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419 5-8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions After determining A, B, k, and C, write the resulting equation. duce a scatter plot in the viewing window. Choose 0 y 80 for the viewing window. (C) Plot the results of parts A and B in the same viewing window. (An improved fit may result by adjusting your value of C slightly.) (B) It appears that a sine curve of the form y k A sin (Bx C) TABLE 2 will closely model this data. The constants k, A, and B are easily determined from Table 2 as follows: A (Max y Min y)/2, B 2/Period, k Min y A. To estimate C, visually estimate to one decimal place the smallest positive phase shift from the plot in part A. SECTION 5-8 x (mos.) 1 2 3 4 5 6 7 8 9 10 11 12 y (temp.) 31 34 43 53 62 71 76 74 67 55 45 35 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions • Graphing y A tan (Bx C ) and y A cot (Bx C ) • Graphing y A sec (Bx C ) and y A csc (Bx C ) In this section the graphing of the more general forms of the tangent, cotangent, secant, and cosecant functions is discussed. Essentially, we follow the same process we developed for graphing y A sin (Bx C ) and y A cos (Bx C ). The process is not difficult if you have a clear understanding of the basic graphs and periodic properties for each of these functions. • Graphing y A tan (Bx C ) and y A cot (Bx C ) For convenient reference, we repeat the graphs that were shown for y tan x and y cot x in Section 5-6 (see Figs. 1 and 2). Graph of y tan x y 2 5 2 FIGURE 1 1 3 2 2 0 1 2 2 3 2 5 2 x Period: Domain: All real numbers except /2 k, k an integer Range: All real numbers Symmetric with respect to the origin Increasing function between consecutive asymptotes Discontinuous at x /2 k, k an integer 420 5 Trigonometric Functions Graph of y cot x y 1 2 3 2 2 0 1 2 x 3 2 2 5 2 3 Period: Domain: All real numbers except k, k an integer Range: All real numbers Symmetric with respect to the origin Decreasing function between consecutive asymptotes Discontinuous at x k, k an integer FIGURE 2 EXPLORE-DISCUSS 1 (A) Match each function to its graph and discuss how the graph compares to the graph of y tan x or y cot x. (1) y 4 tan x (2) y tan 2x (3) y cot (x /2) y y y 10 10 10 5 5 5 0 x 0 x 5 5 0 x 5 10 10 (a) (b) (c) (B) Use a graphing utility to explore the nature of the changes in the graphs of the following functions when the values of A, B, and C are changed. Discuss what happens in each case. y A tan x and y A cot x for different values of A y tan Bx and y cot Bx for different values of B y tan (x C ) and y cot (x C ) for different values of C To quickly sketch the graphs of equations of the form y A tan (Bx C ), you need to know how the constants A, B, and C affect the basic graphs of y tan x and y cot x, respectively. First note that amplitude is not defined for the tangent and cotangent functions. The graphs of both deviate without end from the x axis. The effect of A is to 5-8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions 421 make the graph steeper if A 1 or to make the graph less steep if A 1. If A is negative, the graph is reflected across the x axis. Just as with the sine and cosine functions, the constants B and C, respectively, effect a change in period and phase shift. Since A tan x and A cot x each has a period of , it follows that A tan (Bx C ) and A cot (Bx C ) each completes one cycle as Bx C varies from Bx C 0 to Bx C or (solving for x) as x varies from ↓ x C B Phase shift ↓ x to Period ↓ C B B Thus, y A tan (Bx C ) and y A cot (Bx C ) each has a period of /B and a phase shift of C/B. The basic graph is shifted to the right if C/B is positive and to the left if C/B is negative. As before, you do not need to memorize the formulas for period and phase shift. You only need to