0_007.qd /7/05 :0 AM Page Section.7.7 Inverse Trigonometric Functions Inverse Trigonometric Functions What ou should learn Evaluate and graph the inverse sine function. Evaluate and graph the other inverse trigonometric functions. Evaluate and graph the compositions of trigonometric functions. Inverse Sine Function Recall from Section.9 that, for a function to have an inverse function, it must be one-to-one that is, it must pass the Horizontal Line Test. From Figure.7, ou can see that sin does not pass the test because different values of ield the same -value. = sin Wh ou should learn it You can use inverse trigonometric functions to model and solve real-life problems. For instance, in Eercise 9 on page 5, an inverse trigonometric function can be used to model the angle of elevation from a television camera to a space shuttle launch. sin has an inverse function on this interval. FIGURE.7 However, if ou restrict the domain to the interval (corresponding to the black portion of the graph in Figure.7), the following properties hold.. On the interval,, the function sin is increasing.. On the interval,, sin takes on its full range of values, sin.. On the interval,, sin is one-to-one. So, on the restricted domain, sin has a unique inverse function called the inverse sine function. It is denoted b arcsin NASA or sin. The notation sin is consistent with the inverse function notation f. The arcsin notation (read as the arcsine of ) comes from the association of a central angle with its intercepted arc length on a unit circle. So, arcsin means the angle (or arc) whose sine is. Both notations, arcsin and sin, are commonl used in mathematics, so remember that sin denotes the inverse sine function rather than sin. The values of arcsin lie in the interval arcsin. The graph of arcsin is shown in Eample. Definition of Inverse Sine Function The inverse sine function is defined b When evaluating the inverse sine function, it helps to remember the phrase the arcsine of is the angle (or number) whose sine is. arcsin if and onl if sin where and. The domain of arcsin is,, and the range is,.
0_007.qd /7/05 :0 AM Page Chapter Trigonometr Eample Evaluating the Inverse Sine Function As with the trigonometric functions, much of the work with the inverse trigonometric functions can be done b eact calculations rather than b calculator approimations. Eact calculations help to increase our understanding of the inverse functions b relating them to the right triangle definitions of the trigonometric functions. You ma wish to illustrate the reflections of sin and arcsin about the line. Consider using a graphing utilit to do this. ( ), ( ), ( ), FIGURE.7 (0, 0) (, ) ( ), = arcsin (, ) If possible, find the eact value. a. b. sin c. sin arcsin a. Because sin for it follows that, arcsin Angle whose sine is b. Because sin for it follows that, sin.. Angle whose sine is c. It is not possible to evaluate sin when because there is no angle whose sine is. Remember that the domain of the inverse sine function is,. Now tr Eercise. Eample Graphing the Arcsine Function Sketch a graph of arcsin. B definition, the equations arcsin and sin are equivalent for. So, their graphs are the same. From the interval,, ou can assign values to in the second equation to make a table of values. Then plot the points and draw a smooth curve through the points. 0 sin The resulting graph for arcsin is shown in Figure.7. Note that it is the reflection (in the line ) of the black portion of the graph in Figure.7. Be sure ou see that Figure.7 shows the entire graph of the inverse sine function. Remember that the domain of arcsin is the closed interval, and the range is the closed interval,. Now tr Eercise 7. 0
0_007.qd /7/05 :0 AM Page 5 Section.7 Inverse Trigonometric Functions 5 Other Inverse Trigonometric Functions The cosine function is decreasing and one-to-one on the interval 0 shown in Figure.7., as = cos cos has an inverse function on this interval. FIGURE.7 Consequentl, on this interval the cosine function has an inverse function the inverse cosine function denoted b arccos or cos. Similarl, ou can define an inverse tangent function b restricting the domain of tan to the interval,. The following list summarizes the definitions of the three most common inverse trigonometric functions. The remaining three are defined in Eercises 0 0. You ma need to point out to our students that the range for each of these functions is different. Students should know these ranges well to ensure that their answers are within the correct range. Referencing the graphs of the inverse trigonometric functions ma also be helpful. Definitions of the Inverse Trigonometric Functions Function Domain Range arcsin if and onl if sin arccos if and onl if cos 0 arctan if and onl if tan < < < < The graphs of these three inverse trigonometric functions are shown in Figure.7. = arcsin = arccos = arctan, DOMAIN: RANGE:, FIGURE.7 DOMAIN:, RANGE: 0, DOMAIN: RANGE:,,
