Trigonometric Functions of Any Angle. cos. sin r. cot, r csc, y. y 0

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1 0_00.qd 1 1/7/05 Chapte. 11:05 AM Page 1 Tigonomet Tigonometic Functions of An Angle What ou should lean Evaluate tigonometic functions of an angle. Use efeence angles to evaluate tigonometic functions. Evaluate tigonometic functions of eal numbes. Wh ou should lean it You can use tigonometic functions to model and solve eal-life poblems. Fo instance, in Eecise 87 on page 19, ou can use tigonometic functions to model the monthl nomal tempeatues in New Yok Cit and Faibanks, Alaska. Intoduction In Section., the definitions of tigonometic functions wee esticted to acute angles. In this section, the definitions ae etended to cove an angle. If is an acute angle, these definitions coincide with those given in the peceding section. Definitions of Tigonometic Functions of An Angle Let be an angle in standad position with, a point on the teminal side of and 0. sin tan, cos 0 cot, 0 csc, sec, (, ) 0 0 Because cannot be zeo, it follows that the sine and cosine functions ae defined fo an eal value of. Howeve, if 0, the tangent and secant of ae undefined. Fo eample, the tangent of 90 is undefined. Similal, if 0, the cotangent and cosecant of ae undefined. Eample 1 Evaluating Tigonometic Functions Let, be a point on the teminal side of. Find the sine, cosine, and tangent of. Refeing to Figue.6, ou can see that,, and 5 5. James Ubach/SupeStock (, ) So, ou have the following. sin 5 cos 5 tan 1 FIGURE Now t Eecise 1.

2 0_00.qd 1/7/05 11:05 AM Page 1 Section. Tigonometic Functions of An Angle 1 < < < 0 > 0 0 < < > 0 > 0 The signs of the tigonometic functions in the fou quadants can be detemined easil fom the definitions of the functions. Fo instance, because cos, it follows that cos is positive wheeve > 0, which is in Quadants I and IV. (Remembe, is alwas positive.) In a simila manne, ou can veif the esults shown in Figue.7. < 0 < 0 < < > 0 < 0 < < Eample Evaluating Tigonometic Functions Given tan 5 and cos > 0, find sin and sec. Note that lies in Quadant IV because that is the onl quadant in which the tangent is negative and the cosine is positive. Moeove, using Quadant II sin : + cos : tan : Quadant I sin : + cos : + tan : + tan 5 and the fact that is negative in Quadant IV, ou can let 5and. So, and ou have Quadant III sin : cos : tan : + Quadant IV sin : cos : + tan : sin FIGURE.7 sec Now t Eecise 17. Eample Tigonometic Functions of Quadant Angles (0, 1) Evaluate the cosine and tangent functions at the fou quadant angles 0, and,,. To begin, choose a point on the teminal side of each angle, as shown in Figue.8. Fo each of the fou points, 1, and ou have the following. cos 0 tan 0, 1, ( 1, 0) (1, 0) 0 cos tan 1 0 undefined, 0, 1 FIGURE.8 (0, 1) cos cos tan tan 1 0 undefined, 1, 0, 0, 1 Now t Eecise 9.

3 0_00.qd 1/7/05 11:05 AM Page 1 1 Chapte Tigonomet Sketching seveal angles with thei efeence angles ma help einfoce the fact that the efeence angle is the acute angle fomed with the hoizontal. Refeence Angles The values of the tigonometic functions of angles geate than 90 (o less than 0) can be detemined fom thei values at coesponding acute angles called efeence angles. Definition of Refeence Angle Let be an angle in standad position. Its efeence angle is the acute angle fomed b the teminal side of and the hoizontal ais. Figue.9 shows the efeence angles fo in Quadants II, III, and IV. Quadant II Refeence angle: Refeence angle: Refeence angle: = (adians) = 180 (degees) Quadant III = (adians) = 180 (degees) Quadant IV = (adians) = 60 (degees) FIGURE.9 = 00 FIGURE.0 =. FIGURE.1 = 5 5 FIGURE. = 60 =. 5 and 15 ae coteminal. = 15 Eample Finding Refeence Angles Find the efeence angle. 00 a. b. c. a. Because 00 lies in Quadant IV, the angle it makes with the -ais is Degees Figue.0 shows the angle and its efeence angle b. Because. lies between and it follows that it is in Quadant II and its efeence angle is Radians Figue.1 shows the angle and its efeence angle c. Fist, detemine that 15 is coteminal with 5, which lies in Quadant III. So, the efeence angle is Degees Figue. shows the angle 00. Now t Eecise , 15 and its efeence angle

