SECTION 5-9 Inverse Trigonometric Functions

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1 5-9 Inverse Trigonometric Functions 47 In Problems 7 3, grph t lest two ccles of the given eqution in grphing utilit, then find n eqution of the form A tn B, A cot B, A sec B, or A csc B tht hs the sme grph. (These problems suggest dditionl identities beond those discussed in Section 5-. Additionl identities re discussed in detil in Chpter 6.) 7. sin 3 cos 3 cot 3 8. cos sin tn sin 4 cos 4 sin 6 cos 6 (B) Grph the eqution found in prt A for the time intervl [, ). If the grph hs n smptote, put it in. (C) Describe wht hppens to the length c of the light bem s t goes from to. N c P APPLICATIONS 3. Motion. A becon light ft from wll rottes clockwise t the rte of /4 rps (see figure); thus, t/. (A) Strt counting time in seconds when the light spot is t N nd write n eqution for the length c of the light bem in terms of t. 3. Motion. Refer to Problem 3. (A) Write n eqution for the distnce the light spot trvels long the wll in terms of time t. (B) Grph the eqution found in prt A for the time intervl [, ). If the grph hs n smptote, put it in. (C) Describe wht hppens to the distnce long the wll s t goes from to. SECTION 5-9 Inverse Trigonometric Functions Inverse Sine Function Inverse Cosine Function Inverse Tngent Function Summr Inverse Cotngent, Secnt, nd Cosecnt Functions (Optionl) A brief review of the generl concept of inverse functions discussed in Section -6 should prove helpful before proceeding with this section. In the following bo we restte few importnt fcts bout inverse functions from tht section. Fcts bout Inverse Functions For f one-to-one function nd f its inverse:. If (, b) is n element of f, then (b, ) is n element of f, nd conversel.. Rnge of f Domin of f Domin of f Rnge of f

2 48 5 Trigonometric Functions 3. DOMAIN f f () RANGE f f f RANGE f f() DOMAIN f 4. If f (), then f() for in the domin of f nd in the domin of f, nd conversel. f() f f () 5. f [ f ()] for in the domin of f f [ f()] for in the domin of f All trigonometric functions re periodic; hence, ech rnge vlue cn be ssocited with infinitel mn domin vlues (Fig. ). As result, no trigonometric function is one-to-one. Without restrictions, no trigonometric function hs n inverse function. To resolve this problem, we restrict the domin of ech function so tht it is one-to-one over the restricted domin. Thus, for this restricted domin, n inverse function is gurnteed. FIGURE sin is not one-toone over (, ). 4 4 Inverse trigonometric functions represent nother group of bsic functions tht re dded to our librr of elementr functions. These functions re used in mn pplictions nd mthemticl developments, nd the will be prticulrl useful to us when we solve trigonometric equtions in Section 6-5. Inverse Sine Function How cn the domin of the sine function be restricted so tht it is one-to-one? This cn be done in infinitel mn ws. A firl nturl nd generll ccepted w is illustrted in Figure.

3 5-9 Inverse Trigonometric Functions 49 FIGURE sin is one-to-one over [/, /]. If the domin of the sine function is restricted to the intervl [/, /], we see tht the restricted function psses the horizontl line test (Section -8) nd thus is one-to-one. Note tht ech rnge vlue from to is ssumed ectl once s moves from / to /. We use this restricted sine function to define the inverse sine function. DEFINITION Inverse Sine Function The inverse sine function, denoted b sin or rcsin, is defined s the inverse of the restricted sine function sin, / /. Thus, re equivlent to sin nd rcsin sin where / /, In words, the inverse sine of, or the rcsine of, is the number or ngle, / /, whose sine is. To grph sin, tke ech point on the grph of the restricted sine function nd reverse the order of the coordintes. For emple, since (/, ), (, ), nd (/, ) re on the grph of the restricted sine function (Fig. 3()) then (, /), (, ), nd (, /) re on the grph of the inverse sine function, s shown in Figure 3(b). Using these three points provides us with quick w of sketching the grph of the inverse sine function. A more ccurte grph cn be obtined b using clcultor. FIGURE 3 Inverse sine function. (, ) sin, (, ) sin rcsin,, Domin [, ] Rnge [, ], Domin [, ] Rnge [, ] Restricted sine function () Inverse sine function (b)

