Chapter 6.4, Problem 36E

Solution

To convert: The polar equation to rectangular forms and identifies the graph, also support the answer by graphing the polar equation

The rectangular form of equation is $y=-2$ and graph is a horizontal line.

Given data:

The equation is $r=-2\csc \theta$ .

Formula used:

Use the following conversion relationships between a polar coordinate $\left(r,\theta \right)$ and its corresponding rectangular coordinate $\left(x,y\right)$ .

  $\begin{array}{c}x=r\cos \theta \\ y=r\sin \theta \\ r^2=x^2+y^2\\ \tan \theta =\frac{y}{x}\end{array}$

Calculation:

Divide both sides of the equation $r=-2\csc \theta$ by $\csc \theta$ .

  $\begin{array}{c}\frac{r}{\csc \theta }=\frac{-2\csc \theta }{\csc \theta }\\ \frac{r}{\csc \theta }=-2\\ r\sin \theta =-2\\ y=-2\end{array}$

The graph is a horizontal line.

This is supported by using a graphing calculator in Polar and Radian mode as shown below.

  

Therefore, the rectangular form of equation is $y=-2$ and graph is a horizontal line.