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Chapter 6: Problem 65

Convert each polar equation to a rectangular equation. Then use a rectangular coordinate system to graph the rectangular equation. $$r=4 \csc \theta$$

Short Answer

Expert verified
The rectangular equivalent of the given polar equation \(r=4 \csc \theta\) is \(y = \frac{4}{\sqrt{x^2+y^2}}\). The graph is a half circle with radius 4 above the x-axis.

Step by step solution

01

Converting the Equation to Cartesian Form

Utilizing the relationship between polar and rectangular coordinates, the definition of cosecant (csc) as \(1/ \sin \theta\) and the equation \(r = 4 \csc \theta\), we can rewrite the equation as \(r = \frac{4}{sin \theta}\). Since r is the radius, and so \(r = \sqrt{x^2+y^2}\), it gives us, \(\sqrt{x^2+y^2} = \frac{4}{\sin \theta}\). Replacing \(sin \theta\) with y/r where r = sqrt of (x^2+y^2), we can simplify to get the final rectangular equation, \(y = \frac{4}{\sqrt{x^2+y^2}}\).
02

Drawing the Graph of the Equation

Plot the graph of y = 4/\(\sqrt{x^2+y^2}\). The graph represents a half circle with radius 4, above the x-axis because the y values can only be positive or zero for the given rectangular equation.

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