4.6 Graphs of Other Trigonometric Functions

Size: px

Start display at page:

Download "4.6 Graphs of Other Trigonometric Functions"

Antonia Dean
6 years ago
Views:

.6 Graphs of Other Trigonometric Functions Section.6 Graphs of Other Trigonometric Functions 09 Graph of the Tangent Function Recall that the tangent function is odd. That is, tan tan.

1 .6 Graphs of Other Trigonometric Functions Section.6 Graphs of Other Trigonometric Functions 09 Graph of the Tangent Function Recall that the tangent function is odd. That is, tan tan. Consequentl, the graph of tan is smmetric with respect to the origin. You also know from the identit tan sin cos that the tangent function is undefined at values at which cos 0. Two such values are ± ±.708. tan Undef Undef. tan approaches as tan approaches as approaches from the right. approaches from the left What ou should learn Sketch the graphs of tangent functions. Sketch the graphs of cotangent functions. Sketch the graphs of secant and cosecant functions. Sketch the graphs of damped trigonometric functions. Wh ou should learn it You can use tangent, cotangent, secant, and cosecant functions to model real-life data. For instance, Eercise 6 on page 8 shows ou how a tangent function can be used to model and analze the distance between a television camera and a parade unit. As indicated in the table, tan increases without bound as approaches from the left, and it decreases without bound as approaches from the right. So, the graph of tan has vertical asmptotes at and, as shown in Figure.. Moreover, because the period of the tangent function is, vertical asmptotes also occur at n, where n is an integer. The domain of the tangent function is the set of all real numbers other than n, and the range is the set of all real numbers. = tan Figure. Domain: all real numbers, ecept n Range:, Vertical asmptotes: n Sketching the graph of a tanb c is similar to sketching the graph of a sinb c in that ou locate ke points that identif the intercepts and asmptotes. Two consecutive asmptotes can be found b solving the equations b c and b c. The midpoint between two consecutive asmptotes is an -intercept of the graph. The period of the function a tanb c is the distance between two consecutive asmptotes. The amplitude of a tangent function is not defined. After plotting the asmptotes and the -intercept, plot a few additional points between the two asmptotes and sketch one ccle. Finall, sketch one or two additional ccles to the left and right. A. Rame/PhotoEdit

2 0 Chapter Trigonometric Functions Eample Sketching the Graph of a Tangent Function Sketch the graph of tan b hand. B solving the equations and, ou can see that two consecutive asmptotes occur at and. Between these two asmptotes, plot a few points, including the -intercept, as shown in the table. Three ccles of the graph are shown in Figure.6. Use a graphing utilit to confirm this graph. tan Undef. 0 Undef. 0 Figure.6 Now tr Eercise. Eample Sketching the Graph of a Tangent Function Sketch the graph of tan b hand. B solving the equations and, ou can see that two consecutive asmptotes occur at and. Between these two asmptotes, plot a few points, including the -intercept, as shown in the table. Three complete ccles of the graph are shown in Figure tan Undef. 0 Undef. TECHNOLOGY TIP Your graphing utilit ma connect parts of the graphs of tangent, cotangent, secant, and cosecant functions that are not supposed to be connected. So, in this tet, these functions are graphed on a graphing utilit using the dot mode. A blue curve is placed behind the graphing utilit s displa to indicate where the graph should appear. For instructions on how to use the dot mode, see Appendi A; for specific kestrokes, go to this tetbook s Online Stud Center. Figure.7 Now tr Eercise 7.

3 Section.6 Graphs of Other Trigonometric Functions TECHNOLOGY TIP Graphing utilities are helpful in verifing sketches of trigonometric functions. You can use a graphing utilit set in radian and dot modes to graph the function tan from Eample, as shown in Figure.8. You can use the zero or root feature or the zoom and trace features to approimate the ke points of the graph. = tan B comparing the graphs in Eamples and, ou can see that the graph of a tanb c increases between consecutive vertical asmptotes when a > 0 and decreases between consecutive vertical asmptotes when a < 0. In other words, the graph for a < 0 is a reflection in the -ais of the graph for a > 0. Graph of the Cotangent Function The graph of the cotangent function is similar to the graph of the tangent function. It also has a period of. However, from the identit cot cos sin ou can see that the cotangent function has vertical asmptotes when sin is zero, which occurs at n, where n is an integer. The graph of the cotangent function is shown in Figure.9. Eample Sketching the Graph of a Cotangent Function Sketch the graph of cot b hand. To locate two consecutive vertical asmptotes of the graph, solve the equations 0 and to see that two consecutive asmptotes occur at 0 and. Then, between these two asmptotes, plot a few points, including the -intercept, as shown in the table. Three ccles of the graph are shown in Figure.60. Use a graphing utilit to confirm this graph. Enter the function as tan. Note that the period is, the distance between consecutive asmptotes. Figure.8 = cot Domain: all real numbers, ecept n Range:, Vertical asmptotes: n Figure cot Undef. 0 Undef. Now tr Eercise. Figure.60 Eploration Use a graphing utilit to graph the functions cos and sec cos in the same viewing window. How are the graphs related? What happens to the graph of the secant function as approaches the zeros of the cosine function?

