Section 59 Inverse Trigonometric Functions


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1 46 5 TRIGONOMETRIC FUNCTIONS Section 59 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 4 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 onetoone 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 3. Domain of f Range of f DOMAIN f RANGE f f f () f() RANGE f f DOMAIN f 4. If f ( ), then f() for in the domain of f and in the domain of f, and conversel. f() f f () 5. f( f ()) for in the domain of f f ( f()) for in the domain of f All trigonometric functions are periodic; hence, each range value can be associated with infinitel man domain values (Fig. ). As a result, no trigonometric function is onetoone. Without restrictions, no trigonometric function has an inverse function. To resolve this problem, we restrict the domain of each function so that it is onetoone over the restricted domain. Thus, for this restricted domain, an inverse function is guaranteed.
2 59 Inverse Trigonometric Functions 47 FIGURE sin is not onetoone over (, ). 4 4 Inverse trigonometric functions represent another group of basic functions that are added to our librar of elementar functions. These functions are used in man applications and mathematical developments, and will be particularl useful to us when we solve trigonometric equations in Section 65. FIGURE sin is onetoone over [/, /]. Inverse Sine Function How can the domain of the sine function be restricted so that it is onetoone? This can be done in infinitel man was. A fairl natural and generall accepted wa is illustrated in Figure. If the domain of the sine function is restricted to the interval [/, /], we see that the restricted function passes the horizontal line test (Section 4) and thus is onetoone. Note that each range value from to is assumed eactl once as 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 arcsin, is defined as the inverse of the restricted sine function sin, / /. Thus, sin and arcsin are equivalent to sin where / /, In words, the inverse sine of, or the arcsine of, is the number or angle, / /, whose sine is. To graph sin, take each point on the graph of the restricted sine function and reverse the order of the coordinates. For eample, since (/, ),
3 48 5 TRIGONOMETRIC FUNCTIONS (, ), and (/, ) are on the graph of the restricted sine function (Fig. 3), then (, /), (, ), and (, /) are on the graph of the inverse sine function, as shown in Figure 3. Using these three points provides us with a quick wa of sketching the graph of the inverse sine function. A more accurate graph can be obtained b using a calculator. FIGURE 3 Inverse sine function. (, ) sin, (, ) sin arcsin,, Domain [, ] Range [, ], Domain [, ] Range [, ] Restricted sine function (a) Inverse sine function (b) Eplore/Discuss FIGURE 4 A graphing calculator produced the graph in Figure 4 for sin,, and. (Tr this on our own graphing utilit.) Eplain wh there are no parts of the graph on the intervals [, ) and (, ]. We state the important sine inverse sine identities that follow from the general properties of inverse functions given in the bo at 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 domain of the inverse sine function, and is not in the restricted domain of the sine function. Tr calculating all these eamples with our calculator and see what happens!]
4 59 Inverse Trigonometric Functions 49 EXAMPLE Eact Values Find eact values without using a calculator. (A) arcsin ( (B) sin ) (sin.) (C) cos [sin ( 3 )] Solutions (A) arcsin ( is equivalent to sin 6 arcsin ( ) / b Reference triangle associated with 3 a / [Note: /6, even though sin (/6). must be between / and /, inclusive.] (B) sin (sin.). Sine inverse sine identit, since /. / (C) Let sin ( ; then sin ( 3 ) 3 ), / /. Draw the reference triangle associated with. Then cos cos [sin ( 3 ) ] can be determined directl from the triangle (after finding the third side) without actuall finding. / b 3 c b a b c a 3 5 Since a in quadrant I a a / Thus, cos [sin ( 3 ) ] cos 5/3. MATCHED PROBLEM Find eact values without using a calculator. (A) arcsin (/) (B) sin [sin (.4)] (C) tan [sin (/5)] EXAMPLE Calculator Values Find to four significant digits using a calculator. (A) arcsin (.34) (B) sin.357 (C) cot [sin (.87)]
5 43 5 TRIGONOMETRIC FUNCTIONS Solutions The function kes used to represent inverse trigonometric functions var among different brands of calculators, so read the user s manual for our calculator. Set our calculator in radian mode and follow our manual for ke sequencing. (A) arcsin (.34).39 (B) sin.357 Error.357 is not in the domain of sin (C) cot [sin (.87)] 9.45 MATCHED PROBLEM FIGURE 5 cos is onetoone over [, ]. Find to four significant digits using a calculator. (A) sin.93 (B) arcsin (.35) (C) cot [sin (.3446)] Inverse Cosine Function To restrict the cosine function so that it becomes onetoone, we choose the interval [, ]. Over this interval the restricted function passes the horizontal line test, and each range value is assumed eactl once as moves from to (Fig. 5). We use this restricted cosine function to define the inverse cosine function. DEFINITION INVERSE COSINE FUNCTION The inverse cosine function, denoted b cos or arccos, is defined as the inverse of the restricted cosine function cos,. Thus, cos and arccos are equivalent to cos where, In words, the inverse cosine of, or the arccosine of, is the number or angle,, whose cosine is. Figure 6 compares the graphs of the restricted cosine function and its inverse. Notice that (, ), (/, ), and (, ) are on the restricted cosine graph. Reversing the coordinates gives us three points on the graph of the inverse cosine function.
