Problem 37 - Algebra and Trigonometry

Question

Describe a strategy for graphing \(r=\frac{1}{1+\sin \theta}\)

Step-by-Step Solution

Answer

Plotting strategies for polar functions primarily involve understanding the polar coordinates, converting them into rectangular coordinates to understand the function's behavior better, and then plotting the function using a range of \(\theta\) values. \(r=\frac{1}{1+\sin \theta}\) will appear as outwardly-bulging loops, one for each interval of \(2\pi\) in \(\theta\), similar to circular petals.

Step 1: Understand the Variables

In the equation \(r=\frac{1}{1+\sin \theta}\), \(r\) is the distance of a point from the origin in the plane, and \(\theta\) is the angle measured in radians counter-clockwise from the positive x-axis.

Step 2: Convert the Equation to Rectangular Coordinates

The given equation is in polar coordinates. If we want to graph it precisely, it's useful to convert it into rectangular coordinates. By replacing \(r = x^2 + y^2\) and \(y = r*\sin(\theta)\), the equation can be transformed. However, for graphing purposes, the original form of the equation will also suffice.

Step 3: Generate a Table of Values

Now assign different values to theta within the interval of -\(\pi\) and +\(\pi\). Calculate the corresponding \(r\) values and prepare a table of \(r\) against \(\theta\).

Step 4: Plotting Points and Joining Them

Plot the points with the corresponding \(r\) and \(\theta\) values on the polar graph and then join these points to get a curve.

Step 5: Final analysis

Now, we have a plot of the given equation representing it within the two-dimensional plane.