Question

How do you graph $r=\dfrac{3}{\sin \left( \theta \right)}$?

Answer 1

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.

Complete step by step solution:
We have given that,
$r=\dfrac{3}{\sin \left( \theta \right)}$,
Simplifying the above expression, we get
$r\sin \left( \theta \right)=3$
Writing the polar-coordinates, we get
To convert from polar coordinates to rectangular coordinates, using the formulas from trigonometric functions definitions:
We have,
$y=r\cdot \sin \theta$
Thus, substituting the value of$r\sin \left( \theta \right)=3$, we obtain
$y=3$
Therefore, this is the horizontal line at y = 3.
Plotting the graph at y = 3
And here, x = 0
Thus,
Rectangular coordinate = (x, y) = (0, 3)
Plotting the graph, we have

Hence, it 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 $y=r\cdot \sin \theta$.