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Chapter 7: Problem 68

Use a graphing utility to graph the polar equation. $$r=\frac{3}{\sin \theta}$$

Short Answer

Expert verified
The graph produced is a vertical line crossing the pole with a hole at the pole, showing symmetry with respect to the origin. The hole is due to the undefined points at 胃 equals 0 or 蟺.

Step by step solution

01

Understand the Conversion of Polar Coordinates to Cartesian Coordinates

Firstly, it is important to note the conversion between polar coordinates and cartesian coordinates. For the polar coordinates (r, 胃), r is the distance from the origin and 胃 is the angle from positive x-axis. The cartesian coordinates (x, y) can be found using the formulas: \( x = r \cdot \cos \theta \) and \( y = r \cdot \sin \theta \)
02

Choose Values for 胃 and Calculate Corresponding r

Choose values for \( \theta \) and calculate the corresponding r values using the equation \( r = \frac{3}{\sin \theta} \). For instance, at \( \theta = \frac{\pi}{2} \), r = 3, and at \( \theta = \pi \), r = 0. It's always a good idea to choose values of 胃 between 0 and 2蟺, in steps of \( \frac{\pi}{4} \) for instance. If 胃 equals 0 or 蟺, the function is undefined due to division by zero. They are asymptotes to the graph.
03

Plot the Polar Coordinates

Now plot these coordinates by moving counter-clockwise starting from the positive x-axis, and measure r from the origin. The points obtained represent the graph of the given polar equation.
04

Check Your Result

To check the graph obtained, note that for every point (r, 胃) on the graph, the point (-r, 胃 + 蟺) is also on the graph, because negative r will extend in the opposite direction of positive r. This is referred to as symmetry with respect to the origin. Thus the graph should reflect this symmetry.

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