Question

How do you graph $r=2\cos 3\theta $

Answer 1

Hint: We will have to choose two axes in a plane where we plot the graph of this given function $r=2\cos 3\theta $. We will take the value of angle on one axis and on the other axis we will get the value of required function. Simply take some values of angle and after plotting the values we get the required shape of the graph.

Complete step-by-step solution:
A polar curve is a shape of the figure constructed using the polar coordinate system in any plane. Polar curves are defined by points that are a variable distance from the origin depending on the angle measured off the positive x-axis. Cartesian curves are useful to describe paths in terms of horizontal and vertical distances along the axes, while polar curves are more useful to describe paths that are an absolute distance from a certain point. One important practical application of the use of polar curves is to describe the directional microphone pickup patterns. A directional microphone will pick up different qualities of sound depending on the different locations the sound comes from outside of the microphone. For example, a cardioids microphone has a pickup pattern in the shape of cardioids.
Each point in the polar coordinate system is given by \[(r,\theta )\], where \[r\] is taken as the distance from the pole (origin) to the point, and angle is taken as counter clockwise direction that is made with the point, pole, and the positive \[x\]-axis.
We have known about the Cartesian equations in the $x \text{and} y$ plane.
If we want to convert out Cartesian equation in polar equation we have to do following substitutions,
$\begin{align}
  & x=r\cos \theta \\
 & y=r\sin \theta \\
\end{align}$
Where $\theta ={{\tan }^{-1}}\dfrac{y}{x}$
For calculating the magnitude we will square and add the above equations.
Our given polar equation is $r=2\cos 3\theta $.
Where $\theta $ is the angle and $r$ is the radius.
We will plot the values of angle and we will the required value of function and we plot them to get the required result and corresponding graph of the function is shown below

This is the required graph of $r=2\cos 3\theta $.

Note: The two-dimensional graph of a polar equation in a plane is defined as the set of all points in that plane whose polar coordinates easily satisfy the given polar equation. The graph of the polar equation \[r=1\] consists of those points in any plane whose distance from the pole is taken as \[1\] which signifies that the circle is of radius \[1\] unit and its centre is at the pole.