4 0_00.qd 1/7/05 11:05 AM Page 15 (, ) Section. Tigonometic Functions of An Angle 15 Tigonometic Functions of Real Numbes To see how a efeence angle is used to evaluate a tigonometic function, conside the point, on the teminal side of, as shown in Figue.. B definition, ou know that opp = hp adj opp, adj FIGURE. sin and tan. Fo the ight tiangle with acute angle and sides of lengths and, ou have sin opp hp and So, it follows that sin and sin ae equal, ecept possibl in sign. The same is tue fo tan and tan and fo the othe fou tigonometic functions. In all cases, the sign of the function value can be detemined b the quadant in which lies. tan opp adj. Evaluating Tigonometic Functions of An Angle To find the value of a tigonometic function of an angle : 1. Detemine the function value fo the associated efeence angle.. Depending on the quadant in which lies, affi the appopiate sign to the function value. Leaning the table of values at the ight is woth the effot because doing so will incease both ou efficienc and ou confidence. Hee is a patten fo the sine function that ma help ou emembe the values. sin Revese the ode to get cosine values of the same angles. B using efeence angles and the special angles discussed in the peceding section, ou can geatl etend the scope of eact tigonometic values. Fo instance, knowing the function values of 0 means that ou know the function values of all angles fo which 0 is a efeence angle. Fo convenience, the table below shows the eact values of the tigonometic functions of special angles and quadant angles. Tigonometic Values of Common Angles (degees) (adians) 0 6 sin cos tan 0 1 Undef. 0 Undef

5 0_00.qd 1/7/05 11:05 AM Page Chapte Tigonomet Eample 5 Using Refeence Angles Evaluate each tigonometic function. a. cos b. tan10 c. csc 11 a. Because lies in Quadant III, the efeence angle is, as shown in Figue.. Moeove, the cosine is negative in Quadant III, so cos cos 1. Emphasize the impotance of efeence angles in evaluating tigonometic functions of angles geate than 90. b. Because , it follows that 10 is coteminal with the second-quadant angle 150. So, the efeence angle is , as shown in Figue.5. Finall, because the tangent is negative in Quadant II, ou have tan10 tan 0 c. Because 11, it follows that 11 is coteminal with the second-quadant angle. So, the efeence angle is as shown in Figue.6. Because the cosecant is positive in Quadant II, ou have, csc 11 csc. 1 sin. = = = 0 = 10 = = 11 FIGURE. FIGURE.5 FIGURE.6 Now t Eecise 51.

6 0_00.qd 1/7/05 11:05 AM Page 17 Section. Tigonometic Functions of An Angle 17 Eample 6 Using Tigonometic Identities Let be an angle in Quadant II such that sin 1. Find (a) cos and (b) tan b using tigonometic identities. a. Using the Pthagoean identit sin cos 1, ou obtain 1 cos 1 1 Substitute fo sin. cos Students often have difficult detemining angles, especiall when the functions given ae csc, sec, and/o cot. Have ou students ewite the epession in tems of sin, cos, o tan, whicheve is applicable, befoe evaluating. Activities 1. Detemine the eact values of the si tigonometic functions of the angle in standad position whose teminal side contains the point, 7. Answe: sin 7 58 cos 58 tan 7 cot 7. Fo the angle find the efeence angle, and sketch and in standad position. Answe: 5 = 5. Find two values of, 0 <, that satisf the equation tan 1. Do not use ou calculato. Answe:, 7 15, csc 58 7 sec 58 = 15 Because cos 8 9 in Quadant II, ou can use the negative oot to obtain b. Using the tigonometic identit tan sin, ou obtain cos tan 1 Now t Eecise 59. Substitute fo sin and cos. You can use a calculato to evaluate tigonometic functions, as shown in the net eample. Eample 7 Using a Calculato Use a calculato to evaluate each tigonometic function. a. cot 10 b. sin7 c. sec 9 Function Mode Calculato Kestokes Displa a. cot 10 Degee TAN 10 1 ENTER b. sin7 Radian SIN 7 ENTER cos < c. sec Radian COS ENTER Now t Eecise 69.

7 0_00.qd 1/7/05 11:05 AM Page Chapte Tigonomet. Eecises VOCABULARY CHECK: In Eecises 1 6, let be an angle in standad position, with, a point on the teminal side of and sin.. tan. sec The acute positive angle that is fomed b the teminal side of the angle and the hoizontal ais is called the angle of and is denoted b. PREREQUISITE SKILLS REVIEW: Pactice and eview algeba skills needed fo this section at In Eecises 1, detemine the eact values of the si tigonometic functions of the angle. 1. (a) (b). (a) (b) ( 1, 5). (a) (b) (, 1). (a) (b) (, ) (, 1) ( 1, 1) (, 1) In Eecises 5 10, the point is on the teminal side of an angle in standad position. Detemine the eact values of the si tigonometic functions of the angle. 5. 7, 6. 8, 15 7., , (8, 15) (, ) 9..5, , 7 In Eecises 11 1, state the quadant in which 11. sin < 0 and 1. sin > 0 and 1. sin > 0 and 1. sec > 0 and lies. In Eecises 15, find the values of the si tigonometic functions of with the given constaint. cos < 0 cos > 0 tan < 0 cot < 0 Function Value Constaint 15. sin 5 lies in Quadant II. 16. cos 5 lies in Quadant III. 17. tan cos cot 0. csc 1. sec sin < 0 tan < 0 cos > 0 cot < 0 sin > 0. sin 0 sec 1. cot is undefined.. tan is undefined. In Eecises 5 8, the teminal side of lies on the given line in the specified quadant. Find the values of the si tigonometic functions of b finding a point on the line. Line Quadant 5. II 6. 1 III 7. 0 III 8. 0 IV

