Chapter 2 Trigonometric Functions

Transcription

1 Chaper Trigonomeric Funcions Secion α is a quadran III angle coerminal wih an angle of measure α is a quadran II angle coerminal wih an angle of measure On a TI-8 graphing calculaor, he degree symbol,, and he DMS funcion are locaed in he ANGLE menu α is a quadran IV angle coerminal wih an angle of measure 96.. On a TI-8 graphing calculaor, he degree symbol,, and he DMS funcion are locaed in he ANGLE menu On a TI-8 graphing calculaor, he degree symbol,, and he DMS funcion are locaed in he ANGLE menu A TI-8 calculaor needs o be in degree mode o conver a DMS measure o is equivalen degree measure. On a TI-8 boh he degree symbol,, and he minue symbol, ', are locaed in he ANGLE menu. The second symbol, '', is enered by pressing ALPHA followed by [ '' ] which is locaed on he plus sign, [+], key

2 Secion. 7. A TI-8 calculaor needs o be in degree mode o conver a DMS measure o is equivalen degree measure. On a TI-8 boh he degree symbol,, and he minue symbol, ', are locaed in he ANGLE menu. The second symbol, '', is enered by pressing ALPHA followed by [ '' ] which is locaed on he plus sign, [+], key. 9. A TI-8 calculaor needs o be in degree mode o conver a DMS measure o is equivalen degree measure. On a TI-8 boh he degree symbol,, and he minue symbol, ', are locaed in he ANGLE menu. The second symbol, '', is enered by pressing ALPHA followed by [ '' ] which is locaed on he plus sign, [+], key θ s r θ s r s rθ ( ) in. 67. s rθ ( ) cm 69. θ ( ) 7. θ r θ r ( ) radians or 7 7. ω θ 60 radian/sec 0 7. ω θ 0( ) 60 radians/sec 77. ω θ ( ) 60 0 radians/sec 9.9 radians per second

3 Chaper : Trigonomeric Funcions 79. v ωr mph 8. Speed for ouer ring: Speed for inner ring: v s v s () 60 (8) mph. mph 8. rθ rθ (.)(0 ) (.7) θ 600 θ 00( ) θ The rear gear is making 00 revoluions. The rear gear and ire are making same number of revoluions Tire is inches f s f(00)( ) 88 f 8. s rθ 9,000,000 80,000 mi ( ) The ouer swing has a greaer speed of mph. 87. a. b. ω θ.6 hours.9 radians per hour v s r.6 hours (6 km km).6 hours (699 km).6 hours 7,00 km per hour 89. a. When he rear ire makes one revoluion, he bicycle ravels s rθ r (0 inches) 60 inches. The angular velociy of poin A is ω θ. When he bicycle ravels 60 inches, poin B on he fron ire ravels hough an angle of s 60 inches θ. r 0 inches The angular velociy of poin B is ω θ. Thus, poin B has he greaer angular velociy. 9. a. ( ) ( ) radians ,800 nauical mile s rθ (960 saue miles). saue miles 0,800 b. Poin A and poin B ravel a linear disance of 60 inches in he same amoun of ime. Therefore, boh poins have he same linear velociy. b. Earh s circumference r (960 saue miles) nauical mile 790 nauical miles. saue miles. The quesion, hen, is wha percen of 790 is % 790 /.

4 Secion.... Connecing Conceps 9. A r θ in ( )( ) 9. A r θ ( ) 680 cm Conver o radians. 7 7 radians ,800 s rθ s , To he neares 0 miles, Miami is 780 miles norh of he equaor.... Prepare for Secion. PS. PS. PS. a a a a PS. a a a a PS.. PS Secion.. r + r r y sinθ r cosθ r y anθ y sinθ r 7 cosθ r 7 y anθ r cscθ y r secθ coθ y r 7 cscθ y r 7 7 secθ coθ y

5 6 Chaper : Trigonomeric Funcions. r + r + 9 y 9 sinθ r cosθ r 9 9 y anθ r cscθ y 9 r 9 secθ coθ y 7. ( ) y r 7 sinθ cscθ r 7 7 y 7 r 7 cosθ secθ r 7 7 y anθ coθ y 9. opposie side opp sinθ hyp 6 adj cosθ hyp 6 opp anθ adj hyp 6 cscθ opp hyp 6 secθ adj adj coθ opp. hypoenuse opp sinθ hyp 6 6 adj 6 cosθ hyp 6 6 opp 6 anθ adj hyp 6 cscθ opp 6 hyp 6 secθ adj adj coθ opp 6 y For eercises o, since sin θ, y, r, and. r y. anθ. cosθ r For eercises 6 o 8, since 7. coθ y y an θ, y,, and r +. For eercises 9 o, since sec β r, r,, and y. 9. cos β r. csc β r y For eercises o, since cos θ,, r, and y 9. r. r sec θ

6 Secion. 7. sin + cos + 7. sin 0 cos 60 an 9. sin 0 cos 60 + an + +. sin + cos +. 6 sin + an sec cos an 6 7. csc sec cos 6 9. an cos cos sec an csc h sin h sin h 9. f an in. an9. 7. d + 0. cos8 6 d 6 cos d.f 0 mi d hr d 6 mi h sin 6 d h 6 sin 6 h.7 mi hr 60 min min