4 Chapter Trigonometric Functions Graphs of the Reciprocal Functions The graphs of the two remaining trigonometric functions can be obtained from the graphs of the sine and cosine functions using the reciprocal identities csc and sec sin cos. For instance, at a given value of, the -coordinate for sec is the reciprocal of the -coordinate for cos. Of course, when cos 0, the reciprocal does not eist. Near such values of, the behavior of the secant function is similar to that of the tangent function. In other words, the graphs of tan sin cos and have vertical asmptotes at n, where n is an integer (i.e., the values at which the cosine is zero). Similarl, cot cos sin and sec cos csc sin have vertical asmptotes where sin 0 that is, at n. To sketch the graph of a secant or cosecant function, ou should first make a sketch of its reciprocal function. For instance, to sketch the graph of csc, first sketch the graph of sin. Then take the reciprocals of the -coordinates to obtain points on the graph of csc. You can use this procedure to obtain the graphs shown in Figure.6. Prerequisite Skills To review the reciprocal identities of trigonometric functions, see Section.. = sin = csc = cos = sec Domain: all real numbers, ecept n Range:, ] [, Vertical asmptotes: Smmetr: origin Figure.6 n Domain: all real numbers, ecept n Range:, ] [, Vertical asmptotes: n Smmetr: -ais In comparing the graphs of the cosecant and secant functions with those of the sine and cosine functions, note that the hills and valles are interchanged. For eample, a hill (or maimum point) on the sine curve corresponds to a valle (a local minimum) on the cosecant curve, and a valle (or minimum point) on the

5 Section.6 Graphs of Other Trigonometric Functions sine curve corresponds to a hill (a local maimum) on the cosecant curve, as shown in Figure.6. Additionall, -intercepts of the sine and cosine functions become vertical asmptotes of the cosecant and secant functions, respectivel (see Figure.6). Eample Comparing Trigonometric Graphs Use a graphing utilit to compare the graphs of sin and The two graphs are shown in Figure.6. Note how the hills and valles of the graphs are related. For the function sin, the amplitude is and the period is. B solving the equations and 0 ou can see that one ccle of the sine function corresponds to the interval from to 7. The graph of this sine function is represented b the thick curve in Figure.6. Because the sine function is zero at the endpoints of this interval, the corresponding cosecant function csc sin has vertical asmptotes at,, 7, and so on. Now tr Eercise. csc. Cosecant local minimum Figure.6 Figure.6 Sine maimum Cosecant local maimum ( Sine minimum = sin + = csc ( + ( ( Eample Comparing Trigonometric Graphs Use a graphing utilit to compare the graphs of cos and sec. Begin b graphing cos and sec cos in the same viewing window, as shown in Figure.6. Note that the -intercepts of cos = cos, 0,, 0,, 0,... correspond to the vertical asmptotes,,,... of the graph of sec. Moreover, notice that the period of cos and sec is. Figure.6 = sec Now tr Eercise 7.

6 Chapter Trigonometric Functions Damped Trigonometric Graphs A product of two functions can be graphed using properties of the individual functions. For instance, consider the function f sin as the product of the functions and Using properties of absolute value and the fact that ou have 0 sin. Consequentl, sin. sin, sin which means that the graph of f sin lies between the lines and. Furthermore, because f sin ± at n and f sin 0 at n the graph of f touches the line or the line at n and has -intercepts at n. A sketch of f is shown in Figure.6. In the function f sin, the factor is called the damping factor. = Figure.6 = f() = sin Eample 6 Analze the graph of Analzing a Damped Sine Curve Consider f as the product of the two functions e and sin each of which has the set of real numbers as its domain. For an real number, ou know that and sin. So, e sin e 0 e, which means that Furthermore, because and f e sin. e e sin e. f e sin ±e at f e at n sin 0 the graph of f touches the curves e and e at 6 n and has intercepts at n. The graph is shown in Figure.66. Now tr Eercise. 6 n STUDY TIP Do ou see wh the graph of f sin touches the lines ± at n and wh the graph has -intercepts at n? Recall that the sine function is equal to ± at,,,... (odd multiples of ) and is equal to 0 at,,,... (multiples of ). f() = e sin 6 = e = e 6 Figure.66