6 59 Inverse Trigonometric Functions 43 FIGURE 6 Inverse cosine function. (, ) (, ) cos, cos arccos, (, ) Domain [, ] Range [, ] Restricted cosine function (a) (, ) Domain [, ] Range [, ] Inverse cosine function (b) Eplore/Discuss FIGURE 7 A graphing calculator produced the graph in Figure 7 for cos,, and 4. (Tr this on our own graphing utilit.) Eplain wh there are no parts of the graph on the intervals [, ) and (, ]. 4 We complete the discussion b giving the cosine inverse cosine identities: COSINE INVERSE COSINE IDENTITIES cos (cos ) f(f ()) cos (cos ) f (f()) Eplore/Discuss 3 Evaluate each of the following with a calculator. Which illustrate a cosine inverse cosine identit and which do not? Discuss wh. (A) cos (cos.) (B) cos [cos ()] (C) cos (cos ) (D) cos [cos (3)]
7 43 5 TRIGONOMETRIC FUNCTIONS EXAMPLE 3 Eact Values Find eact values without using a calculator. (A) arccos (3/) (B) cos (cos.7) (C) sin [cos ( 3 )] Solutions (A) arccos (3/) is equivalent to cos arccos 3 Reference triangle associated with b 3 a [Note: 5/6, even though cos (5/6) 3/. must be between and, inclusive.] (B) cos (cos.7).7 Cosine inverse cosine identit, since.7 (C) Let cos ( ; then cos 3 ) 3,. Draw a reference triangle associated with. Then sin sin [cos ( 3 ) ] can be determined directl from the triangle (after finding the third side) without actuall finding. b a b c a c 3 b c b 3 ( ) 8 Since b in quadrant II a a Thus, sin [cos ( 3 ) ] sin /3. MATCHED PROBLEM 3 Find eact values without using a calculator. (A) arccos (/) (B) cos (cos 3.5) (C) cot [cos (/5)] EXAMPLE 4 Calculator Values Find to four significant digits using a calculator. (A) arccos.435 (B) cos.37 (C) csc [cos (.349)]
8 59 Inverse Trigonometric Functions 433 Solutions Set our calculator in radian mode. (A) arccos (B) cos.37 Error.37 is not in the domain of cos (C) csc [cos (.349)]. MATCHED PROBLEM 4 FIGURE 8 tan is onetoone over (/, /). Find to four significant digits using a calculator. (A) cos (.6773) (B) arccos (.3) (C) cot [cos (.536)] Inverse Tangent Function To restrict the tangent function so that it becomes onetoone, we choose the interval (/, /). Over this interval the restricted function passes the horizontal line test, and each range value is assumed eactl once as moves across this restricted domain (Fig. 8). We use this restricted tangent function to define the inverse tangent function. tan 3 3 DEFINITION 3 INVERSE TANGENT FUNCTION The inverse tangent function, denoted b tan or arctan, is defined as the inverse of the restricted tangent function tan, / /. Thus, tan and arctan are equivalent to tan where / / and is a real number In words, the inverse tangent of, or the arctangent of, is the number or angle, / /, whose tangent is.
