Mathematics
Graphing Trigonometric Functions
Graphing trigonometric functions involves plotting the values of sine, cosine, tangent, and other trigonometric functions on a coordinate plane. These functions create periodic wave-like patterns due to their repetitive nature. Understanding the properties of these graphs, such as amplitude, period, and phase shift, is essential for analyzing and solving trigonometric equations and real-world problems.
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9 Key excerpts on "Graphing 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- 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 - PDF- David Cohen, Theodore Lee, David Sklar, , David Cohen, Theodore Lee, David Sklar(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
Even before making any calculations, we can gain strong intuitive insight into how the graph must look by carrying out the following experiment. After drawing the unit circle, x 2 y 2 1, place your fingertip at the point (1, 0) and then move your finger counterclockwise around the circle. As you do this, keep track of what happens to the y -coordinate of your fingertip. (The y -coordinate is sin u .) If we think of u as the radian measure of an angle, the y -coordinate of your fingertip is sin u . 4 ( 4) 2 4, 544 CHAPTER 8 Graphs of the Trigonometric Functions 6800 K 5400 K 5.367 days Temp. Time x Figure 2 The surface temperature of the star Delta Cephei is a periodic function of time. The period is 5.367 days. The amplitude is (6800 5400) 2 700 degrees Kelvin. (a) Voltage (mV) (b) Diastolic Systolic 1.0 Arterial pressure (mm Hg) 120 100 80 .5 0 -.5 Figure 3 Electrical activity of the heart and blood pressure as periodic functions of time. The figure shows (a) a typical ECG (electrocardiogram) and (b) the corresponding graph of arterial blood pressure. [Adapted from Physics for the Health Sciences by C. R. Nave and B. C. Nave (Philadelphia: W. B. Saunders Co., 1975)] Copyright 201 Cengage Learning. 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 545 Figure 5 tells us a great deal about the sine function: where the function is in-creasing and decreasing, where the graph crosses the x -axis, and where the high and low points of the graph occur.
- Lawrence S. Leff, Christina Pawlowski-Polanish, Barron's Educational Series, Elizabeth Waite(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
x = π.GRAPHING A TRIGONOMETRIC FUNCTION WITH A CALCULATOR
To graph a trigonometric function using your graphing calculator, set the angular mode to radians, enter the function, and then press . The graph will be displayed in the interval –2π < x < 2π using the following preset values:Because Xscl is set to , each tic mark on the x-axis is an integer multiple of . If you need to view the graph in a different interval of x, press and selectively change the values of Xmin, Xmax, and Xscl.FIGURE 10.9 Graph of y = 3 sin 2x (–π < x < π)For example, to graph y = 3 sin 2x over the interval –π < x < π, pressDivide Xmin, Xmax, and Xscl by 2 so that consecutive tic marks are radian apart in the interval –π < x < π. Because the period of y = 3 sin 2x is π radians, one full cycle will be displayed on either side of the y-axis, as shown in Figure 10.9 .GRAPHING y = tan x
The graph of the tangent function looks very different from the graphs of the sine and cosine functions. The function y = tan x is not defined at or at any odd-integer multiple of , so the graph has vertical asymptotes through these x-values, as shown in Figure 10.10 .Here are key facts about the tangent function that you can observe from the graph: •period = π •amplitude: none•domain = {real numbers except odd multiples of•range = (−∞, ∞)•x-intercepts: at integer multiples of π•vertical asymptotes: at odd-integer multiples of
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,
- 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- Cynthia Y. Young(Author)
- 2017(Publication Date)
- Wiley(Publisher)
STUDY TIP The graph of 2f (x) is the graph of f (x) reflected about the x-axis. 4.2 Translations of the Sine and Cosine Functions: Addition of Ordinates 185 186 CHAPTER 4 Graphing Trigonometric Functions VERTICAL TRANSLATIONS (SHIFTS) OF SINUSOIDAL FUNCTIONS To graph sinusoidal functions of the form y 5 A sin 1Bx2 6 k or y 5 A cos 1Bx2 6 k, where k . 0, start with the graphs of y 5 A sin 1Bx2 or y 5 A cos 1Bx2 and shift them up 112 or down 122 k units. [ CONCEPT CHECK] In the graph of y 5 k 1 A cos(Bx), if |A| . 1 then the sinusoidal graph is stretched vertically and depending on the sign of k is shifted vertically by k units: (A) if k . 0 the graph is shifted up (B) if k . 0 the graph is shifted down. ANSWER A ▼ EXAMPLE 1 Graphing Functions of the Form y 5 k 1 A sin (Bx ) Graph y 5 23 1 2 sin 1 px 2 , 22 # x # 2. Solution: STEP 1 Graph y 5 2 sin 1 px 2 over one period. The amplitude is 2, the period is 2p p 5 2. STUDY TIP Note that the graphs of f (x) 5 cos(2x) and f (x) 5 cos x coincide. x y 1.0 –1.0 0.5 –0.5 –2π –π π 2π f (x) = cos(–x) = cos(x) Graph y 5 2 cos 1 px 2 . Reflect around the x-axis. Vertical Shifts Recall that graphing functions using vertical shifts occur in the following way 1k . 02: ■ ■ To graph ƒ1 x 2 1 k, shift the graph of ƒ1 x 2 up k units. ■ ■ To graph ƒ1 x 2 2 k, shift the graph of ƒ1 x 2 down k units. Therefore, functions like y 5 k 1 A sin 1 Bx 2 or y 5 k 1 A cos 1 Bx 2 are graphed by shifting the graphs of y 5 A sin 1 Bx 2 or y 5 A cos 1 Bx 2 vertically (up or down) k units. Although we found with reflection that it was just as easy to plot points and graph as opposed to first graphing one function and then reflecting about the x-axis, in this case it is much easier to first graph the simpler function and then perform a vertical shift. –2 –1 1 2 –2 –1 1 2 x y –2 –1 1 2 –2 –1 1 2 x y 2cos(π x) y 2cos(π x) y Divide 2 into four equal parts. The step size is 1 2 .
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