How do you graph \[r=2\cos \left( 3\dfrac{\theta }{2} \right)\]?

Hint:
In this given question, we have been asked to plot the graph for a given polar form. The polar form of a complex number is the other way to represent a given complex number, the polar form is represented by z = a + bi. When we need to plot the points in the plane, we need to have rectangular coordinate i.e. (x, y). For conversion from polar coordinates to rectangular coordinates, we will need to use the formulas from trigonometric functions. The relation between the polar coordinates \[\left( r,\theta \right)\] and the rectangular coordinates (x, y) is \[x=r\cdot \cos \theta \] and\[y=r\cdot \sin \theta \].

Complete step by step solution:
We have given that,
\[r=2\cos \left( 3\dfrac{\theta }{2} \right)\],
As we know that,
\[r\ge 0,\ \Rightarrow \dfrac{3\theta }{2}\in \left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]\ thus\ \Rightarrow \theta \in \left[ -\dfrac{\pi }{3},\dfrac{\pi }{3} \right]\]
Therefore,
Simplify the given expression according to the given condition, we get
\[r=2\cos \left( 3\dfrac{\theta }{2} \right)=2\sqrt{\dfrac{1}{2}\left( 1+\cos 3\theta \right)}\ge 0\]
To convert from polar coordinates to rectangular coordinates, using the formulas from trigonometric functions definitions:
We have,
\[x=r\cdot \cos \theta \] And \[y=r\cdot \sin \theta \]
So we have that,
\[{{r}^{2}}={{x}^{2}}+{{y}^{2}}\]
For converting the Cartesian (x, y) = \[r\left( \cos \theta ,\sin \theta \right)\]
Thus,
\[{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{2}}=\sqrt{2}\sqrt{{{\left( {{x}^{2}}+{{y}^{2}} \right)}^{\dfrac{3}{2}}}+{{x}^{3}}-3x{{y}^{2}}}\]
Plotting the graph of the above equation,

Hence, this is the required graph.

Note:
Students need to remember that polar coordinates are those coordinates that can be plotted into a circular grid. On the other side, rectangular coordinates are those coordinates that can be plotted into a plane i.e. graph and the rectangular coordinates are represented in the form of (x, y). Rectangular coordinates include only numerical values as they represent only horizontal axis and vertical axis. The relation between the polar coordinates \[\left( r,\theta \right)\] and the rectangular coordinates (x, y) is \[x=r\cdot \cos \theta \] and\[y=r\cdot \sin \theta \].