Mathematics
Graphs of Trigonometric Functions
Graphs of trigonometric functions represent the relationships between angles and the values of trigonometric ratios such as sine, cosine, and tangent. These functions produce periodic wave-like patterns, with specific characteristics such as amplitude, period, and phase shift. Understanding these graphs is essential for analyzing and solving problems in fields like physics, engineering, and mathematics.
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10 Key excerpts on "Graphs of Trigonometric Functions"
eBook - PDF- Mary Jane Sterling(Author)
- 2023(Publication Date)
- For Dummies(Publisher)
5 The Graphs of Trig Functions IN THIS PART . . . Create the basic graphs of the six trig functions. Use the basic graphs of sine and cosine to more easily graph cosecant and secant. Perform transformations on graphs of trig functions to make them fit a particular situation. Use trig functions to model periodic applications — things occurring over and over as time goes by. CHAPTER 18 Graphing Sine and Cosine 303 Chapter 18 Graphing Sine and Cosine T he graphs of the sine and cosine functions are very similar. If you look at them without a coordinate axis for reference, you can’t tell them apart. They keep repeating the same values over and over — and the values, or outputs, are the same for the two functions. These two graphs are the most rec- ognizable and useful for modeling real-life situations. The sine and cosine curves can represent anything tied to seasons — the weather, shopping, hunting, and daylight. The equations and graphs of the curves are helpful in describing what happens during those seasons. You also find the curves used in predator-prey scenarios and physical cycles. The ABCs of Graphing You can graph trig functions in a snap — well, maybe not that fast — but you can do it quickly and efficiently with just a few pointers. If you set up the axes properly and have a general understanding of the different functions’ shapes, then you’re in business. Different kinds of values represent the two axes in trig graphs. The x-axis is in angle measures, and the y-axis is in plain old numbers. The x-axis is labeled in IN THIS CHAPTER » Looking at the basic graphs of sine and cosine » Working with variations of the graphs » Using sine and cosine curves to make predictions 304 PART 5 The Graphs of Trig Functions either degrees or radians. Often, a graph represents the values from 2 to 2 to accommodate two complete cycles of the sine, cosine, secant, or cosecant func- tions (or four complete cycles of the tangent or cotangent functions).
eBook - PDF- Cynthia Y. Young(Author)
- 2021(Publication Date)
- Wiley(Publisher)
• Solve harmonic motion problems. • Graph sums of trigonometric and other algebraic functions. CONCEPTUAL OBJECTIVES • Understand why the graphs of the sine and cosine functions are called sinusoidal graphs. • Rewriting the sinusoidal function in standard form makes identifying the phase shift easier. • Visualize harmonic motion as a sinusoidal function. • Understand that the y-coordinates of the combined function are found by adding the y-coordinates of the individual functions. 6.8.1 The Graphs of Sinusoidal Functions 6.8.1 Skill Determine the amplitude and period of sinusoidal functions. 6.8.1 Conceptual Understand why the graphs of the sine and cosine functions are called sinusoidal graphs. The following are examples of things that repeat in a predictable way (i.e., they are roughly periodic): • heartbeat • tide levels • time of sunrise • average outdoor temperature for the time of year The trigonometric functions are strictly periodic. In the unit circle, the value of any of the trigonometric functions is the same for any coterminal angle (same initial and terminal sides no matter how many full rotations the angle makes). For example, if we add (or subtract) multiples of 2π to the angle θ, the values for sine and cosine are unchanged. sin(θ + 2nπ) = sin θ or cos (θ + 2nπ) = cos θ (n is any integer) A function ƒ is called a periodic function if there is a positive number p such that f (x + p) = f (x) for all x in the domain of f If p is the smallest such number for which this equation holds, then p is called the fundamental period. Periodic Function 6.8 Graphs of Sine and Cosine Functions 609 You will see in this section and the next that sine, cosine, secant, and cosecant functions have fundamental period 2π, but that tangent and cotangent functions have fundamental period π. The Graph of f (x) = sin x Let us start by point-plotting the sine function.