8 0_00.qd 1/7/05 11:05 AM Page 19 Section. Tigonometic Functions of An Angle 19 In Eecises 9 6, evaluate the tigonometic function of the quadant angle. 9. sin sec. sec. sin. cot 5. csc 6. cot In Eecises 7, find the efeence angle, and sketch and in standad position In Eecises 5 58, evaluate the sine, cosine, and tangent of the angle without using a calculato csc In Eecises 59 6, find the indicated tigonometic value in the specified quadant. Function Quadant Tigonometic Value 59. sin 5 IV cos 60. cot II sin 61. tan III sec 6. csc IV cot 6. cos 5 8 I sec 6. sec 9 III tan In Eecises 65 80, use a calculato to evaluate the tigonometic function. Round ou answe to fou decimal places. (Be sue the calculato is set in the coect angle mode.) 65. sin sec cos csc0 69. tan cot sec 7 7. tan tan.5 7. cot tan 76. tan sin sec cot In Eecises 81 86, find two solutions of the equation. Give ou answes in degees 0 < 60 and in adians 0 <. Do not use a calculato. 81. (a) sin 1 (b) sin 1 8. (a) cos (b) cos 8. (a) csc (b) cot 1 8. (a) sec (b) sec 85. (a) tan 1 (b) cot 86. (a) sin (b) sin Model It 9 csc Data Analsis: Meteoolog The table shows the monthl nomal tempeatues (in degees Fahenheit) fo selected months fo New Yok Cit N and Faibanks, Alaska F. (Souce: National Climatic Data Cente) Month New Yok Faibanks, Cit, N F Janua 10 Apil 5 Jul 77 6 Octobe 58 Decembe 8 6 (a) Use the egession featue of a gaphing utilit to find a model of the fom a sinbt c d fo each cit. Let t epesent the month, with t 1 coesponding to Janua.

9 0_00.qd 1/7/05 11:05 AM Page 0 0 Chapte Tigonomet Model It (continued) (b) Use the models fom pat (a) to find the monthl nomal tempeatues fo the two cities in Febua, Mach, Ma, June, August, Septembe, and Novembe. (c) Compae the models fo the two cities. d 6 mi Not dawn to scale 88. Sales A compan that poduces snowboads, which ae seasonal poducts, foecasts monthl sales ove the net eas to be t S.1 0.t. cos 6 whee S is measued in thousands of units and t is the time in months, with t 1 epesenting Janua 006. Pedict sales fo each of the following months. (a) Febua 006 (b) Febua 007 (c) June 006 (d) June Hamonic Motion The displacement fom equilibium of an oscillating weight suspended b a sping is given b t cos 6t whee is the displacement (in centimetes) and t is the time (in seconds). Find the displacement when (a) t 0, (b) t 1 and (c) t 1,. 90. Hamonic Motion The displacement fom equilibium of an oscillating weight suspended b a sping and subject to the damping effect of fiction is given b t e t cos 6t whee is the displacement (in centimetes) and t is the time (in seconds). Find the displacement when (a) t 0, (b) t 1 and (c) t 1,. 91. Electic Cicuits The cuent I (in ampees) when 100 volts is applied to a cicuit is given b I 5e t sin t whee t is the time (in seconds) afte the voltage is applied. Appoimate the cuent at t 0.7 second afte the voltage is applied. 9. Distance An aiplane, fling at an altitude of 6 miles, is on a flight path that passes diectl ove an obseve (see figue). If is the angle of elevation fom the obseve to the plane, find the distance d fom the obseve to the plane when (a) (b) and (c) 0, 90, 10. FIGURE FOR 9 Snthesis Tue o False? In Eecises 9 and 9, detemine whethe the statement is tue o false. Justif ou answe. 9. In each of the fou quadants, the signs of the secant function and sine function will be the same. 9. To find the efeence angle fo an angle (given in degees), find the intege n such that 0 60n 60. The diffeence 60n is the efeence angle. 95. Witing Conside an angle in standad position with 1 centimetes, as shown in the figue. Wite a shot paagaph descibing the changes in the values of,, sin, cos, and tan as inceases continuousl fom 0 to Witing Eplain how efeence angles ae used to find the tigonometic functions of obtuse angles. Skills Review 1 cm (, ) In Eecises , gaph the function. Identif the domain and an intecepts and asmptotes of the function f g h 1 f ln 106. log 10