Problem 103 - Calculus Early Transcendentals

Question

Points at which the graphs of \(r=f(\theta)\) and \(r=g(\theta)\) intersect must be determined carefully. Solving \(f(\theta)=g(\theta)\) identifies some-but perhaps not all-intersection points. The reason is that the curves may pass through the same point for different values of \(\theta .\) Use analytical methods and a graphing utility to find all the intersection points of the following curves. \(r=2 \cos \theta\) and \(r=1+\cos \theta\)

Step-by-Step Solution

Answer

Question: Find the intersection points of the polar equations \(r = 2 \cos \theta\) and \(r = 1 + \cos \theta\). Answer: The intersection points occur when \(\theta = 2n\pi\) for any integer value of \(n\), with corresponding Cartesian coordinates \(x = 2\) and \(y = 0\).

Step 1: Solve the Equations for Intersection Points

Solve the equation \(2 \cos \theta= 1 + \cos \theta\) by subtracting \(\cos \theta\) from both sides: \[\cos \theta = 1\] The intersection points occur when \(\cos \theta = 1\). This only happens when \(\theta = 2n\pi\) for any integer value of \(n\) because the cosine of any multiple of \(2\pi\) is equal to 1.

Step 2: Analyze the Cartesian Coordinates

We can consider the Cartesian representation of these polar equations to check if any other intersection points exist: \[x = r\cos\theta\] \[y = r\sin\theta\] The Cartesian representations of the given polar equations are: \[x = 2\cos^2\theta\] \[y = 2\cos\theta\sin\theta\] and \[x = (1+\cos\theta)\cos\theta\] \[y = (1+\cos\theta)\sin\theta\] Now compare the values of \(x\) and \(y\) coordinates when \(\theta = 2n\pi\): both equations should yield the same values at these points. For \(r=2\cos\theta\): \[x = 2\cos^2(2n\pi)\] \[y = 2\cos(2n\pi)\sin(2n\pi)\] For \(r=1+\cos\theta\): \[x = (1+\cos(2n\pi))\cos(2n\pi)\] \[y = (1+\cos(2n\pi))\sin(2n\pi)\] Since \(\cos(2n\pi) = 1\) and \(\sin(2n\pi) = 0\), the intersection points when \(\theta = 2n\pi\) are given by: \[x = 2\cos^2(2n\pi) = 2\cdot 1^2 = 2\] \[y = 2\cos(2n\pi)\sin(2n\pi) = 2\cdot 1\cdot 0 = 0\] for the polar equation \(r = 2\cos\theta\), and \[x = (1+\cos(2n\pi))\cos(2n\pi) = (1 + 1)\cdot 1 = 2\] \[y = (1+\cos(2n\pi))\sin(2n\pi) = (1 + 1)\cdot 0 = 0\] for the polar equation \(r = 1+\cos\theta\). We can see that both sets of coordinates yield the same values for intersection points, so we have found all the intersection points.

Step 3: Verification using a Graphing Utility

We can verify our solution by using a graphing utility to plot the polar equation \(r=2\cos\theta\) (polar plot) and superimpose the plot of the polar equation \(r=1+\cos\theta\). The intersection points should coincide with our solution: \(\theta = 2n\pi\) for any integer value of \(n\). After examining the graph, we confirm that our findings are correct and complete.

Key Concepts

Polar Coordinates

Polar coordinates are one of the coordinate system alternatives used in mathematics to specify the location of a point. Unlike the familiar Cartesian coordinates which use a grid of horizontal and vertical lines, polar coordinates rely on angles and distances from a central point, called the pole.

In this system, a point is located by the radial distance from the pole, denoted as r, and an angle, θ, measured in radians. The angle usually starts from the positive x-axis and increases counterclockwise. This unique system is particularly useful in scenarios where the relationship between points is more naturally expressed in terms of angles and distances, such as in cases of circular or spiral patterns.

As with the Cartesian system, you can convert between polar and Cartesian coordinates. To find the Cartesian coordinates (x,y) from polar coordinates (r,θ), you use the equations:
\[ x = r\cos(θ) \]
\[ y = r\sin(θ) \]
These conversions are crucial for understanding how curves plotted in polar coordinates correspond to shapes and intersections in the more ubiquitous Cartesian plane.

Cosine Function

The cosine function is a fundamental trigonometric function that arises frequently in both pure and applied mathematics. Cosine describes the ratio of the adjacent side to the hypotenuse in a right-angled triangle. In terms of the unit circle, it gives the x-coordinate of a point on the circle's circumference, given an angle θ from the positive x-axis.

Mathematically, the cosine function is periodic, which means it repeats its values in a regular interval of 2π radians. Its maximum value is 1, and it occurs when θ is an even multiple of π radians (0, 2π, 4π, ...), which corresponds to points lying on the positive x-axis of the Cartesian plane. The cosine function also plays a vital role in defining the shape of curves in polar coordinates, influencing their size and orientation when involved in polar equations.

Analytical Methods in Calculus

Calculus is a branch of mathematics concerned with the study of change and motion. Within calculus, analytical methods involve the rigorous and systematic approach to solving equations and evaluating functions. In polar equations, finding intersection points analytically involves setting the equations equal to each other and solving for the angles at which the radial distances are the same.

These methods also extend to include the use of derivatives for understanding the behavior of functions, and integrals for calculating areas and volumes. The thorough analytical approach allows for the verification of intersection points by exploring the different possible values of θ and confirming these solutions through transformation to Cartesian coordinates or by other means. Such solutions must be critically evaluated to determine if they include all possible intersection points or if further investigation is necessary.

Graphing Utility

A graphing utility is an invaluable digital tool in mathematics, especially when dealing with complex functions and equations. It allows for the visualization of equations in various coordinate systems, including polar coordinates.

Graphing utilities can plot the curves described by polar equations, which is helpful for identifying intersection points that may not be immediately apparent through algebraic methods. When using a graphing utility to analyze problems involving polar coordinates, one can overlay multiple plots to visually inspect for intersections and compare the graphical findings with analytical results. Such cross-verification ensures the accuracy of the solutions and provides additional insight into the behavior of the functions involved in the equations.