eBook - PDF- Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
Graphs of the Trigonometric Functions ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to: • Identify and calculate the characteristics of periodic waves and the sine curve. • Calculate the amplitude, period, and phase angle from the general equation y = a sin(bx + c). • Draw a quick sketch of a sine curve. • Draw a quick sketch of other trigonometric functions. • Calculate the amplitude, time period, and phase shift for a sine wave as a function of time. • Graph on a polar coordinate system. • Graph parametric equations. If a picture is worth a thousand words, a graph is worth a thousand calculations. For example, a graph can instantaneously show thousands of points that make up a solid line. The graph can also visually demonstrate many of the characteristics of the equation that defines the graph. In this chapter, we look at the Graphs of Trigonometric Functions. You already have an idea of how these functions work; the graphs allow us to see how the functions vary through all pos- sible angles. We also introduce periodic waves (think of the heartbeat monitor showing basically the same waveform repeated over and over). These functions will describe many physical and mathematical properties that repeat regu- larly, in devices such as pendulums, electrical generators, or mechanical levers and gears. 17–1 The Sine Curve Periodic Functions A curve that repeats its shape over and over is called a periodic curve or periodic waveform, as shown in Fig. 17-1. The function that has this graph is thus called a periodic function. Each repeated portion of the curve is called a cycle. 1 Period 1 Cycle Amplitude Amplitude 0 y x FIGURE 17-1 A periodic waveform. 17 361 Section 17–1 ◆ The Sine Curve The x axis represents either an angle (in degrees or radians) or time (usually seconds or milliseconds).
eBook - PDF- David Cohen, Theodore Lee, David Sklar, , David Cohen, Theodore Lee, David Sklar(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 8.2 Graphs of the Sine and Cosine Functions 543 P Q R S T A B C D E F 12 5π 12 π 3 π 4 π 6 π y y x x 8.2 GRAPHS OF THE SINE AND COSINE FUNCTIONS Our focus in this section is on the sine and cosine functions. As preparation for the discussion, we want to understand what is meant by the term “periodic function.” By way of example both of the functions in Figure 1 are periodic. That is, their graphs display patterns that repeat themselves at regular intervals. In Figure 1(a) the graph of the function f repeats itself every six units. We say that the period of f is 6. Similarly, the period of g in Figure 1(b) is 2 p . In both cases, no-tice that the period represents the minimum number of units that we must travel along the horizontal axis before the graph begins to repeat itself. With these examples in mind, we can state the definition of a periodic function. (a) The graph of f -6 6 6 4 2 12 (b) The graph of g -π -4 -2π π 4 2π y x y x Figure 1 Definition A Periodic Function and Its Period A nonconstant function f is said to be periodic if there is a number p 0 such that f ( x p ) f ( x ) for all x in the domain of f . The smallest such number p is called the period of f . We also want to define the term “amplitude” as it applies to periodic functions. For a function such as g in Figure 1(b), in which the graph is centered about the hor-izontal axis, the amplitude is simply the maximum height of the graph above the horizontal axis. Thus the amplitude of g is 4. More generally, we define the amplitude of any periodic function.
eBook - PDF- Paul A. Calter, Michael A. Calter(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
422 15 Trigonometric, Parametric, and Polar Graphs ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter you should be able to • Graph the sine wave, by calculator or manually. • Find the amplitude, period, frequency, and phase shift for a sine wave. • Find roots or instantaneous values on a sine wave. • Write the equation of a given sine wave. • Graph and analyze a sine wave as a function of time. • Graph the cosine, tangent, cotangent, secant, and cosecant functions. • Graph the inverse trigonometric functions. • Graph parametric equations. • Graph points and equations in polar coordinates. • Convert between polar and rectangular form. ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ So far we have dealt with curves that rise, or fall, or perhaps rise and fall a few times. Now we will introduce curves that oscillate, repeating the same shape indefi- nitely, the periodic functions. These are the sort of curves we find in alternating cur- rent, or the mechanical vibrations that could cause a bridge to collapse. We find pe- riodic motion in mechanical devices, such as the pistons in an automobile engine, the motions of the celestial bodies, sound waves, and in radio, radar, and television signals. Periodic signals are crucial to the operation of many of the exciting techno- logical devices of the twenty-first century, from computers to satellite telephones. In this chapter we give a small introduction to the world of periodic functions. Our main focus will be on the sine function, which has wide applications to alternat- ing current, mechanical vibrations, and so forth. Our task in this chapter will be to graph such functions, building upon our earlier methods for graphing and to extract useful information from the function. For example, given the alternating current I 37.5(sin 284t 22°) Section 1 ◆ Graphing the Sine Wave by Calculator 423 Screen for Example 1.
eBook - ePub- Mary Jane Sterling(Author)
- 2019(Publication Date)
- For Dummies(Publisher)
parent trig graph — the most basic, unaltered graph — makes the task of graphing more complex graphs much easier. In this chapter, you see the parent graph of each trig function and some of its transformations.Getting a Grip on Periodic Graphs
Periodic graphs are like other graphs of functions that keep going, and going, and going. But these graphs have a really special property: They keep repeating and repeating the same set of values over and over. Just remember that trig functions are periodic graphs, and the steps to graphing them will be easy! Because they repeat their values over and over again, you just need to figure out one period (or cycle), and then you can repeat it as many times as you like.The key to graphing trig functions is to graph just one period. You start by graphing the parent graph, and then perform any necessary transformations (just like you do in Chapter 3 for other types of graphs). As with other graphs, the same transformations can be applied to trig graphs:- For trig functions, vertical stretches and flattening are achieved by simply multiplying the parent function by a constant. For example, is the same as the parent graph, only its wave goes up to a value of 2 and down to . Multiplying the parent graph by a negative constant simply flips the graph upside down, or reflects it over the x -axis.