0_007.qd /8/05 8:5 AM Page Chapter Trigonometr Eample Evaluating Inverse Trigonometric Functions Find the eact value. a. arccos b. cos c. arctan 0 d. tan a. Because cos, and lies in 0,, it follows that arccos. Angle whose cosine is b. Because cos, and cos. lies in 0,, it follows that Angle whose cosine is c. Because tan 0 0, and 0 lies in,, it follows that arctan 0 0. Angle whose tangent is 0 d. Because tan, and lies in,, it follows that tan Angle whose tangent is. Now tr Eercise. Eample Calculators and Inverse Trigonometric Functions It is important to remember that the domain of the inverse sine function and the inverse cosine function is,, as indicated in Eample (c). Use a calculator to approimate the value (if possible). a. arctan8.5 b. sin 0.7 c. arccos Function Mode Calculator Kestrokes a. arctan8.5 Radian 8.5 From the displa, it follows that arctan8.5.500. b. sin 0.7 Radian SIN 0.7 From the displa, it follows that sin 0.7 0.70. c. arccos Radian COS TAN ENTER ENTER ENTER In real number mode, the calculator should displa an error message because the domain of the inverse cosine function is,. Now tr Eercise 5. In Eample, if ou had set the calculator to degree mode, the displas would have been in degrees rather than radians. This convention is peculiar to calculators. B definition, the values of inverse trigonometric functions are alwas in radians.
0_007.qd /7/05 :0 AM Page 7 Compositions of Functions Section.7 Inverse Trigonometric Functions 7 Recall from Section.9 that for all in the domains of f and f, inverse functions have the properties f f and f f. Inverse Properties of Trigonometric Functions If and, then sinarcsin and arcsinsin. If and 0, then cosarccos and arccoscos. If is a real number and < <, then tanarctan and arctantan. Activities. Evaluate arccos. 5 Answer:. Use a calculator to evaluate arctan.. Answer:.8. Write an algebraic epression that is equivalent to sinarctan. Answer: 9 Keep in mind that these inverse properties do not appl for arbitrar values of and. For instance, arcsin sin In other words, the propert arcsinsin is not valid for values of outside the interval,. Eample 5 Using Inverse Properties If possible, find the eact value. a. tanarctan5 b. arcsin sin 5 c. coscos a. Because 5 lies in the domain of the arctan function, the inverse propert applies, and ou have tanarctan5 5. b. In this case, 5 does not lie within the range of the arcsine function,. However, 5 is coterminal with 5 arcsin which does lie in the range of the arcsine function, and ou have arcsin sin 5 arcsin sin.. c. The epression coscos is not defined because cos is not defined. Remember that the domain of the inverse cosine function is,. Now tr Eercise.
0_007.qd /7/05 :0 AM Page 8 8 Chapter Trigonometr Eample shows how to use right triangles to find eact values of compositions of inverse functions. Then, Eample 7 shows how to use right triangles to convert a trigonometric epression into an algebraic epression. This conversion technique is used frequentl in calculus. u = arccos Angle whose cosine is FIGURE.75 = 5 5 ( ) = u = arcsin 5 5 Angle whose sine is 5 FIGURE.7 ( ( Eample Find the eact value. a. b. tan arccos Evaluating Compositions of Functions a. If ou let u arccos then cos u,. Because cos u is positive, u is a firstquadrant angle. You can sketch and label angle u as shown in Figure.75. Consequentl, tan arccos b. If ou let u arcsin 5, then sin u 5. Because sin u is negative, u is a fourth-quadrant angle. You can sketch and label angle u as shown in Figure.7. Consequentl, cos arcsin 5 cos arcsin 5 tan u opp adj cos u adj hp 5. Now tr Eercise 5. 5. Eample 7 Some Problems from Calculus Write each of the following as an algebraic epression in. a. sinarccos, 0 b. cotarccos, 0 < u = arccos Angle whose cosine is FIGURE.77 () You ma want to review with students how to rationalize the denominator of a fractional epression. If ou let u arccos, then cos u, where. Because ou can sketch a right triangle with acute angle u, as shown in Figure.77. From this triangle, ou can easil convert each epression to algebraic form. a. b. cos u adj hp sinarccos sin u opp hp 9, cotarccos cot u adj opp 9, Now tr Eercise 59. 0 0 < In Eample 7, similar arguments can be made for interval, 0. -values ling in the
