SECTION 5-8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions

Transcription

1 -8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions 9 duce a scatter plot in the viewing window. Choose 8 for the viewing window. (B) It appears that a sine curve of the form k A sin (B C) After determining A, B, k, and C, write the resulting equation. (C) Plot the results of parts A and B in the same viewing window. (An improved fit ma result b adjusting our value of C slightl.) will closel model this data. The constants k, A, and B are easil determined from Table as follows: A (Ma Min )/, B /Period, k Min A. To estimate C, visuall estimate to one decimal place the smallest positive phase shift from the plot in part A. TABLE (mos.) (temp.) SECTION -8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions Graphing A tan (B C) and A cot (B C) Graphing A sec (B C) and A csc (B C) In this section the graphing of the more general forms of the tangent, cotangent, secant, and cosecant functions is discussed. Essentiall, we follow the same process we developed for graphing A sin (B C) and A cos (B C). The process is not difficult if ou have a clear understanding of the basic graphs and periodic properties for each of these functions. Graphing A tan (B C ) and A cot (B C ) For convenient reference, we repeat the graphs that were shown for tan and cot in Section -6 (see Figs. and ). Graph of tan Period: Domain: All real numbers ecept / k, k an integer Range: All real numbers Smmetric with respect to the origin Increasing function between consecutive asmptotes Discontinuous at / k, k an integer FIGURE

2 Trigonometric Functions Graph of cot Period: Domain: All real numbers ecept k, k an integer Range: All real numbers Smmetric with respect to the origin Decreasing function between consecutive asmptotes Discontinuous at k, k an integer FIGURE EXPLORE-DISCUSS (A) Match each function to its graph and discuss how the graph compares to the graph of tan or cot. () tan () tan () cot ( /) (a) (b) (c) (B) Use a graphing utilit to eplore 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. A tan and A cot for different values of A tan B and cot B for different values of B tan ( C) and cot ( C) for different values of C To quickl sketch the graphs of equations of the form A tan (B C), ou need to know how the constants A, B, and C affect the basic graphs of tan and cot, respectivel. First note that amplitude is not defined for the tangent and cotangent functions. The graphs of both deviate without end from the ais. The effect of A is to

3 -8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions A make the graph steeper if or to make the graph less steep if. If A is negative, the graph is reflected across the ais. Just as with the sine and cosine functions, the constants B and C, respectivel, effect a change in period and phase shift. Since A tan and A cot each has a period of, it follows that A tan (B C) and A cot (B C) each completes one ccle as B C varies from or (solving for ) as varies from B C to B C Phase shift Period A C B to C B B Thus, A tan (B C) and A cot (B 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, ou do not need to memorize the formulas for period and phase shift. You onl need to remember the process used to obtain the formulas. EXAMPLE Graphing an Equation of the Form A cot (B C ) Find the period and phase shift for cot (/), then sketch its graph for. Solution One ccle of cot (/) is completed as / varies from to. Solve each equation for : Phase shift Period In general, if C, there is no phase shift. The graph is sketched for one period, (, ), then etended over the interval (, ) as in Figure. FIGURE

4 Trigonometric Functions Matched Problem Find the period and phase shift for tan (/), then sketch its graph for. EXAMPLE Graphing an Equation of the Form A cot (B C ) Find the period and phase shift for cot ( /), then sketch the graph for /. Solution Step. Find the period and phase shift b solving B C and B C for : Phase shift Period FIGURE FIGURE Step. Sketch one period of the graph starting at / (the phase shift) and ending at / / (the phase shift plus one period) Figure. Step. Etend the graph over the interval (/, ) Figure. Matched Problem Find the period and phase shift for tan (/ /), then sketch the graph for. Graphing A sec (B C ) and A csc (B C ) For convenient reference, we repeat the graphs that were shown for csc and sec in Section -6 (see Figs. 6 and 7).

