Problem 84 - Essential Calculus

Question

Think About It 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 graph of the polar equation shifts along with the changes in the value of \(\phi\). The equation converted as a function of \(\sin \theta\) for \(\phi = \pi / 2\) is \(r = 6[1 + \cos(\theta) \sin(\pi / 2) + \sin(\theta) \cos(\pi / 2)]\).

Step 1: Graph Polar Equation

First, graph the polar equation \(r=6[1+\cos(\theta-\phi)]\) using a graphing utility for when \(\phi = 0\), \(\phi = \pi / 4\) and \(\phi = \pi / 2\). Observe how the graph changes as the value of \(\phi\) changes.

Step 2: Analyze the Graphs

Study the three graphs produced by different \(\phi\) values. Notice if the graph shifts, rotates, or changes shape with different \(\phi\) values. The changes observed give an understanding of the effect of angle \(\phi\) on the graph.

Step 3: Convert the Equation

When \(\phi = \pi / 2\), the term \(\cos(\theta - \phi)\) can be rewritten using the identity \(\cos(A - B) = \sin A \cos B + \cos A \sin B\). Substituting \(\sin \theta\) for \(\cos \theta\) in the polar equation results in the equation in terms of \(\sin \theta\).

Key Concepts

Polar Coordinate System

Polar coordinates offer a different way to locate points on a plane, compared to the familiar Cartesian coordinate system. In the polar system, every point is determined by a distance from a central point—known as the pole, akin to the origin in Cartesian coordinates—and an angle measured from the positive x-axis.

In mathematical terms, a polar coordinate is given as \( (r, \theta) \), where \( r \) is the radial distance from the pole, and \( \theta \) is the angular coordinate, often measured in radians. Unlike Cartesian coordinates, which use a grid, polar coordinates are based on circles centered at the pole, with each circle corresponding to a value of \( r \) and radial lines from the pole representing values of \( \theta \).

Understanding \( r = 6[1 + \cos(\theta - \phi)] \) in Polar Coordinates

The exercise involves graphing the polar equation \( r = 6[1 + \cos(\theta - \phi)] \), which includes \( \phi \) as a parameter affecting the radius \( r \). In this context, graphing means plotting points that satisfy the equation for various angles \( \theta \), and observing changes for different values of \( \phi \).By altering \( \phi \), you're effectively rotating or shifting the polar graph. Observing how these changes affect the graph can bolster your understanding of the relationship between the equation and its geometric representation.

Graphing Utility Usage

Graphing utilities are technological tools—like software or calculators—that provide a visual representation of equations. They're extremely helpful in understanding complex mathematical concepts, particularly in subjects like trigonometry and calculus, where visualization can be challenging.

When using a graphing utility to plot polar equations, you often have to enter the equation in the format the tool requires, choose appropriate scales for \( r \) and \( \theta \) axes, and then analyze the resulting graph.

Advantages of Graphing Utilities

• Instant visualization of equations, providing an intuitive understanding of their shapes and transformations.
• Ability to adjust parameters on-the-fly to see real-time changes in the graph.
• Helpful for verifying manual calculations and understanding the effects of altering different variables in an equation.

In our specific exercise, the graphing utility would allow you to quickly observe how the parameter \( \phi \) influences the polar equation \( r = 6[1 + \cos(\theta - \phi)] \) by creating different graphs for \( \phi = 0 \), \( \phi = \pi / 4 \) and \( \phi = \pi / 2 \). This kind of dynamic visual aid is what makes graphing utilities an essential educational resource.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the involved variables. These identities simplify expressions and solve equations that would otherwise be much more complex.

A fundamental set of trigonometric identities involves the cosine and sine functions, such as the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \), and angle sum and difference identities, for example, \( \cos(A - B) = \cos A\cos B + \sin A\sin B \) and \( \sin(A + B) = \sin A\cos B + \cos A\sin B \).

Application in Polar Equations

In the context of the exercise, the trigonometric identity for \( \cos(A - B) \) is used to rewrite the polar equation \( r = 6[1 + \cos(\theta - \phi)] \) when \( \phi = \pi / 2 \). This demonstrates how identities can be applied to manipulate equations into a preferred form—for instance, representing the polar equation solely in terms of \( \sin \theta \)—which can greatly assist in understanding and solving more complex equations.

Comprehending these identities helps students delve deeper into the essential nature of trigonometric functions and their numerous applications across various fields including physics, engineering, and computer science.