Chapter 11.3, Problem 21E

To determine

To determine: The equivalent Cartesian equation for the given polar equation. Also, identify the graph without using a grapher.

Answer to Problem 21E

The required equation is $y=4$ and the equation is a horizontal line.

Explanation of Solution

Given information:

The polar equation: $r=4\csc \theta$

Calculation:

The given equation is $r=4\csc \theta$ .

To find the required Cartesian equation, use the formulas $x=r\cos \theta$ , $y=r\sin \theta$ , $r^2=x^2+y^2$ and $\tan \theta =\frac{y}{x}$ .

Rewrite the given equation as:

  $\begin{array}{c}r=4\csc \theta \\ r=\frac{4}{\sin \theta }\\ r\sin \theta =4\end{array}$

Substitute y for $r\sin \theta$ in the above equation.

  $\begin{array}{c}r\sin \theta =4\\ y=4\end{array}$

The above equation is an equation of a horizontal line.

Hence, the required equation is $y=4$ and the equation is a horizontal line.