7 Figure.67 summarizes the si basic trigonometric functions. Section.6 Graphs of Other Trigonometric Functions = sin = cos Domain: all real numbers Range: [, ] Domain: all real numbers Range: [, ] = tan = cot = tan Domain: all real numbers, Range: ecept n, Domain: all real numbers, ecept n Range:, = csc = sin = sec = cos Domain: all real numbers, ecept n Range:, ] [, Figure.67 Domain: all real numbers, ecept n Range:, ] [,

8 6 Chapter Trigonometric Functions.6 Eercises See for worked-out solutions to odd-numbered eercises. Vocabular Check Fill in the blanks.. The graphs of the tangent, cotangent, secant, and cosecant functions have asmptotes.. To sketch the graph of a secant or cosecant function, first make a sketch of its function.. For the function f g sin, g is called the factor of the function. Librar of Parent Functions In Eercises, use the graph of the function to answer the following. (a) Find all -intercepts of the graph of f. (b) Find all -intercepts of the graph of f. (c) Find the intervals on which the graph f is increasing and the intervals on which the graph f is decreasing. (d) Find all relative etrema, if an, of the graph of f. (e) Find all vertical asmptotes, if an, of the graph of f.. f tan. f cot 7. tan 8. tan 9. sec 0. sec. csc. csc. cot. cot In Eercises 0, use a graphing utilit to graph the function (include two full periods). Graph the corresponding reciprocal function and compare the two graphs. Describe our viewing window.. csc 6. csc 7. sec 8. sec csc sec. f sec. f csc In Eercises, use a graph of the function to approimate the solution to the equation on the interval [,.. tan. cot. sec. csc In Eercises, sketch the graph of the function. (Include two full periods.) Use a graphing utilit to verif our result.. tan 6. tan 7. tan 8. tan 9. sec 0. sec. sec. sec. csc. csc. 6. cot cot In Eercises 8, use the graph of the function to determine whether the function is even, odd, or neither.. f sec 6. f tan 7. f csc 8. f cot In Eercises 9, use a graphing utilit to graph the two equations in the same viewing window. Use the graphs to determine whether the epressions are equivalent. Verif the results algebraicall. 9. sin csc, 0. sin sec, tan. cos sin, cot. sec, tan

9 Section.6 Graphs of Other Trigonometric Functions 7 In Eercises 6, match the function with its graph. Describe the behavior of the function as approaches zero. [The graphs are labeled (a), (b), (c), and (d).] (a) (c) (b) (d). f cos.. 6. Conjecture In Eercises 7 0, use a graphing utilit to graph the functions f and g. Use the graphs to make a conjecture about the relationship between the functions g sin f sin, f cos, f sin cos f sin cos In Eercises, use a graphing utilit to graph the function and the damping factor of the function in the same viewing window. Describe the behavior of the function as increases without bound.. f e cos. f e sin. h e cos. g e sin,, Eploration In Eercises and 6, use a graphing utilit to graph the function. Use the graph to determine the behavior of the function as c. (a) as approaches from the right (b) as approaches from the left (c) as approaches from the right (d) as approaches from the left. f tan 6. f sec g 0 g cos g sin g cos f sin g cos Eploration In Eercises 7 and 8, use a graphing utilit to graph the function. Use the graph to determine the behavior of the function as c. (a) As 0, the value of f. (b) As 0, the value of f. (c) As, the value of f. (d) As, the value of f. 7. f cot 8. f csc 9. Predator-Pre Model The population P of cootes (a predator) at time t (in months) in a region is estimated to be P 0, sin t and the population p of rabbits (its pre) is estimated to be p, cos t. Use the graph of the models to eplain the oscillations in the size of each population. Population 0,000,000 0,000, Time (in months) 60. Meteorolog The normal monthl high temperatures H (in degrees Fahrenheit) for Erie, Pennslvania are approimated b t t Ht. 0.8 cos.69 sin 6 6 and the normal monthl low temperatures L are approimated b t t Lt cos.6 sin 6 6 p P where t is the time (in months), with t corresponding to Januar. (Source: National Oceanic and Atmospheric Association) (a) Use a graphing utilit to graph each function. What is the period of each function? (b) During what part of the ear is the difference between the normal high and normal low temperatures greatest? When is it smallest? (c) The sun is the farthest north in the sk around June, but the graph shows the warmest temperatures at a later date. Approimate the lag time of the temperatures relative to the position of the sun. t