5 -8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions Graph of csc sin csc sin Period: Domain: All real numbers ecept k, k an integer Range: All real numbers such that or Smmetric with respect to the origin Discontinuous at k, k an integer FIGURE 6 Graph of sec cos sec cos Period: Domain: All real numbers ecept / k, k an integer Range: All real numbers such that or Smmetric with respect to the ais Discontinuous at / k, k an integer FIGURE 7 EXPLORE-DISCUSS (A) Match each function to its graph, and discuss how the graph compares to the graph of csc or sec. () csc () sec () csc ( /) (a) (b) (c)

6 Trigonometric Functions (B) Use a graphing utilit to eplore 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. A sec and A csc for different values of A sec B and csc B for different values of B sec ( C) and csc ( 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, we find the period and phase shift for each b solving B C and B C. To graph either A sec (B C) or A csc (B C), ou will probabl find it easier to graph (/A) cos (B C) or (/A) sin (B C) with a dashed curve, then take reciprocals. An eample should help to make the process clear. EXAMPLE Graphing an Equation of the Form A sec (B C ) Find the period and phase shift for / /. sec ( ), then sketch the graph for Solution Step. Find the period and phase shift b solving B C and B C for : Phase shift Period Step. Since sec ( ) cos ( ) we graph cos ( ) for one ccle from / to /, and then take reciprocals. Notice that we also place vertical asmptotes through the intercepts of the cosine graph to guide us when we sketch the secant function Figure 8.

7 -8 Graphing More General Tangent, Cotangent, Secant, and Cosecant Functions FIGURE 8 FIGURE 9 Step. Etend the graph over the required interval (/, /) Figure 9. Matched Problem Find the period and phase shift for csc (/ ), then sketch the graph for. Answers to Matched Problems. Period, phase shift. Period, phase shift

8 6 Trigonometric Functions. Period, phase shift 6 8 EXERCISE -8 A In Problems 8, find the period of each function, and graph the function for the indicated interval.. cot, /. tan,. tan 8,. cot,. csc (/), 6. sec,.. 7. sec, 8. csc (/), 8 B In Problems 9, find the period and phase shift, then graph each function. 9. cot,. tan,. tan ( ),. cot ( ),. sec,. csc, In Problems 8, determine whether the statement is true or false. If true, eplain wh. If false, give a countereample.. The graphs of cos () and csc () have infinitel man intersection points. 6. The graphs of sin () and csc () have infinitel man intersection points. 7. Ever horizontal line intersects the graph of. sec ( ) infinitel man times. 8. The maimum deviation of the graph of 7 tan ( ) from the ais is 7. In Problems 9, graph at least two ccles of the given equation in a graphing utilit, then find an equation of the form A tan B, A cot B, A sec B, or A csc B that has the same graph. (These problems suggest additional identities beond those discussed in Section -. Additional identities are discussed in detail in Chapter 6.) 9. cot tan. cot tan. csc cot. csc cot C In Problems 6, find the period and phase shift, then graph each function.. tan, 7. cot ( ),. csc, 6. sec,

9 -9 Inverse Trigonometric Functions 7 In Problems 7, graph at least two ccles of the given equation in a graphing utilit, then find an equation of the form A tan B, A cot B, A sec B, or A csc B that has the same graph. (These problems suggest additional identities beond those discussed in Section -. Additional identities are discussed in detail in Chapter 6.) 7. sin cos cot 8. cos sin tan 9.. sin cos sin 6 cos 6 (B) Graph the equation found in part A for the time interval [, ). If the graph has an asmptote, put it in. (C) Describe what happens to the length c of the light beam as t goes from to. N a c P APPLICATIONS. Motion. A beacon light ft from a wall rotates clockwise at the rate of / rps (see figure); thus, t/. (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.. Motion. Refer to Problem. (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 [, ). If the graph has an asmptote, put it in. (C) Describe what happens to the distance a along the wall as t goes from to. SECTION -9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tangent Function Summar Inverse Cotangent, Secant, and Cosecant Functions (Optional) A brief review of the general concept of inverse functions discussed in Section -6 should prove helpful before proceeding with this section. In the following bo 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 its inverse:. If (a, b) is an element of f, then (b, a) is an element of f, and conversel.. Range of f Domain of f Domain of f Range of f