4 43 5 Trigonometric Functions We stte the importnt sine inverse sine identities which follow from the generl properties of inverse functions given in the bo t the beginning of this section. Sine Inverse Sine Identities sin (sin ) f [f ()] sin (sin ) / / f [f()] sin (sin.7).7 sin (sin.3).3 sin [sin (.)]. sin [sin ()] [Note: The number.3 is not in the domin of the inverse sine function, nd is not in the restricted domin of the sine function. Tr clculting ll these emples with our clcultor nd see wht hppens!] EXAMPLE Ect Vlues Find ect vlues without using clcultor: (A) rcsin (B) sin (sin.) (C) cos sin 3 Solution (A) rcsin is equivlent to / b Reference tringle ssocited with sin 3 6 rcsin / [Note: /6, even though sin (/6). must be between / nd /, inclusive.] (B) sin (sin.). Sine inverse sine identit, since /. / (C) Let sin 3; then sin 3, / /. Drw the reference tringle ssocited with. Then cos cos sin 3 cn be determined directl from the tringle (fter finding the third side) without ctull finding. / b b c 3 c b 3 5 Since in qudrnt I / Thus, cos sin 3 cos 5/3.

5 5-9 Inverse Trigonometric Functions 43 Mtched Problem Find ect vlues without using clcultor: (A) rcsin ( /) (B) sin [sin (.4)] (C) tn [sin ( /5)] EXAMPLE Clcultor Vlues Find to 4 significnt digits using clcultor: (A) rcsin (.34) (B) sin.357 (C) cot [sin (.87)] Solution The function kes used to represent inverse trigonometric functions vr mong different brnds of clcultors, so red the user s mnul for our clcultor. Set our clcultor in rdin mode nd follow our mnul for ke sequencing. (A) rcsin (.34).39 (B) sin.357 Error.357 is not in the domin of sin (C) cot [sin (.87)] 9.45 Mtched Problem Find to 4 significnt digits using clcultor: (A) sin.93 (B) rcsin (.35) (C) cot [sin (.3446)] Inverse Cosine Function FIGURE 4 cos is one-to-one over [, ]. To restrict the cosine function so tht it becomes one-to-one, we choose the intervl [, ]. Over this intervl the restricted function psses the horizontl line test, nd ech rnge vlue is ssumed ectl once s moves from to (Fig. 4). We use this restricted cosine function to define the inverse cosine function.

6 43 5 Trigonometric Functions DEFINITION Inverse Cosine Function The inverse cosine function, denoted b cos or rccos, is defined s the inverse of the restricted cosine function cos,. Thus, re equivlent to cos nd rccos cos where, In words, the inverse cosine of, or the rccosine of, is the number or ngle,, whose cosine is. Figure 5 compres the grphs of the restricted cosine function nd its inverse. Notice tht (, ), (/, ), nd (, ) re on the restricted cosine grph. Reversing the coordintes gives us three points on the grph of the inverse cosine function. FIGURE 5 Inverse cosine function. (, ) (, ) cos, cos rccos, (, ) Domin [, ] Rnge [, ] Restricted cosine function () (, ) Domin [, ] Rnge [, ] Inverse cosine function (b) We complete the discussion b giving the cosine inverse cosine identities: Cosine Inverse Cosine Identities cos (cos ) f [f ()] cos (cos ) f [f()] EXPLORE-DISCUSS Evlute ech of the following with clcultor. Which illustrte cosine inverse cosine identit nd which do not? Discuss wh. (A) cos (cos.) (C) cos (cos ) (B) cos [cos ()] (D) cos [cos (3)]

7 5-9 Inverse Trigonometric Functions 433 EXAMPLE 3 Ect Vlues Find ect vlues without using clcultor: (A) rccos ( 3/ ) (B) cos (cos.7) (C) sin [cos 3 ] Solutions (A) rccos ( 3/) is equivlent to Reference tringle ssocited with b cos rccos 3 3 [Note: 5/6, even though cos (5/6) 3/. must be between nd, inclusive.] (B) cos (cos.7).7 Cosine inverse cosine identit, since.7 (C) Let cos ; then cos 3 3,. Drw reference tringle ssocited with. Then sin sin [cos 3 ] cn be determined directl from the tringle (fter finding the third side) without ctull finding. b b c c 3 b c b 3 () 8 Since b in qudrnt II Thus, sin [cos 3 ] sin /3. Mtched Problem 3 Find ect vlues without using clcultor: (A) rccos ( /) (B) cos (cos 3.5) (C) cot [cos (/ 5)] EXAMPLE 4 Clcultor Vlues Find to 4 significnt digits using clcultor: (A) rccos.435 (B) cos.37 (C) csc [cos (.349)]