10 8 Chapter Trigonometric Functions 6. Distance A plane fling at an altitude of miles over level ground will pass directl over a radar antenna (see figure). Let d be the ground distance from the antenna to the point directl under the plane and let be the angle of elevation to the plane from the antenna. ( d is positive as the plane approaches the antenna.) Write d as a function of and graph the function over the interval 0 < <. 6. Numerical and Graphical Reasoning A crossed belt connects a 0-centimeter pulle on an electric motor with a 0-centimeter pulle on a saw arbor (see figure). The electric motor runs at 700 revolutions per minute. 0 cm 0 cm mi φ d 6. Television Coverage A television camera is on a reviewing platform 6 meters from the street on which a parade will be passing from left to right (see figure). Write the distance d from the camera to a particular unit in the parade as a function of the angle, and graph the function over the interval < <. (Consider as negative when a unit in the parade approaches from the left.) Not drawn to scale Not drawn to scale (a) Determine the number of revolutions per minute of the saw. (b) How does crossing the belt affect the saw in relation to the motor? (c) Let L be the total length of the belt. Write L as a function of, where is measured in radians. What is the domain of the function? (Hint: Add the lengths of the straight sections of the belt and the length of belt around each pulle.) (d) Use a graphing utilit to complete the table. 6 m d L Camera 6. Harmonic Motion An object weighing W pounds is suspended from a ceiling b a steel spring (see figure). The weight is pulled downward (positive direction) from its equilibrium position and released. The resulting motion of the weight is described b the function et cos t, where is the distance in feet and t is the time in seconds t > 0. Equilibrium (a) Use a graphing utilit to graph the function. (b) Describe the behavior of the displacement function for increasing values of time t. (e) As increases, do the lengths of the straight sections of the belt change faster or slower than the lengths of the belts around each pulle? (f) Use a graphing utilit to graph the function over the appropriate domain. Snthesis True or False? In Eercises 6 and 66, determine whether the statement is true or false. Justif our answer. 6. The graph of has an asmptote at. 8 tan 66. For the graph of sin, as approaches, approaches Graphical Reasoning Consider the functions f sin and g csc on the interval 0,. (a) Use a graphing utilit to graph f and g in the same viewing window.

11 Section.6 Graphs of Other Trigonometric Functions 9 (b) Approimate the interval in which f > g. (c) Describe the behavior of each of the functions as approaches. How is the behavior of g related to the behavior of f as approaches? 68. Pattern Recognition (a) Use a graphing utilit to graph each function. sin sin sin sin sin (b) Identif the pattern in part (a) and find a function that continues the pattern one more term. Use a graphing utilit to graph. (c) The graphs in parts (a) and (b) approimate the - periodic function in the figure. Find a function that is a better approimation. Eploration In Eercises 69 and 70, use a graphing utilit to eplore the ratio f, which appears in calculus. (a) Complete the table. Round our results to four decimal places f f (b) Use a graphing utilit to graph the function f. Use the zoom and trace features to describe the behavior of the graph as approaches 0. (c) Write a brief statement regarding the value of the ratio based on our results in parts (a) and (b). 69. f tan tan 70. f Librar of Parent Functions In Eercises 7 and 7, determine which function is represented b the graph. Do not use a calculator (a) f tan (b) f tan (b) f csc (c) f tan (c) f csc (d) f tan (d) f sec (e) f tan (e) f csc 7. Approimation Using calculus, it can be shown that the tangent function can be approimated b the polnomial tan! where is in radians. Use a graphing utilit to graph the tangent function and its polnomial approimation in the same viewing window. How do the graphs compare? 7. Approimation Using calculus, it can be shown that the secant function can be approimated b the polnomial sec!! where is in radians. Use a graphing utilit to graph the secant function and its polnomial approimation in the same viewing window. How do the graphs compare? Skills Review 6! In Eercises 7 78, identif the rule of algebra illustrated b the statement. 7. a 9 a a b 0 a b 0 In Eercises 79 8, determine whether the function is one-to-one. If it is, find its inverse function. 79. f f 7 8. f 8. f (a) f sec

Essential Question What are the characteristics of the graph of the tangent function?

8.5 Graphing Other Trigonometric Functions Essential Question What are the characteristics of the graph of the tangent function? Graphing the Tangent Function Work with a partner. a. Complete the table