Problem 64 - Contemporary Precalculus

Question

Explore various ways in which a calculator can produce inaccurate graphs of trigonometric functions. These exercises also provide examples of two functions, with different graphs, whose graphs appear identical in certain viewing windows. Find a viewing window in which the graphs of \(y=\cos x\) and \(y=.54\) appear identical. [Hint: See the chart in Exercise 61 and note that \(\cos 1 \approx .54 .]\)

Step-by-Step Solution

Answer

Answer: The viewing window is \(1-2\pi \le x \le 1+2\pi\) for the x-axis and \(0.54 - \epsilon \le y \le 0.54 + \epsilon\) for the y-axis, where \(\epsilon\) is a small value (e.g. 0.01).

Step 1: Review given information and find values for x that give \(\cos x = 0.54\)

We are given that \(\cos 1 \approx 0.54\). Let's find other values of x for which \(\cos x = 0.54\). We know that the cosine function is periodic with the period \(2\pi\), so the x values can be found using the formula \(x = 1 + 2\pi k\), where \(k\) is an integer.

Step 2: Create a table with values of x and the corresponding cosine values

To determine the viewing window for which the graphs of \(y=\cos x\) and \(y=0.54\) will appear identical, we will need to consider some values of x around the given value x=1. Let's create a table with these values and the corresponding cosine values. |-| x | \(\cos x\) | |--|----|---------| |1|1-2π| 0.54 | |2| 1 | 0.54 | |3|1+2π| 0.54 |

Step 3: Determine the viewing window

Since the values of \(\cos x\) are relatively close to each other at these specific values of x, we can determine a viewing window for which the calculator can't distinguish between the graphs of \(y=\cos x\) and \(y=0.54\). A suitable viewing window, given the table above, would be: \- x-axis: \(1-2\pi \le x \le 1+2\pi\) \- y-axis: \(0.54 - \epsilon \le y \le 0.54 + \epsilon\), where \(\epsilon\) is a small value (e.g. 0.01) This viewing window will make the graphs of \(y=\cos x\) and \(y=0.54\) appear identical, illustrating how calculators can produce inaccurate graphs of trigonometric functions.

Key Concepts

cosine function

The cosine function is a fundamental trigonometric function that plays a significant role in mathematics. It is often denoted as \(\cos x\), where \(x\) is an angle measured in radians. This function is defined using the unit circle, where it represents the x-coordinate of a point on the circle.

The cosine function is periodic with a period of \(2\pi\), which means it repeats its values every \(2\pi\) units. For instance, \(\cos 0 = \cos 2\pi=1\). Because of this inherent periodicity, cosine values recur at regular intervals, making it predictable in behavior.

Some important properties of cosine function include:

• Range: The values of \(\cos x\) lie within \(-1\) to \(1\).
• Symmetry: The cosine function is an even function, meaning \(\cos(-x) = \cos x\).
• Key Values: \(\cos 0 = 1\), \(\cos \frac{\pi}{2} = 0\), \(\cos \pi = -1\), \(\cos \frac{3\pi}{2} = 0\).

These properties help us understand its graph and behavior, essential when evaluating and visualizing trigonometric functions.

graphing calculators

Graphing calculators are powerful tools that help visualize mathematical functions. They are particularly useful in graphing trigonometric functions like the cosine function. These devices plot graphs by computing a series of points over a specified range for the given function, and then connecting these points where appropriate. However, it's crucial to be aware of potential inaccuracies in the graphs produced by these calculators.

The accuracy of a graph from a calculator heavily depends on the viewing window set. The viewing window determines the range of x and y values displayed on the screen. If the window is too zoomed in or out, graphs of distinctly different functions might appear identical. For example, the graphs of \(y = \cos x\) and a constant function \(y = 0.54\) can look the same within a poorly-chosen range.

To avoid such errors:

• Choose a viewing window carefully, considering the period and amplitude of the functions involved.
• Be mindful of the calculator's pixel limitations, as they can affect precision.
• Perform additional calculations or analyses if unexpected graph behaviors occur.

By understanding these limitations, students can effectively use graphing calculators as a learning aid rather than a source of confusion.

visualization of periodic functions

Visualizing periodic functions such as the cosine function is an essential aspect of understanding trigonometry. These functions exhibit repeating patterns over intervals, making them predictable and useful in modeling real-world phenomena like sound and light waves.

When visualizing periodic functions on a calculator or graph, it's important to choose the correct viewing window to capture the function's essential characteristics. For instance, a period of the cosine function can be illustrated well within an x-range of \(-2\pi\) to \(2\pi\), since it encompasses full cycles, showing the repeating nature of the function.

Key aspects to visualize in periodic functions include:

• Period: The length of one full cycle of the function. For \(y = \cos x\), this is \(2\pi\).
• Amplitude: The height from the midline to the peak or trough of the function. For cosine, it's 1.
• Phase Shift: A horizontal shift of the function, which can occur if the function changes form. Not present in \(y=\cos x\).

Understanding how these factors affect the shape and behavior of the graph enables a deeper comprehension of periodic functions. Utilizing graphing tools with these adjustments helps prevent the misinterpretation of trigonometric graphs.