begin{gathered} r={\cos }^2\theta \\ r=\frac{1+\cos 2\theta }{2} \\ r=\frac{1}{2}+\frac{1}{2}\cos 2\theta \end{gathered}$$

Step 3. Graph the equation in θr-plane.

The value of $\theta$lies on the horizontal axis and $r$lies on the vertical axis.

The graph of $r=\frac{1}{2}+\frac{1}{2}\cos 2\theta$  is obtained by compressing the the graph of  $r=\cos \theta$ by factor of $\frac{1}{2}$horizontally then compress the graph vertcally by a factor of $\frac{1}{2}$ then vertcally shift the graph by $\frac{1}{2}$ units upward.