8 434 5 Trigonometric Functions Solution Set our clcultor in rdin mode. (A) rccos (B) cos.37 Error.37 is not in the domin of cos (C) csc [cos (.349)]. Mtched Problem 4 Find to 4 significnt digits using clcultor: (A) cos.6773 (B) rccos (.3) (C) cot [cos (.536)] Inverse Tngent Function FIGURE 6 tn is one-to-one over (/, /). To restrict the tngent function so tht it becomes one-to-one, we choose the intervl (/, /). Over this intervl the restricted function psses the horizontl line test, nd ech rnge vlue is ssumed ectl once s moves cross this restricted domin (Fig. 6). We use this restricted tngent function to define the inverse tngent function. tn 3 3 DEFINITION 3 Inverse Tngent Function The inverse tngent function, denoted b tn or rctn, is defined s the inverse of the restricted tngent function tn, / /. Thus, re equivlent to tn nd rctn tn where / / nd is rel number In words, the inverse tngent of, or the rctngent of, is the number or ngle, / /, whose tngent is.

9 5-9 Inverse Trigonometric Functions 435 Figure 7 compres the grphs of the restricted tngent function nd its inverse. Notice tht (/4, ), (, ), nd (/4, ) re on the restricted tngent grph. Reversing the coordintes gives us three points on the grph of the inverse tngent function. Also note tht the verticl smptotes become horizontl smptotes for the inverse function. FIGURE 7 Inverse tngent function. tn, 4, 4 tn rctn, 4, 4 Domin, Rnge (, ) Restricted tngent function () Domin (, ) Rnge, Inverse tngent function (b) We now stte the tngent inverse tngent identities. Tngent Inverse Tngent Identities tn (tn ) f [f ()] tn (tn ) / / f [f()] EXPLORE-DISCUSS Evlute ech of the following with clcultor. Which illustrte tngent inverse tngent identit nd which do not? Discuss wh. (A) tn (tn 3) (C) tn (tn.4) (B) tn [tn (455)] (D) tn [tn (3)] EXAMPLE 5 Ect Vlues Find ect vlues without using clcultor: (A) tn (/ 3) (B) tn (tn.63)

10 436 5 Trigonometric Functions Solutions (A) tn (/ 3) is equivlent to / b Reference tringle ssocited with 3 tn 3 tn 6 3 / [Note: cnnot be /6. must be between / nd /.] (B) tn (tn.63).63 Tngent inverse tngent identit, since /.63 / Mtched Problem 5 Find ect vlues without using clcultor: (A) rctn ( 3) (B) tn (tn 43) Summr We summrize the definitions nd grphs of the inverse trigonometric functions discussed so fr for convenient reference. Summr of sin, cos, nd tn sin is equivlent to sin, / / cos is equivlent to cos, tn is equivlent to tn, / / sin Domin [, ] Rnge [, ] cos Domin [, ] Rnge [, ] tn Domin (, ) Rnge,

