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Chapter 7: Problem 39

Plot the following eurves:\(y=\cos x+\cos 2 x\)

Short Answer

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Question: Sketch the curve of the function \(y=\cos x+\cos 2x\) and identify its key features, such as x-intercepts, local maximums, and minimums. Answer: The curve of the function \(y=\cos x+\cos 2x\) has a domain of \((-\infty, \infty)\). It features multiple x-intercepts, local maximums, and minimums, which can be observed from the plot. One way to sketch the curve these points is by using a graphing calculator or software to create a table of values and plotting the points on a graph. The final plot should resemble the following curve: ![Graph of y = cos(x) + cos(2x)](https://i.stack.imgur.com/uQSAa.png)

Step by step solution

01

Define the domain of the function

Since both cosine functions have a domain of \((-\infty, \infty)\), their sum will also have a domain of \((-\infty, \infty)\). Therefore, we can plot the function in the entire real line.
02

Determine the key features of the function

We will check for some important points: - Points of intersection: We find the x-intercepts where \(y=\cos x+\cos 2x=0\). Unfortunately, there is no standard algebraic method to solve for x in this case. But we can try it graphically in the next steps. - Maximum and minimum points: By analyzing the first and second derivatives, we could find the local maximums and minimums. However, since we are using a graphical approach, we can observe these points directly from the curve plot.
03

Create a table of values for key points

For this step, we use a graphing calculator or graphing software (e.g. Desmos, GeoGebra, or Wolfram Alpha). These tools will provide the x-intercepts, and we can select points within different periods to understand the shape of the curve better. We can then construct the table as follows (where \(x\) is in radians): | x | y | |---|---| | -6 | 1.35040 | | -3 | 1.89929 | | 0 | 2 | | 3 | 1.89929 | | 6 | 1.35040 | (Of course, more points could be added to get a better understanding of the curve shape.)
04

Sketch the curve

Using the table of values and the key points, sketch the curve of \(y=\cos x+\cos 2x\). Make sure to include the important points (x-intercepts, local maximums/minimums, and any other critical points). The final plot should resemble the curve created by the graphing calculator or software: ![Graph of y = cos(x) + cos(2x)](https://i.stack.imgur.com/uQSAa.png) In this graph, we can see the x-intercepts, maximum and minimum points, as well as the overall shape of the curve \(y=\cos x+\cos 2x\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Function
The cosine function is a fundamental concept in trigonometry used to represent periodic oscillations. It's one of the trigonometric functions, such as sine and tangent. Here are some characteristics of the cosine function, usually denoted as \(y = \cos(x)\):

  • It has a range of \[ -1, 1 \], meaning the output values of \(\cos(x)\) are confined within these limits.
  • The function is periodic with a period of \(2\pi\) radians, meaning it repeats its values every \(2\pi\) radians.
  • At \(x = 0\), and at every interval of \(2\pi\), the cosine function has a value of 1, which is its maximum value.
  • \cos(x)\ is an even function, which implies that \(\cos(-x) = \cos(x)\). This reflects as symmetry about the y-axis in its graph.
  • Critical points of the function, where the slope is zero, occur at \(\frac{\pi}{2} + k\pi\) for any integer \(k\), corresponding to peak and trough of the wave-like graph.
When plotting \(y=\cos(x)\), graphing software or knowledge of these properties will help you create a correctly shaped curve. Recognizing the traits of the cosine function is essential for understanding more complex trigonometric expressions built on it, such as \(y=\cos(x) + \cos(2x)\) in the exercise above.
Graphical Solution of Equations
When faced with equations that are not easily solvable algebraically, such as \(y=\cos(x) + \cos(2x) = 0\), a graphical solution can be your go-to method. This approach entails plotting the equation on a graph and visually identifying the points of interest.

  • Finding X-Intercepts: To determine where the function crosses the x-axis, i.e., find the roots of the equation, plot the graph and look for points where \(y=0\).
  • Analyzing Behavior: The graph can also show the behavior of the function, indicating where it increases or decreases, providing insights into its derivative.
  • Identifying Maxima and Minima: Without calculus, the peak and trough points on the graph highlight where the function reaches local maximums or minimums.
This technique is particularly helpful in trigonometry, where equations often involve periodic functions that are more intuitively analyzed through their graphs rather than by solving equations.
Trigonometric Curve Sketching
Sketching trigonometric curves means drawing the graphs of trigonometric functions while bearing in mind their unique properties. Key steps in sketching include:

  • Understanding the Function: Know the general shape and characteristics of the basic trigonometric functions.
  • Determining Amplitude and Period: Identify any changes in the wave's height (amplitude) and the distance between repetitions (period).
  • Finding Critical Points: Calculate or visually identify important points such as maxima, minima, and x-intercepts.
  • Plotting Points: Creating a table of values at regular intervals can help plot points that define the curve's shape.
  • Drawing the Curve: Connect the plotted points smoothly, considering the continuous nature of trigonometric functions.
For example, in the exercise \(y=\cos(x) + \cos(2x)\), recognizing that \(\cos(2x)\) has twice the frequency of \(\cos(x)\) is key to sketching the combined curve accurately. Each function contributes to the overall shape, which can be depicted using a graphing tool or by hand with knowledge of trigonometric sketching techniques.

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