Problem 72 - Precalculus with Limits

Question

GRAPHICAL REASONING Use a graphing utility to graph the polar equation \(r = 6[1+\cos(\theta - \phi)]\) for (a) \(\phi = 0\), (b) \(\phi = \pi/4\), and (c) \(\phi = \pi/2\). Use the graphs to describe the effect of the angle \(\phi\). Write the equation as a function of \(\sin\ \theta\) for part (c).

Step-by-Step Solution

Answer

The effect of varying \(\phi\) in the given polar equation is a counterclockwise rotation of the graph through an angle of \(\phi\) radians. For part (c), the rewritten equation is \(r = 6[1+\sin(\theta)]\).

Step 1: Graphing \(r = 6[1+cos(\theta)]\)

First, graph the polar equation \(r = 6[1+cos(\theta)]\). This is a limaçon, a special polar curve of which the graph takes on a snail-like shape. The relation of the cos(\(\theta\)) function here molds the fundamental figure.

Step 2: Graphing \(r = 6[1+cos(\theta - \pi/4)]\)

For part (b), \(\phi\) is set to \(\pi/4\). Graph the polar equation \(r = 6[1+cos(\theta - \pi/4)]\). The primary difference from the previous graph is a rotation of the figure counterclockwise by \(\pi/4\) radians as resulting from the shift of the argument of the cosine function.

Step 3: Graphing \(r = 6[1+cos(\theta - \pi/2)]\)

For part (c), set \(\phi\) to \(\pi/2\). Now, graph the polar equation \(r = 6[1+cos(\theta - \pi/2)]\). In this graph, a rotation of the figure counterclockwise by \(\pi/2\) radians is observed, a similar effect to the one previously noted.

Step 4: Rewriting the equation in terms of \(sin\ \theta\)

Write the polar equation \(r = 6[1+cos(\theta - \pi/2)]\) as a function of \(sin\ \theta\). To do this, use the trigonometric identity \(sin(90-\theta) = cos(\theta)\) to convert cosines into sines. This results in the equation \(r = 6[1+\sin(\theta)]\).

Key Concepts

limaçon

A limaçon is a fascinating and unique type of polar curve. The term "limaçon" comes from the French word for "snail," and its shape can indeed look somewhat like the shell of a snail. Mathematically, it is described by the polar equation of the form \(r = a + b \cos \theta \) or \(r = a + b \sin \theta \). The graph of it varies significantly based on the values of \(a\) and \(b\). For instance:

• If \(a = b\), it forms a cardioid shape.
• If \(a > b\), it produces a dimpled limaçon.
• If \(b > a\), you see a limaçon with a loop.

These curves can describe motions, model periodic phenomena, and are visual treats due to their symmetric properties. When plotting a limaçon, seeing how it transforms based on parameter changes can be quite engaging and insightful.

graphing utility

Using a graphing utility makes visualizing complex equations, like polar equations, straightforward and accessible. These tools allow you to input a polar equation and instantly see its graphical representation. Understanding the tool's interface can target specific needs, like adjusting scale or rotating angles. For our equation \(r = 6[1+\cos(\theta - \phi)]\), entering different values of \(\phi\) quickly shows how the graph changes.

Graphing utilities often include features such as:

• Zooming and panning to explore different parts of the graph.
• Plotting multiple equations at once to compare their shapes and sizes.
• Interactive elements to manipulate parameters dynamically.

These features make graphing utilities essential tools in studying polar coordinates, enabling a practical exploration of abstract mathematical concepts.

trigonometric identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They play a crucial role in simplifying expressions and solving trigonometric equations. In our exercise, using trigonometric identities allows us to rewrite the equation involving \(\cos\) and transform it to involve \(\sin\).

For example, the identity \( \sin(\frac{\pi}{2} - \theta) = \cos(\theta) \) was used to rewrite \( r = 6[1+\cos(\theta - \frac{\pi}{2})] \) as \( r = 6[1+\sin(\theta)] \). This effectively swaps the cosine curve's shape into a sine curve one, maintaining the same frequency and amplitude but changing its orientation.
By understanding these identities, you can manipulate and solve various trigonometric problems with ease.

angle rotation

Angle rotation refers to shifting the angle used in trigonometric functions, resulting in a rotated graph. In polar equations, this is particularly evident when examining expressions like \( \theta - \phi \). By altering \( \phi \), the graph rotates, changing the orientation without modifying the intrinsic shape.

In our task, rotation is accomplished by setting different values of \( \phi \):

• \(\phi = 0\): The initial graph's orientation.
• \(\phi = \frac{\pi}{4}\): Rotated counterclockwise by \(45^\circ\).
• \(\phi = \frac{\pi}{2}\): Rotated counterclockwise by \(90^\circ\).

These adjustments illustrate the graph rotating along a central point, providing visual clarity on how different angle shifts affect polar graphs.