remember the process used to obtain the formulas. EXAMPLE 1 Graphing an Equation of the Form y A cot (Bx C ) Find the period and phase shift for y 2 cot (x/2), then sketch its graph for 2 x 2. Solution One cycle of y 2 cot (x/2) is completed as x/2 varies from 0 to . Solve each equation for x: x 0 2 x 2 x0 x 0 2 Phase shift 0 Period 2 In general, if C 0, there is no phase shift. The graph is sketched for one period, (0, 2), then extended over the interval (2, 2) as in Figure 3. FIGURE 3 y 2 0 y 2 x 2 0 2 x 422 5 Trigonometric Functions Matched Problem 1 EXAMPLE 2 Find the period and phase shift for y 3 tan (x/2), then sketch its graph for 3 x 3. Graphing an Equation of the Form y A cot (Bx C ) Find the period and phase shift for y cot (2x /2), then sketch the graph for /2 x . Solution Step 1. Find the period and phase shift by solving Bx C 0 and Bx C for x: 2x y 4 4 2 2 2x 2 x 4 x 4 2 Phase shift 4 x 5 FIGURE 4 2x 2x 5 0 2 Period 2 Step 2. Sketch one period of the graph starting at x /4 (the phase shift) and ending at x /4 /2 (the phase shift plus one period)—Figure 4. Step 3. Extend the graph over the interval (/2, )—Figure 5. y FIGURE 5 5 2 4 4 2 3 4 x 5 Matched Problem 2 Find the period and phase shift for y tan (x/2 /4), then sketch the graph for 3 x 3. • Graphing y A sec (Bx C ) and y A csc (Bx C ) For convenient reference, we repeat the graphs that were shown for y csc x and y sec x in Section 5-6 (see Figs. 6 and 7). 423 5-8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions Graph of y csc x y y csc x Period: 2 Domain: All real numbers except k, k an integer Range: All real numbers y such that y 1 or y 1 Symmetric with respect to the origin Discontinuous at x k, k an integer 1 sin x y sin x 2 3 2 2 3 2 1 0 1 2 x 2 FIGURE 6 Graph of y sec x y y sec x Period: 2 Domain: All real numbers except /2 k, k an integer Range: All real numbers y such that y 1 or y 1 Symmetric with respect to the y axis Discontinuous at x /2 k, k an integer 1 cos x y cos x 1 2 3 2 2 0 1 2 3 2 x 2 FIGURE 7 EXPLORE-DISCUSS 2 (A) Match each function to its graph, and discuss how the graph compares to the graph of y csc x or y sec x. (1) y 21 csc x (2) y sec x (3) y csc (x /2) y y 5 y 5 3 2 1 1 2 0 x 1 2 3 2 2 0 2 x 3 2 1 2 5 5 (a) 3 (b) (c) x 2 424 5 Trigonometric Functions (B) Use a graphing utility to explore the nature of the changes in the graphs of the following functions when the values of A, B, and C are changed. Discuss what happens in each case. y A sec x and y A csc x for different values of A y sec Bx and y csc Bx for different values of B y sec (x C ) and y csc (x C ) for different values of C As with the tangent and cotangent functions, amplitude is not defined for either the secant or the cosecant functions. Since both functions have a period of 2, we find the period and phase shift for each by solving Bx C 0 and Bx C 2. To graph either y A sec (Bx C ) or y A csc (Bx C ), you will probably find it easier to graph y (1/A) cos (Bx C ) or y (1/A) sin (Bx C ) with a dashed curve, then take reciprocals. An example should help to make the process clear. EXAMPLE 3 Graphing an Equation of the Form y A sec (Bx C ) Find the period and phase shift for y 3/4 x 3/4. Solution 1 2 sec (2x ), then sketch the graph for Step 1. Find the period and phase shift by solving Bx C 0 and Bx C 2 for x: 2x 0 2x 2 2x 2x 2 x 2 x Phase shift 2 Period 2 Step 2. Since 1 1 sec (2x ) 2 2 cos (2x ) we graph y 2 cos (2x ) for one cycle from /2 to /2 , and then take reciprocals. Notice that we also place vertical asymptotes through the x intercepts of the cosine graph to guide us when we sketch the secant function—Figure 8. 