0_007.qd /7/05 :0 AM Page 9 Section.7 Inverse Trigonometric Functions 9.7 Eercises VOCABULARY CHECK: Fill in the blanks. Function Alternative Notation Domain Range. arcsin. cos. arctan PREREQUISITE SKILLS REVIEW: Practice and review algebra skills needed for this section at www.eduspace.com. In Eercises, evaluate the epression without using a calculator.. arcsin. arcsin 0. arccos. arccos 0 5. arctan. arctan 7. cos 8. 9. arctan 0. arctan arccos... sin. 5. tan 0. cos In Eercises 7 and 8, use a graphing utilit to graph f, g, and in the same viewing window to verif geometricall that g is the inverse function of f. (Be sure to restrict the domain of f properl.) 7. f sin, 8. f tan, g arcsin g arctan sin arcsin tan In Eercises 9, use a calculator to evaluate the epression. Round our result to two decimal places. 9. arccos 0.8 0. arcsin 0.5. arcsin0.75. arccos0.7. arctan. arctan 5 5. sin 0.. cos 0. 7. arccos0. 8. arcsin0.5 9. arctan 0.9 0. arctan.8. arcsin. arccos. tan 7. tan 95 7 In Eercises 5 and, determine the missing coordinates of the points on the graph of the function. 5.. = arctan (, ) ( = arccos, ) (, ) In Eercises 7, use an inverse trigonometric function to write as a function of. 7. 8. 9. 0. 5.. +, ( ) + (, ) (, ) In Eercises 8, use the properties of inverse trigonometric functions to evaluate the epression.. sinarcsin 0.. tanarctan 5 5. cosarccos0.. sinarcsin0. 7. arcsinsin 8. arccos cos 7 0 +
0_007.qd /7/05 :0 AM Page 50 50 Chapter Trigonometr In Eercises 9 58, find the eact value of the epression. (Hint: Sketch a right triangle.) 9. sinarctan 50. secarcsin 5 5. costan 5. sin 5 cos 5 5. cosarcsin 5 5. cscarctan 5 55. secarctan 5 5. tanarcsin 57. sinarccos 58. cotarctan 5 8 In Eercises 59 8, write an algebraic epression that is equivalent to the epression. (Hint: Sketch a right triangle, as demonstrated in Eample 7.) 59. cotarctan 0. sinarctan. cosarcsin. secarctan. sinarccos. secarcsin 5.. 7. csc arctan 8. In Eercises 9 and 70, use a graphing utilit to graph f and g in the same viewing window to verif that the two functions are equal. Eplain wh the are equal. Identif an asmptotes of the graphs. 9. 70. In Eercises 7 7, fill in the blank. 7. arctan 9 arcsin, 7. arcsin arccos, 0 7. arccos 0 arcsin 7. tan arccos cot arctan cos arcsin h r 0 arccos arctan, f sinarctan, f tan arccos, g g In Eercises 75 and 7, sketch a graph of the function and compare the graph of g with the graph of f arcsin. In Eercises 77 8, sketch a graph of the function. 77. arccos 78. gt arccost 79. f ) arctan 80. f arctan 8. hv tanarccos v 8. In Eercises 8 88, use a graphing utilit to graph the function. 8. f arccos 8. f arcsin 85. f arctan 8. 87. 88. f arccos f arctan f sin f cos In Eercises 89 and 90, write the function in terms of the sine function b using the identit A cos t B sin t A B sin t arctan A B. Use a graphing utilit to graph both forms of the function. What does the graph impl? 89. f t cos t sin t 90. f t cos t sin t 9. Docking a Boat A boat is pulled in b means of a winch located on a dock 5 feet above the deck of the boat (see figure). Let be the angle of elevation from the boat to the winch and let s be the length of the rope from the winch to the boat. 5 ft (a) Write as a function of s. (b) Find when s 0 feet and s 0 feet. s 75. g arcsin 7. g arcsin
