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Chapter 6: Problem 74

Use a graphing utility to graph the polar equation. $$r=\frac{1}{1-\sin \theta}$$

Short Answer

Expert verified
The final graph should correspond to the polar equation \(r=\frac{1}{1-\sin \theta}\) and should correctly depict the relationship between the variables in this equation.

Step by step solution

01

Initialize the \(\theta\) Values

Start from \(\theta = 0\) and iterate up by a small increment (like \(\pi/6\) for example) up to \(2\pi\), which is a full circle in polar coordinates. Now you will have a series of \(\theta\) values to use.
02

Calculate Corresponding r Values

Substitute each of the \(\theta\) values from Step 1 into the polar equation \(r=\frac{1}{1-\sin \theta}\) to calculate the corresponding \(r\) values.
03

Convert Polar Coordinates to Cartesian Coordinates

The radius \(r\) and angle \(\theta\) must be converted to Cartesian coordinates using the relations \(x = r \cdot cos(\theta)\) and \(y = r \cdot sin(\theta)\), for each pair of \(r\) and \(\theta\) values obtained in the previous steps. This will give a collection of (x, y) points.
04

Plot the Points

Graph all the (x, y) points obtained in Step 3 in Cartesian space. As \(\theta\) varies from 0 to \(2\pi\), the graph should form a complete shape.
05

Interpret the Graph

Check if the resultant graph represents the polar equation given. The shape of the graph is determined by the given polar equation, and can provide insight into the behaviour of the equation that might not necessarily be apparent from the equation itself.

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Most popular questions from this chapter

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