Problem 24 - Algebra and Trigonometry

Question

Test for symmetry and then graph each polar equation. $$ r=2+4 \sin \theta $$

Step-by-Step Solution

Answer

The polar equation \( r = 2 + 4\sin\theta \) doesn't show symmetry with respect to the origin or the polar axis. The graph would be a circle-like curve above the x-axis (polar axis), shifted due to the terms in the given equation.

Step 1: Test for Symmetry

Test symmetry with respect to the origin. For this, we replace \( r \) with \( -r \) and \( \theta \) with \( \theta + \pi \) in the equation. This gives us \( -r = 2 + 4 \sin(\theta + \pi) \). Simplifying the right side, we get \( -r = 2 - 4 \sin(\theta) \). So, the original equation \( r = 2+4\sin\theta \) is not symmetric about the origin.

Step 2: Test for more Symmetry

We now test for symmetry with respect to the polar axis. For this, we replace \( \theta \) with \( -\theta \) in the equation, so \( r = 2 + 4\sin(-\theta) \). By properties of sine, this simplifies to \( r = 2 - 4\sin\theta \). So, the original equation \( r = 2+4\sin\theta \) is not symmetric about the polar axis either.

Step 3: Graph the Polar Equation

Having found no symmetry, now we graph the equation. For \(\theta = 0\), \(r = 2 + 4 \sin 0 = 2\). This would be a point on the polar axis (positive x-axis). Graph some more points by plugging in other values of \(\theta\) and obtaining corresponding values for \(r\). For different values of \(\theta\) in the range of \(0\) to \(2\pi\), we can find corresponding \(r\) values. Then, plot the graph by setting \(\theta\) as the angle from the x-axis (polar axis) and \(r\) as the distance from the origin.

Key Concepts

Symmetry in Polar Coordinates

Understanding symmetry in polar coordinates can substantially simplify the process of graphing polar equations. Symmetry in this context refers to an equation producing a graph that looks the same after a certain transformation is applied, such as rotating the graph or reflecting it across a line.

When testing for symmetry with respect to the origin, you replace the radius variable, represented as \( r \), with \( -r \), and the angle variable, \( \theta \), with \( \theta + \text{\pi} \). If the resulting equation is equivalent to the original one, the graph is symmetric about the origin. Think of this symmetry as rotating the graph 180 degrees. In the given exercise, however, the equation did not display this type of symmetry.

Another common symmetry is about the polar axis. This is tested by replacing \( \theta \) with \( -\theta \). If an equation remains unchanged, then the graph will mirror across the polar axis. Unfortunately, the exercise showed that the graph of the given equation does not possess this symmetry either.

Why Is Symmetry Helpful?

Identifying symmetry can reduce the workload by half or more, as plotting the points for one portion of the graph automatically gives you points for the symmetric parts. However, even without symmetry, plotting points at various angles can successfully graph the equation.

Polar Axis

In polar coordinates, the polar axis is the reference line from which angles are measured. It corresponds to zero degrees and is typically represented by the positive x-axis in a Cartesian coordinate system.

As you graph polar equations like \( r = 2+4\sin\theta \), the polar axis serves as the starting point. You measure the angle \( \theta \) from it. For example, when \( \theta = 0 \), you're at the polar axis, and the radius \( r \) gives the distance from the origin. If \( r \) is positive, you’ll move in the direction of the polar axis; if negative, in the exact opposite direction.

Plotting Points Along the Polar Axis

During the graphing process, when you plot points at \( \theta = 0 \), this is straightforward as you just measure along the polar axis. With the equation from our exercise, when \( \theta = 0 \), we found that \( r = 2 \), meaning we would plot a point 2 units along the polar axis. Each point you plot represents the pairing of an angle with a distance from the origin — a fundamental concept in understanding polar graphs.

Properties of Sine

Sine is one of the basic trigonometric functions with properties critical to graphing polar equations. Some important properties include the fact that sine is an odd function, meaning that \( \sin(-\theta) = -\sin(\theta) \). This property was utilized when testing for symmetry about the polar axis in the exercise.

More properties of the sine function are its periodicity, with a period of \( 2\text{\pi} \), and its value range between -1 and 1. Due to its periodic nature, when graphing polar equations, we often only look at the interval from \( 0 \) to \( 2\text{\pi} \) because this captures a full cycle of the sine function.

Applying Sine in Polar Equations

When utilizing sine in polar equations, remember that its values dictate the radial distance from the origin based on the angle. For example, with \( r = 2+4\sin\theta \), the sine of \( \theta \) modifies the radius, thus changing the graph's shape. The properties of sine play a significant role, especially when determining the maximum and minimum radial distances, which occur when \( \sin(\theta) \) is at its extremities, 1 and -1 respectively.