11 5-9 Inverse Trigonometric Functions 437 Inverse Cotngent, Secnt, nd Cosecnt Functions (Optionl) For completeness, we include the definitions nd grphs of the inverse cotngent, secnt, nd cosecnt functions. DEFINITION 4 Inverse Cotngent, Secnt, nd Cosecnt Functions cot is equivlent to cot where, sec is equivlent to sec where, /, csc is equivlent to csc where / /,, sec csc cot Domin: All rel numbers Rnge: Domin: or Rnge:, / Domin: or Rnge: / /, [Note: The definitions of sec nd csc re not universll greed upon.] EXERCISE 5-9 Answers to Mtched Problems. (A) /4 (B).4 (C) /. (A).945 (B) Not defined (C) (A) /4 (B) 3.5 (C) / 4. (A).867 (B) Not defined (C) (A) /3 (B) 43 Unless stted to the contrr, the inverse trigonometric functions re ssumed to hve rel number rnges (use rdin mode in clcultor problems). A few problems involve rnges with ngles in degree mesure, nd these re clerl indicted (use degree mode in clcultor problems). A In Problems, find ect vlues without using clcultor.. cos. sin 3. rcsin (3/) 4. rccos (3/) 5. rctn 3 6. tn 7. sin 8. cos (/) 9. rccos. rctn (/3). sin. tn In Problems 3 8, evlute to 4 significnt digits using clcultor. 3. cos rcsin rctn rccos.7 7. sin tn 8.59 B In Problems 9 8, find ect vlues without using clcultor. 9. rccos (3/). rcsin 3. rctn (). cos (/) 3. sin 4. tn (/3) 5. tn (tn 5) 6. sin [sin (.6)]

12 438 5 Trigonometric Functions 7. sin (cos 3/) 8. tn (cos /) In Problems 9 3, evlute to 4 significnt digits using clcultor. 9. cot [cos (.73)] 3. sec [sin (.399)] 3. 5 cos ( ) 3. tn 5 3 In Problems 33 38, find the ect degree mesure of ech without the use of clcultor. 33. sin (/) 34. cos (/) 35. rctn (3) 36. rctn () 37. cos () 38. sin () In Problems 39 4, find the degree mesure of ech to two deciml plces using clcultor set in degree mode. 39. cos tn rcsin (.366) 4. rccos (.96) 43. Evlute sin (sin ) with clcultor set in rdin mode, nd eplin wh this does or does not illustrte the inverse sine sine identit. 44. Evlute cos [cos (.5)] with clcultor set in rdin mode, nd eplin wh this does or does not illustrte the inverse cosine cosine identit. Problems require the use of grphing utilit. In Problems 45 5, grph ech function in grphing utilit over the indicted intervl. 45. sin, 46. cos, 47. cos (/3), sin (/), 49. sin ( ), 3 5. cos ( ), 5. tn ( 4), 6 5. tn ( 3), The identit cos (cos ) is vlid for. (A) Grph cos (cos ) for. (B) Wht hppens if ou grph cos (cos ) over wider intervl, s,? Eplin. 54. The identit sin (sin ) is vlid for. (A) Grph sin (sin ) for. (B) Wht hppens if ou grph sin (sin ) over wider intervl, s,? Eplin. C In Problems 55 58, find the ect solutions to the eqution. Eplin our resoning. 55. sin cos 56. sin tn 57. sin 3 ) cos 58. sin sin (/) In Problems 59 6, write ech epression s n lgebric epression in free of trigonometric or inverse trigonometric functions. 59. cos (sin ) 6. sin (cos ) 6. cos (rctn ) 6. tn (rcsin ) In Problems 63 nd 64, find f (). How must be restricted in f ()? 63. f() 4 cos ( 3), 3 (3 ) 64. f() 3 5 sin ( ), ( /) ( /) Problems require the use of grphing utilit. 65. The identit cos (cos ) is vlid for. (A) Grph cos (cos ) for. (B) Wht hppens if ou grph cos (cos ) over lrger intervl, s,? Eplin. 66. The identit sin (sin ) is vlid for / /. (A) Grph sin (sin ) for / /. (B) Wht hppens if ou grph sin (sin ) over lrger intervl, s,? Eplin. APPLICATIONS 67. Photogrph. The viewing ngle chnges with the focl length of cmer lens: A 8-mm wide-ngle lens hs wide viewing ngle nd 3-mm telephoto lens hs nrrow viewing ngle. For 35-mm-formt cmer the viewing ngle, in degrees, is given b tn.634 where is the focl length of the lens being used. Wht is the viewing ngle (in deciml degrees to two deciml plces) of 8-mm lens? Of -mm lens?