- Horizontal stretches and compressions occur by changing the period of the graph. For sine and cosine parent graphs, the period is . The same is true for cosecant and secant graphs. For tangent and cotangent graphs, the period is . Multiplying the angle in the function by a constant transforms the period. For example,
- Lawrence S. Leff, Christina Pawlowski-Polanish, Barron's Educational Series, Elizabeth Waite(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
y-coordinate represents its trigonometric value. Using the unit circle, you will be able to create a table of values for each trigonometric function to then translate onto their graph.FIGURE 10.2Plotting the points from the table of values onto the coordinate plane gives the graphs of y = cos x and y = sin x, as shown in Figures 10.3 and 10.4 . Notice the graphs have the same basic shape except that the cosine curve is out of phase with the sine curve.For both y = cos x and y = sin x:•period = 2π •amplitude = 1 •domain = (–∞,∞) •range = [–1, 1]FIGURE 10.3FIGURE 10.4FREQUENCY
The frequency of a trigonometric function is the number of cycles that its graph completes in an interval of 2π radians. Because the sine curve and the cosine curve each complete one cycle every 2π radians, the frequency of each curve is 1.EXERCISE 1If 0 ≤ x ≤ 2π, determine the interval on which the graph of y = sin x is decreasing and, at the same time, the graph of y = cos x is increasing.SOLUTIONSketch the graphs of y = sin x and y = cos x on the same set of axes, as shown in the accompanying figure. The sine curve is decreasing and the cosine curve is increasing on .AMPLITUDE AND PERIOD OF y = a sin bx AND y = a cos bx
In the equations y = a sin bx and y = a cos bx, the number a affects the amplitude and the number b determines the period. For each of these functions:•The amplitude is |a|. For example, the maximum value of y = 2 sin x is +2 and its minimum value is −2, so the amplitude of y = 2 sin x is . Figure 10.5 compares the graphs of y = sin x, y = 2 sin x, and sin x over the interval 0 ≤ x ≤ 2π.•The period is . If y = cos 2x, then b = 2, so the period is . Therefore, the graph of y = cos 2x completes one full cycle in π radians. If , then , so the period is . Figure 10.6 compares the graphs of y = cos x, y = cos 2x, and over the interval 0 ≤ x ≤ 2π. Because the period of
eBook - ePubDiscrete and Continuous Fourier Transforms
Analysis, Applications and Fast Algorithms
- Eleanor Chu(Author)
- 2008(Publication Date)
- Chapman and Hall/CRC(Publisher)
Chapter 1Analytical and Graphical Representation of Function Contents
Our objective in this chapter is to introduce the fundamental concepts and graphical tools for analyzing time-domain and frequency-domain function contents. Our initial discussion will be restricted to linear combinations of explicitly given sine and cosine functions, and we will show how the various representations of their frequency contents are connected to the Fourier series representation of periodic functions in general.1.1Time and Frequency Contents of a FunctionLet us consider a familiar trigonometric function x (t ) = 5 cos(2πt ).Byplotting x (t ) versus t over the interval 0 ≤ t ≤ 4, one obtains the following diagram.Figure 1.1 A time-domain plot of x (t ) = 5 cos(2πt ) versus t.The graph is the time-domain representation of x (t ). We observe that when t varies from 0to 1,the angle θ = 2πt goes from 0 radians to 2π radians, and the cosine function completes one cycle. The same cycle repeats for each following time intervals: t ∈ [1, 2], t ∈ [2, 3],and so on. The time it takes for a periodic function x (t ) to complete one cycle is called the period, and it is denoted by T . In this case, we have T =1 unit of time (appropriate units may be used to suit the application in hand), and x (t + T ) = x (t ) for t ≥ 0.While the function x (t ) is fully speci ed in its analytical form, the graph of x (t ) reveals how the numerical function values change with time. Since a graph is plotted from a table of pre-computed function values, the cont ents of the graph are the numbers in the table. However, compared to reading a large table of data, reading the graph is a much more convenient and effective way to s ee the trend or pattern represented by the data, the approximate locations of minimum, maximum, or zero function values. With this understanding, the time-domain (or time) content of x (t ) (in this simple case) is the graph which plots x (t ) versus t
- Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
65 Chapter 3 Graphing Trigonometric Functions EXERCISE 3.1 Basic Graphs 1. 2, 2, 3. Since sin x represents the y coordinate of a point moving on the unit circle from (1, 0), the intercepts of the graph of y = sin x occur when the point is on the x axis, that is, when the point has moved 0, , 2, … and –, –2, … Since cos x represents the x coordinate of the point, the intercepts of its graph occur when the point is on the y axis, that is, when the point has moved 2 , 3 2 , 5 2 , … and – 2 , – 3 2 , …. 5. Draw vertical asymptotes through the x intercepts of the graph of y = sin x (at x = n, n any integer). Note the points x = 2 + 2n, y = 1 and x = – 2 + 2n, y = –1. Take reciprocal values for a few points between – and 0, and between 0 and and sketch the graph between the asymptotes. 7. (A) 1 unit (B) Indefinitely far (C) Indefinitely far 9. – 3 2 , – 2 , 2 , 3 2 11. –2, –, 0, , 2 13. The graph has no x intercepts; sec x is never 0. 15. (A) None; sin x is always defined. (B) –2, –, 0, , 2 (C) – 3 2 , – 2 , 2 , 3 2 17. x 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 cos x 1 1.0 0.98 0.96 0.92 0.88 0.83 0.76 0.70 x 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 cos x 0.62 0.54 0.45 0.36 0.27 0.17 0.07 –0.03 19. 21. Exercise 3.1 Basic Graphs 66 23. The dashed line shows y = sin x in this interval. The solid line is y = csc x. 25. (A) (B) (C) The mode setting is crucial. Degree mode will make the graph totally different. 27. The range of the cosine function is the set of all real numbers between –1 and 1: –1 cos x 1 Multiply both inequalities by 5 –5 5 cos x 5 Therefore, –5 y 5 and Max y = 5, Min y = –5. 29. The range of the sine function is the set of all real numbers between –1 and 1: –1 sin x 1 Add 9 to both inequalities 9 – 1 9 + sin x 9 + 1 Simplify 8 9 + sin x 10 Therefore, 8 y 10 and Max y = 10, Min y = 8. 31. The cosecant function has no maximum and no minimum, so Max y and Min y do not exist.
eBook - PDF- Raymond A. Barnett, Michael R. Ziegler, Karl E. Byleen(Authors)
- 2012(Publication Date)
- Wiley(Publisher)
Figures 6 and 7 illustrate these phenomena. 194 3 GRAPHING TRIGONOMETRIC FUNCTIONS 1. 2. Answers to Matched Problems 2p 2p 3p p p x y y sin x y sin x sin 3x 3 y sin x sin 3x 3 sin 5x 5 d d FIGURE 6 Square wave x y y sin p x y sin p x sin 2p x 2 y sin p x sin 2p x 2 sin 3p x 3 2 1 0 1 2 3 4 q q FIGURE 7 Sawtooth wave p p 2p x y 0 5 1 0 2p 3. (A) (B) The period appears to be 2p. 4. EXERCISE 3.5 y = y 1 + y 2 + y 3 y 3 = 0.22 sin(1,320pt) y 2 = 0.28 sin(880pt) y 1 = 0.30 sin(440pt) 3.5 Graphing the Sum of Functions 195 0.6 0.6 0.008 0 0.8 0.8 0.008 0 4 4 0 4p 0.6 0.6 0.008 0 0.6 0.6 0.008 0 1. Explain how to graph the sum of two functions using addition of ordinates. 2. What is a fundamental tone? 3. How are the frequencies of overtones related to the fre- quency of the fundamental tone? 4. Explain why the frequency of a harmonic tone is equal to the frequency of the fundamental tone. In Problems 5–12, sketch the graph of each equation using addition of ordinates. 5. 6. 7. 8. y = x + sin x, 0 … x … 2p y = x + cos x, 0 … x … 5p/2 y = 1 + cos x, - p … x … p y = 1 + sin x, - p … x … p 9. 10. 11. 12. In Problems 13–20, sketch the graph of each equation using addition of ordinates. 13. 14. 15. 16. 17. 18. y = 3 cos x + cos 3x, 0 … x … 2p y = 3 cos x + sin 2x, 0 … x … 3p y = 3 sin x + 2 cos x, 0 … x … 2p y = 3 sin x + cos x, 0 … x … 2p y = sin x + 2 cos x, 0 … x … 2p y = sin x + cos x, 0 … x … 2p y = x - sin px, - 2 … x … 2 y = x - cos 2px, - 1 … x … 1 y = x/2 - sin 2px, 0 … x … 2 y = x/2 + cos px, 0 … x … 3 A B 196 3 GRAPHING TRIGONOMETRIC FUNCTIONS 19. 20. Problems 21–24 require the use of a graphing calculator. 21. Graph the following equation for and (compare to the square wave in Fig. 6, page 194). 22. Graph the following equation for and (compare to the sawtooth wave in Fig. 7, page 194).
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