425 5-8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions FIGURE 8 y 4 3 2 1 2 2 1 x 3 4 Step 3. Extend the graph over the required interval (3/4, 3/4)—Figure 9. y FIGURE 9 4 3 2 1 2 x 2 1 3 4 Matched Problem 3 Find the period and phase shift for y 2 csc (x/2 ), then sketch the graph for 2 x 10. Answers to Matched Problems 2. Period 2, phase shift 21 1. Period 2, phase shift 0 y y 6 3 2 6 x 1 1 6 2 3 1 3 1 2 x 2 6 3 426 5 Trigonometric Functions 3. Period 4, phase shift 2 y 1 2 EXERCISE x 1 2 4 6 8 10 5-8 A In Problems 15–18, determine whether the statement is true or false. If true, explain why. If false, give a counterexample. In Problems 1–8, find the period of each function, and graph the function for the indicated interval. 15. The graphs of y cos (x) and y csc (x) have infinitely many intersection points. 1. y 2 cot 4x, 0 x /2 2. y 3 tan 2x, x 3. y 14 tan 8x, 0 x 21 4. y 12 cot 2x, 0 x 1 5. y csc (x/2), 3 x 3 6. y sec x, 1.5 x 3.5 7. y 2 sec x, 1 x 3 8. y 2 csc (x/2), 0 x 8 B In Problems 9–14, find the period and phase shift, then graph each function. 2 , 2 x 32 10. y tan x , x 2 16. The graphs of y sin (x) and y csc (x) have infinitely many intersection points. 17. Every horizontal line intersects the graph of y 0.1 sec (5x 1) infinitely many times. 18. The maximum deviation of the graph of y 7 tan (3x 2) from the x axis is 7. In Problems 19–22, graph at least two cycles of the given equation in a graphing utility, then find an equation of the form y A tan Bx, y A cot Bx, y A sec Bx, or y A csc Bx that has the same graph. (These problems suggest additional identities beyond those discussed in Section 5-2. Additional identities are discussed in detail in Chapter 6.) 19. y cot x tan x 20. y cot x tan x 21. y csc x cot x 22. y csc x cot x 9. y cot x 3 3 x 11. y tan (2x ), 4 4 12. y cot (2x ), x 2 2 14. y csc x , 1 x 1 2 13. y sec x , 1 x 1 2 C In Problems 23–26, find the period and phase shift, then graph each function. 23. y 2 tan 4 x 4 , 1 x 7 24. y 3 cot (x ), 2 x 2 2 x 2 , 1 x 3 26. y 2 sec x , 1 x 3 2 25. y 3 csc 5-9 Inverse Trigonometric Functions (B) Graph the equation found in part A for the time interval [0, 1). If the graph has an asymptote, put it in. In Problems 27–30, graph at least two cycles of the given equation in a graphing utility, then find an equation of the form y A tan Bx, y A cot Bx, y A sec Bx, or y A csc Bx that has the same graph. (These problems suggest additional identities beyond those discussed in Section 5-2. Additional identities are discussed in detail in Chapter 6.) (C) Describe what happens to the length c of the light beam as t goes from 0 to 1. 27. y sin 3x cos 3x cot 3x P 28. y cos 2x sin 2x tan 2x c 20 sin 6x 1 cos 6x ★ APPLICATIONS ★ a N sin 4x 29. y 1 cos 4x 30. y 427 31. Motion. A beacon light 20 ft from a wall rotates clockwise at the rate of 1/4 rps (see figure); thus, t/2. (A) Start counting time in seconds when the light spot is at N and write an equation for the length c of the light beam in terms of t. SECTION 5-9 32. Motion. Refer to Problem 31. (A) Write an equation for the distance a the light spot travels along the wall in terms of time t. (B) Graph the equation found in part A for the time interval [0, 1). If the graph has an asymptote, put it in. (C) Describe what happens to the distance a along the wall as t goes from 0 to 1. Inverse Trigonometric Functions • • • • • Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summary Inverse Cotangent, Secant, and Cosecant Functions (Optional) A brief review of the general concept of inverse functions discussed in Section 2-6 should prove helpful before proceeding with this section. In the following box we restate a few important facts about inverse functions from that section. Facts about Inverse Functions For f a one-to-one function and f 1 its inverse: 1. If (a, b) is an element of f, then (b, a) is an element of f 1, and conversely. 2. Range of f Domain of f 1 Domain of f Range of f 1