9 434 5 TRIGONOMETRIC FUNCTIONS Figure 9 compares the graphs of the restricted tangent function and its inverse. Notice that (/4, ), (, ), and (/4, ) are on the restricted tangent graph. Reversing the coordinates gives us three points on the graph of the inverse tangent function. Also note that the vertical asmptotes become horizontal asmptotes for the inverse function. FIGURE 9 Inverse tangent function. tan, 4, 4 tan arctan, 4, 4 Domain, Range (, ) Restricted tangent function (a) Domain (, ) Range, Inverse tangent function (b) We now state the tangent inverse tangent identities. TANGENT INVERSE TANGENT IDENTITIES tan (tan ) f(f ()) tan (tan ) / / f (f()) Eplore/Discuss 4 Evaluate each of the following with a calculator. Which illustrate a tangent inverse tangent identit and which do not? Discuss wh. (A) tan (tan 3) (B) tan [tan (455)] (C) tan (tan.4) (D) tan [tan (3)] EXAMPLE 5 Eact Values Find eact values without using a calculator. (A) tan (/3) (B) tan (tan.63)
10 Solutions (A) tan (/3) is equivalent to 59 Inverse Trigonometric Functions 435 tan 3 6 tan 3 / b Reference triangle associated with 3 a / [Note: cannot be /6. must be between / and /.] (B) tan (tan.63).63 Tangent inverse tangent identit, since /.63 / MATCHED PROBLEM 5 Find eact values without using a calculator. (A) arctan (3) (B) tan (tan 43) Summar We summarize the definitions and graphs of the inverse trigonometric functions discussed so far for convenient reference. SUMMARY OF sin, cos, AND tan sin is equivalent to sin where, / / cos is equivalent to cos where, tan is equivalent to tan where, / / sin Domain [, ] Range [, ] cos Domain [, ] Range [, ] tan Domain (, ) Range,
11 436 5 TRIGONOMETRIC FUNCTIONS Inverse Cotangent, Secant, and Cosecant Functions (Optional) For completeness, we include the definitions and graphs of the inverse cotangent, secant, and cosecant functions. DEFINITION 4 INVERSE COTANGENT, SECANT, AND COSECANT FUNCTIONS cot is equivalent to cot where, sec is equivalent to sec where, /, csc is equivalent to csc where / /,, sec cot Domain: All real numbers Range: Domain: or Range:, / csc Domain: or Range: / /, [Note: The definitions of sec and csc are not universall agreed upon.] Answers to Matched 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
12 59 Inverse Trigonometric Functions 437 EXERCISE cos ( ) 4. tan 3 5 Unless stated to the contrar, the inverse trigonometric functions are assumed to have real number ranges (use radian mode in calculator problems). A few problems involve ranges with angles in degree measure, and these are clearl indicated (use degree mode in calculator problems). A In Problems, find eact values without using a calculator.. cos. sin 3. arcsin (3/) 4. arccos (3/) 5. arctan 3 6. tan 7. sin (/) 8. cos ( ) 9. arccos.. sin (. tan ) In Problems 3 8, evaluate to four significant digits using a calculator. 3. sin cos arctan tan arccos arcsin.3 B In Problems 9 34, find eact values without using a calculator. 9. arcsin (/). arccos ( ). tan (3). tan () 3. cos () sin () tan (tan 5) 8. sin [sin (.6)] 9. cos (cos.3) 3. tan [tan (.5)] 3. sin (cos 3/) 3. tan (cos ) 33. csc [tan ()] 34. arctan (/3) sin (3/) cos (3/) cos [sin (/)] In Problems 35 4, evaluate to four significant digits using a calculator. 35. arctan (.4) 36. tan (4.38) 37. cot [cos (.73)] 38. sec [sin (.399)] In Problems 4 46, find the eact degree measure of each without the use of a calculator. 4. sin (/) 4. cos ( ) 43. arctan (3) 44. arctan () 45. cos () 46. sin () In Problems 47 5, find the degree measure of each to two decimal places using a calculator set in degree mode. 47. cos tan arcsin (.366) 5. arccos (.96) 5. tan (837) 5. sin (.77) 53. Evaluate sin (sin ) with a calculator set in radian mode, and eplain wh this does or does not illustrate the inverse sine sine identit. 54. Evaluate cos [cos (.5)] with a calculator set in radian mode, and eplain wh this does or does not illustrate the inverse cosine cosine identit. Problems require the use of a graphing utilit. In Problems 55 6, graph each function in a graphing utilit over the indicated interval. 55. sin, 56. cos, 57. cos (/3), sin (/), 59. sin ( ), 3 6. cos ( ), 6. tan ( 4), 6 6. tan ( 3), The identit cos (cos ) is valid for. (A) Graph cos (cos ) for. (B) What happens if ou graph cos (cos ) over a wider interval, sa? Eplain. 64. The identit sin (sin ) is valid for. (A) Graph sin (sin ) for. (B) What happens if ou graph sin (sin ) over a wider interval, sa? Eplain.