7 8 Chaper : Trigonomeric Funcions an 7.8 an f heigh f A bh ( a sin θ )( a cos θ ) a sinθcosθ ,900 sin 0.06 d 670,900 d km sin ,000,000 km 67. d an an.9 d an d an.9 0an6. d an 6. an.9 d 6 f d d an.9 h h an an h h an 7.8 an 00 + h an 00an7.8 h an 7.8 an.0 h.60 0 f h 60 f

8 Secion a. b. ( AC) + 9 AC + 9 AC 8, AC 69. fee h an.6 h an.6 h 9 fee an an. 9 fee... Connecing Conceps Consider he righ riangle formed by A, B and he midpoin of AC. r r r 6 7 r. m if θ d d sinθ d d sin 6 d d sin d 8. f... Prepare for Secion. PS. PS. 0 PS PS PS. PS6. ( ) + ( ) 9+

9 60 Chaper : Trigonomeric Funcions Secion.., y, r + y sinθ r cosθ anθ r y cscθ secθ coθ. 8, y, c ( 8) + ( ) 89 y sin θ cscθ r cos θ secθ r y 8 an θ coθ 8 8., y, r ( ) + ( ) y sinθ r cosθ anθ r y cscθ secθ coθ 7., y 0, r ( ) + ( 0) y 0 sinθ 0 r csc θ is undefined cosθ r secθ y 0 anθ 0 co θ is undefined 9. sin80 0. an80 0. csc90. cos 0 7. an undefined 9. sin. sin θ > 0 in quadrans I and II. cos θ > 0 in quadrans I and IV. quadran I. cos θ > 0 in quadrans I and IV. an θ < 0 in quadrans II and IV. quadran IV. sin θ < 0 in quadrans III and IV. cos θ < 0 in quadrans II and III. quadran III y sinθ, r 7., y, r, ± ( ) ±, in quadran III y y an θ cscθ r, r, y, ± ±, in quadran II, coθ y 9. ( ). is in quadran IV, sin y θ θ, y, r,, r anθ. cos θ, θ is in quadran I or IV. anθ, θ is in quadran I or III. θ is in quadran I,, y, r r csc θ y. cos θ, θ is in quadran II or III. sinθ θ is in quadran II,, y coθ, θ is in quadran I or II. y, r

10 Secion θ 60 Since 90 < θ < 80, θ + θ 80 θ 0 9. θ Since 70 < θ < 60, θ θ 60 θ 9. θ 0 θ >, 0 θ is coerminal wihα. Since 0 < α <, α α θ θ. 8 θ Since < θ <, θ + θ 8 θ

11 6 Chaper : Trigonomeric Funcions. θ θ is coerminal wih α 6. Since 70 < α < 60, α + α 60 α θ 7. θ 7 60 θ is coerminal wih α Since 80 < α < 70, α + 80 α α 80 α 6 θ 6 9. θ is in quadran III. 80 so θ. Thus, sin sin.. θ 0 is in quadran I soθ. Thus, an 0 an.. θ is in quadran III. so θ. Thus, csc sin / sin /. /

12 Secion θ + is coerminal wih in quadran I andθ, 7 so cos cos. 7. θ is coerminal wih in quadran I and θ, so sec 76 sec cos /. 9. θ is coerminal cos80 wih 80, so co 0 co80 sin80 which is undefined., 0 6. sin cos( 6 ) sec sin sec (.) csc sin 0 cos 0 an sin 0 + cos sin an cos 77. ( )( ) 79. sin + cos + +

13 6 Chaper : Trigonomeric Funcions... Connecing Conceps 8. sin θ, θ is in quadran I or quadran II θ 0, 0 8. cos θ, θ is in quadran II or quadran III θ 0,0 8. cscθ θ is in quadran III or IV, θ 87. anθ θ is in quadran II or IV θ, anθ θ is in quadran II or IV θ, sinθ θ is in quadran I or II θ, 9. + an θ sec θ y + y + r sec θ 9. cos(90 θ ) sinθ y y r r... Prepare for Secion. PS. + y (0) + () Yes PS. C r () PS. + y PS. even + + Yes PS. + y + + No PS6. neiher Secion.. 6 y sin sin 6 cos cos y sin 7 sin 6 cos 7 cos 6 The poin on he uni circle corresponding o is 6,. The poin on he uni circle corresponding o 7 is 6,.

