Problem 68 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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 θ equals 0 or π.

Step 1: 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, θ), r is the distance from the origin and θ is the angle from positive x-axis. The cartesian coordinates (x, y) can be found using the formulas: \( x = r \cdot \cos \theta \) and \( y = r \cdot \sin \theta \)

Step 2: Choose Values for θ 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 θ between 0 and 2π, in steps of \( \frac{\pi}{4} \) for instance. If θ equals 0 or π, the function is undefined due to division by zero. They are asymptotes to the graph.

Step 3: 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.

Step 4: Check Your Result

To check the graph obtained, note that for every point (r, θ) on the graph, the point (-r, θ + π) 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.