13 438 5 TRIGONOMETRIC FUNCTIONS C In Problems 65 68, write each epression as an algebraic epression in free of trigonometric or inverse trigonometric functions. 65. cos (sin ) 66. sin (cos ) 67. cos (arctan ) 68. tan (arcsin ) In Problems 69 and 7, find f (). How must be restricted in f ()? 69. f() 4 cos ( 3), 3 (3 ) 7. f() 3 5 sin ( ), ( /) ( /) Problems 7 and 7 require the use of a graphing utilit. 7. The identit cos (cos ) is valid for. (A) Graph cos (cos ) for. (B) What happens if ou graph cos (cos ) over a larger interval, sa? Eplain. 7. The identit sin (sin ) is valid for / /. (A) Graph sin (sin ) for / /. (B) What happens if ou graph sin (sin ) over a larger interval, sa? Eplain. APPLICATIONS 74. Photograph. Referring to Problem 73, what is the viewing angle (in decimal degrees to two decimal places) of a 7mm lens? Of a 7mm lens? 75. (A) Graph the function in Problem 73 in a graphing utilit using degree mode. The graph should cover lenses with focal lengths from mm to mm. (B) What focallength lens, to two decimal places, would have a viewing angle of 4? Solve b graphing 4 and tan (.634/) in the same viewing window and finding the point of intersection using an approimation routine. 76. (A) Graph the function in Problem 73 in a graphing utilit, in degree mode, with the graph covering lenses with focal lengths from mm to, mm. (B) What focal length lens, to two decimal places, would have a viewing angle of? Solve b graphing and tan (.634/) in the same viewing window and finding the point of intersection using an approimation routine. 77. Engineering. The length of the belt around the two pulles in the figure is given b L D (d D) C sin where (in radians) is given b cos D d C Verif these formulas, and find the length of the belt to two decimal places if D 4 inches, d inches, and C 6 inches. 73. Photograph. The viewing angle changes with the focal length of a camera lens. A 8mm wideangle lens has a wide viewing angle and a 3mm telephoto lens has a narrow viewing angle. For a 35mm format camera the viewing angle, in degrees, is given b C D.634 tan where is the focal length of the lens being used. What is the viewing angle (in decimal degrees to two decimal places) of a 8mm lens? Of a mm lens? d D d 78. Engineering. For Problem 77, find the length of the belt if D 6 inches, d 4 inches, and C inches. 79. Engineering. The function 4 cos sin cos represents the length of the belt around the two pulles in Problem 77 when the centers of the pulles are inches apart.
14 59 Inverse Trigonometric Functions 439 (A) Graph in a graphing utilit (in radian mode), with the graph covering pulles with their centers from 3 to inches apart. (B) How far, to two decimal places, should the centers of the two pulles be placed to use a belt 4 inches long? Solve b graphing and 4 in the same viewing window and finding the point of intersection using an approimation routine. 8. Engineering. The function 6 cos sin cos E C r r Shadow d A D represents the length of the belt around the two pulles in Problem 78 when the centers of the pulles are inches apart. (A) Graph in a graphing utilit (in radian mode), with the graph covering pulles with their centers from 3 to inches apart. (B) How far, to two decimal places, should the centers of the two pulles be placed to use a belt 36 inches long? Solve b graphing and 36 in the same viewing window and finding the point of intersection using an approimation routine. 8. Motion. The figure represents a circular courtard surrounded b a high stone wall. A floodlight located at E shines into the courtard. (A) If a person walks feet awa from the center along DC, show that the person s shadow will move a distance given b d r r tan r where is in radians. [Hint: Draw a line from A to C.] (B) Find d to two decimal places if r feet and 4 feet. 8. Motion. In Problem 8, find d for r 5 feet and 5 feet.
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