Problem 53 - Precalculus with Limits

Question

In Exercises 49-58, use a graphing utility to graph the polar equation. Describe your viewing window. \(r=8\ \cos\ \theta\)

Step-by-Step Solution

Answer

The graph of the equation \(r = 8\cos\theta\) is a circle with a radius of 8. The viewing window for \(\theta\) should be between 0 and 2π (or 0 to 360 degrees), and for r, it should at least be -10 to 10.

Step 1: Understand the polar coordinate

Polar coordinates are set up quite differently from the rectangular Cartesian coordinates that are more commonly used in mathematics. In Cartesian coordinates, numbers on a graph are represented by horizontal and vertical distances from the origin. Polar coordinates, on the other hand, represent numbers on a graph using a distance and an angle: the distance r from the origin (the pole) and the angle theta from the polar axis (usually the x-axis). In this particular equation, \(r=8\cos\theta\), the radius (r) depends on the angle and the cosine function.

Step 2: Identify the characteristics of the equation

The given equation is a transformed cosine function as it is of the form \(r=a\cos\theta\). Here, the amplitude is 8. Thus, the radius varies as the cosine of the angle, leading to a circular shape due to cosine's cycle of -1 to 1.

Step 3: Choose an appropriate viewing window and plot the equation

Now that the basics are understood, the next task is to plot the function using a graphing utility, like a TI-84 graphing calculator or Desmos. It is crucial to remember that the window for theta is usually between 0 and \(2\pi\) (or 360 degrees). Given the nature of the function, the maximum value of r will be when cos(theta) equals 1, meaning r will also peak at 8. So the window for r should go at least -10 to 10 to capture the whole graph. Plot the equation in this window.

Step 4: Evaluate the graph

Examining the plotted graph, it should appear as a circle - this is due to the consistent amplitude of the cosine function. The maximum distance from the origin (r=8) occurs when cos(theta) is at its maximum value of 1, and the minimum occurs when cos(theta) is at its lowest point of -1. This gives the regular nature of the circle.

Key Concepts

Cartesian Coordinates

When we visualize mathematical problems, Cartesian coordinates are often our go-to system. They use two or three dimensions represented by perpendicular axes: typically x and y in 2D, or x, y, and z in 3D. Each point on the plane is defined by distances along these axes.

In this system, a point is identified by its horizontal position (x) and vertical position (y) from the origin. This makes graphing lines, curves, and shapes straightforward because each value corresponds to a specific point on the grid. Think of it like city streets laid out in a grid, where moving left or right changes one value (x), and moving up or down changes another (y).

While Cartesian coordinates are incredibly useful, polar coordinates offer a different perspective. Instead of using perpendicular axes, they use angles and radii, providing unique solutions, especially in circular motion and periodic phenomena.

Cosine Function

The cosine function is integral in understanding periodic behavior, like waves and circles. It operates within the trigonometric family functions and is defined by its pattern: repeating every \(2\pi\) radians or 360 degrees. The cosine of an angle \(\theta\) gives us the x-coordinate of a point on the unit circle.

Mathematically, the cosine function is represented as \(\cos(\theta)\). Key points of cosine's curve include:

• Maximum at \(\cos(0) = 1\)
• Zero at \(\cos(\frac{\pi}{2}) = 0\)
• Minimum at \(\cos(\pi) = -1\)

The equation \(r = 8\cos\theta\) uses the cosine function to determine the radius, dictating a circular path as \(\theta\) changes. The value of r fluctuates from -8 to 8, modeling a circle due to cosine's consistency from -1 to 1.

Graphing Calculator

Graphing calculators like the TI-84 and digital tools such as Desmos have become vital in visual learning. These tools allow us to visualize complex equations, experiment with functions, and validate answers quickly.

For graphing polar equations like \(r = 8\cos\theta\), setting the right viewing window is crucial:

• For \(\theta\): Typically set from 0 to \(2\pi\) (360 degrees) to capture a full cycle.
• For r-values: Given the equation's transformation, allow for a window encompassing -10 to 10 to display the complete circle.

Using a graphing calculator simplifies plotting by automating the calculations. With just inputs, you see how \(r\) varies with \(\theta\), providing an instant visual for the relationship and behavior of equations like ours, showcasing their true nature.