0_007.qd /7/05 :0 AM Page 5 Section.7 Inverse Trigonometric Functions 5 9. Photograph A television camera at ground level is filming the lift-off of a space shuttle at a point 750 meters from the launch pad (see figure). Let be the angle of elevation to the shuttle and let s be the height of the shuttle. 9. Granular Angle of Repose Different tpes of granular substances naturall settle at different angles when stored in cone-shaped piles. This angle is called the angle of repose (see figure). When rock salt is stored in a coneshaped pile feet high, the diameter of the pile s base is about feet. (Source: Bulk-Store Structures, Inc.) (a) Write as a function of s. (b) Find when s 00 meters and s 00 meters. 750 m s Not drawn to scale 9. Photograph A photographer is taking a picture of a three-foot-tall painting hung in an art galler. The camera lens is foot below the lower edge of the painting (see figure). The angle subtended b the camera lens feet from the painting is arctan, Model It > 0. (a) Find the angle of repose for rock salt. (b) How tall is a pile of rock salt that has a base diameter of 0 feet? 95. Granular Angle of Repose When whole corn is stored in a cone-shaped pile 0 feet high, the diameter of the pile s base is about 8 feet. (a) Find the angle of repose for whole corn. (b) How tall is a pile of corn that has a base diameter of 00 feet? 9. Angle of Elevation An airplane flies at an altitude of miles toward a point directl over an observer. Consider and as shown in the figure. 7 ft ft mi Not drawn to scale ft ft α β (a) Write as a function of. (b) Find when 7 miles and mile. 97. Securit Patrol A securit car with its spotlight on is parked 0 meters from a warehouse. Consider and as shown in the figure. Not drawn to scale (a) Use a graphing utilit to graph as a function of. (b) Move the cursor along the graph to approimate the distance from the picture when is maimum. (c) Identif the asmptote of the graph and discuss its meaning in the contet of the problem. (a) Write 0 m Not drawn to scale as a function of. (b) Find when 5 meters and meters.
0_007.qd /7/05 :0 AM Page 5 5 Chapter Trigonometr Snthesis True or False? In Eercises 98 00, determine whether the statement is true or false. Justif our answer. 98. 99. 00. sin 5 tan 5 arctan arcsin arccos 0. Define the inverse cotangent function b restricting the domain of the cotangent function to the interval 0,, and sketch its graph. 0. Define the inverse secant function b restricting the domain of the secant function to the intervals 0, and,, and sketch its graph. 0. Define the inverse cosecant function b restricting the domain of the cosecant function to the intervals, 0 and 0,, and sketch its graph. 0. Use the results of Eercises 0 0 to evaluate each epression without using a calculator. (a) arcsec (b) arcsec (c) arccot (d) arccsc 05. Area In calculus, it is shown that the area of the region bounded b the graphs of 0,, a, and b is given b Area arctan b arctan a arcsin (see figure). Find the area for the following values of a and b. (a) a 0, b (b) a, b (c) a 0, b (d) a, b a b 0. Think About It Use a graphing utilit to graph the functions f and g arctan. = 5 arctan 5 + For > 0, it appears that g > f. Eplain wh ou know that there eists a positive real number a such that g < f for > a. Approimate the number a. 07. Think About It Consider the functions given b f sin and (a) Use a graphing utilit to graph the composite functions f f and f f. (b) Eplain wh the graphs in part (a) are not the graph of the line. Wh do the graphs of f f and f f differ? 08. Proof Prove each identit. (a) arcsin arcsin (b) arctan arctan (c) arctan arctan, (d) arcsin arccos (e) arcsin arctan Skills Review f arcsin. In Eercises 09, evaluate the epression. Round our result to three decimal places. 09. 8.. 0. 0.. 50. In Eercises, sketch a right triangle corresponding to the trigonometric function of the acute angle. Use the Pthagorean Theorem to determine the third side. Then find the other five trigonometric functions of.. sin. tan 5. cos 5. sec 7. Partnership Costs A group of people agree to share equall in the cost of a $50,000 endowment to a college. If the could find two more people to join the group, each person s share of the cost would decrease b $50. How man people are presentl in the group? 8. Speed A boat travels at a speed of 8 miles per hour in still water. It travels 5 miles upstream and then returns to the starting point in a total of hours. Find the speed of the current. 9. Compound Interest A total of $5,000 is invested in an account that pas an annual interest rate of.5%. Find the balance in the account after 0 ears, if interest is compounded (a) quarterl, (b) monthl, (c) dail, and (d) continuousl. 0. Profit Because of a slump in the econom, a department store finds that its annual profits have dropped from $7,000 in 00 to $,000 in 00. The profit follows an eponential pattern of decline. What is the epected profit for 008? (Let t represent 00.) > 0