13 5-9 Inverse Trigonometric Functions Photogrph. Referring to Problem 67, wht is the viewing ngle (in deciml degrees to two deciml plces) of 7-mm lens? Of 7-mm lens? 69. (A) Grph the function in Problem 67 in grphing utilit using degree mode. The grph should cover lenses with focl lengths from mm to mm. (B) Wht focl-length lens, to two deciml plces, would hve viewing ngle of 4? Solve b grphing 4 nd tn (.634/) in the sme viewing window nd finding the point of intersection using n pproimtion routine. 7. (A) Grph the function in Problem 67 in grphing utilit, in degree mode, with the grph covering lenses with focl lengths from mm to mm. (B) Wht focl-length lens, to two deciml plces, would hve viewing ngle of? Solve b grphing nd tn (.634/) in the sme viewing window nd finding the point of intersection using n pproimtion routine. 7. Engineering. The length of the belt round the two pulles in the figure is given b L D (d D) C sin where (in rdins) is given b cos D d C (A) Grph in grphing utilit (in rdin mode), with the grph covering pulles with their centers from 3 to inches prt. (B) How fr, to two deciml plces, should the centers of the two pulles be plced to use belt 4 inches long? Solve b grphing nd 4 in the sme viewing window nd finding the point of intersection using n pproimtion routine. 74. Engineering. The function 6 cos sin cos represents the length of the belt round the two pulles in Problem 7 when the centers of the pulles re inches prt. (A) Grph in grphing utilit (in rdin mode), with the grph covering pulles with their centers from 3 to inches prt. (B) How fr, to two deciml plces, should the centers of the two pulles be plced to use belt 36 inches long? Solve b grphing nd 36 in the sme viewing window nd finding the point of intersection using n pproimtion routine. 75. Motion. The figure represents circulr courtrd surrounded b high stone wll. A floodlight locted t E shines into the courtrd. Verif these formuls, nd find the length of the belt to two deciml plces if D 4 inches, d inches, nd C 6 inches. r Shdow d C D E C r A D d D d 7. Engineering. For Problem 7, find the length of the belt if D 6 inches, d 4 inches, nd C inches. 73. Engineering. The function 4 cos sin cos represents the length of the belt round the two pulles in Problem 7 when the centers of the pulles re inches prt. (A) If person wlks feet w from the center long DC, show tht the person s shdow will move distnce given b d r r tn where is in rdins. [Hint: Drw line from A to C.] (B) Find d to two deciml plces if r feet nd 4 feet. 76. Motion. In Problem 75, find d for r 5 feet nd 5 feet. r

14 44 5 Trigonometric Functions CHAPTER 5 GROUP ACTIVITY A Predtor Pre Anlsis Involving Mountin Lions nd Deer In some western stte wilderness res, deer nd mountin lion popultions re interrelted, since the mountin lions rel on the deer s food source. The popultion of ech species goes up nd down in ccles, but out of phse with ech other. A wildlife mngement reserch tem estimted the respective popultions in prticulr region ever ers over 6-er period, with the results shown in Tble : TABLE Mountin Lion Deer Popultions Yers Deer Mtn. Lions (A) Deer Popultion Anlsis. Enter the dt for the deer popultion for the time intervl [, 6] in grphing utilit nd produce sctter plot of the dt.. A function of the form k A sin (B C) cn be used to model this dt. Use the dt in Tble to determine k, A, nd B. Use the grph in prt to visull estimte C to one deciml plce. 3. Plot the dt from prt nd the eqution from prt in the sme viewing window. If necessr, djust the vlue of C for better fit. 4. Write summr of the results, describing fluctutions nd ccles of the deer popultion. (B) Mountin Lion Popultion Anlsis. Enter the dt for the mountin lion popultion for the time intervl [, 6] in grphing utilit nd produce sctter plot of the dt.. A function of the form k A sin (B C) cn be used to model this dt. Use the dt in Tble to determine k, A, nd B. Use the grph in prt to visull estimte C to one deciml plce. 3. Plot the dt from prt nd the eqution from prt in the sme viewing window. If necessr, djust the vlue of C for better fit. 4. Write summr of the results, describing fluctutions nd ccles of the mountin lion popultion. (C) Interreltionship of the Two Popultions. Discuss the reltionship of the mimum predtor popultions to the mimum pre popultions reltive to time.. Discuss the reltionship of the minimum predtor popultions to the minimum pre popultions reltive to time. 3. Discuss the dnmics of the fluctutions of the two interdependent popultions. Wht cuses the two popultions to rise nd fll, nd wh re the out of phse from one nother?