14 Secion. 6. y sin sin cos cos 7. 6 y sin sin 6 cos cos 6 The poin on he uni circle corresponding o is,. 9. y sin sin cos cos 0 The poin on he uni circle corresponding o is (,0).. The poin on he uni circle corresponding o is 6,. y sin sin sin cos cos cos The poin on he uni circle corresponding o is,.. an an. cos cos csc csc 9. sin sin. 7 ( ). csc(.0).8 7. an sec sec. sin ( ) cos sec a. sin 0.9 b. cos 0.. a. sin. 0.8 b. cos sin 0. when 0. or.7 9. sin 0. when. or 6.0. f ( ) sin( ) sin f ( ) The funcion defined by f ( ) sin is an odd funcion.. G( ) sin( ) + cos( ) sin + cos The funcion defined by G + is neiher ( ) sin cos an even nor an odd funcion.. S( ) sin( ) sin sin S ( ) The funcion defined by sin( ) S( ) is an even funcion. 7. v( ) sin( ) cos( ) sincos ( ) v The funcion defined by v is an odd funcion. ( ) sin cos

15 66 Chaper : Trigonomeric Funcions cos cos cos ( ) ( ) cos cos cos( + ) cos cos( + ) sin sin ( ) y sin y ( ) sin 6. sin ancos cos cos sin 6. csc sin co cos sin sin sin cos sec cos 6. sec cos cos cos sin an cos 67. an sec an + an an an an an an an an co an co 69. cos sin an sin cos sin cos sin cos cos+ cos cos + cos ( cos)( + cos) cos csc sin 7. sin cos + an + co cos sin an sin cos sin + cos cos sin cos sin sin + cos sin sin csc 7. ( ) ( sin + co sin csc sin sin ) 77. sin + cos sin cos sin ± cos Because 0 < <,sin is posiive. Thus, sin cos.

16 Secion csc + co csc ± + co Because < <, csc is posiive. Thus, csc + co. 8. d ( ) ( ) 970cos 6 d() 970cos 6 970cos 8 70 miles cos cos sin cos cos cos co + co + csc sin sin cscsec co co co cos sin cos sin cos sin cos sin ( sin ) sin + sin 89. (sin cos ) sin sin cos+ cos sincos 9. ( sin )( + sin ) sin cos 9. sin ( sin )( sin ) ( cos )( cos ) + + cos cos sin sin( + cos) sin + + cos+ cos sin( + cos) cos ( cos) + + sin( + cos) sin( + cos) csc sin cos sin cos sin cos+ sin ) 97. an an 6 ( an+ )( an ) 9. ( )( 99. sin sin ( sin+ )( sin )... Connecing Conceps 0. csc, 0 < < sin csc cos + sin cos ± sin cos is posiive in quadran I. cos cos 0. sin, < < sin an cos sin an ± sin Because < <,an is negaive. an ( ) an an

17 68 Chaper : Trigonomeric Funcions 0. sin + cos sin sin csc 07. ( cos )( cos+ ) cos ( ) cos sin... Prepare for Secion. PS. sin 0.7 PS. cos 0.7 PS. Reflec he graph of y f () across he -ais o produce y f (). PS. Compress each poin on he graph of y f () oward he y-ais by a facor of. PS. 6 PS6.. y sin a, p 7. y sin a, p /. y cos a, p 6 / Secion.. y sin a, p 9. y cos a, p. y cos a, p /. y sin a, p. y cos a, p 8 / 7. y.7sin 0.8 a.7, p y sin, a, p. y cos, a, p. y 7 cos, a 7 7, p. y sin, a, p 7. y cos, a, p 9. y sin, a, p. y cos, a, p. y sin, a, p. y cos, a, p /

18 Secion y cos, a, p 6 9. y sin, a, p /. y cos, a, p. y sin, a, p / /. y cos, a, p 7. y sin., a, p. 9. y sin. y cos. y sin. y cos 7., b, a b y cos 9., b, a b y sin 6., b, a b y cos 6. a. Ampliude, a, period, b V sin, 0 8 ms b. Frequency cycles cycle/ms 8 ms 6. ( ) f sin, a, p 67. y cos, a, p / / y cos, a, p

19 70 Chaper : Trigonomeric Funcions 69. y sin 7. y cos 7. sin y e sin The maimum value of e is e. sin The minimum value of e is e sin The funcion defined by y e is periodic wih a period of.... Connecing Conceps 7. a p b b y sin 77. a. p. b b 8 y.sin a p. b b y cos... Prepare for Secion.6 PS. an.7 PS. co 0.6 PS. Srech each poin on he graph of y f () away from he -ais by a facor of o produce y f (). PS. Shif he graph of y f () unis o he righ and up unis. PS. PS6. / Secion.6. y an is undefined for + k, k an ineger.. y sec is undefined for + k, k an ineger.. p 7. p 9. p. / p

20 Secion.6 7. p. p 8 7. p 9. p / /. p 8.. y an, p. /. y co, p 7. y( ) sec, p 9. y csc, p. y an, p. y co, p / /. y csc, p 6 7. / y sec, p 9. y sec, p. y an, p. y csc, p. y sec, p 7. y co, p 9. y an, p., b b y co., b b y csc

21 7 Chaper : Trigonomeric Funcions. 8, b b y sec 7. y an 9. y csc 6. a. an h b. cos.. d h. an d..sec cos c. d. The graph of d is above he graph of h, however, he disance beween he graphs approaches 0 as approaches.... Connecing Conceps 6., b b y an 6., b 8 b 8 y sec 67., b b y co 69.., b b y csc... Prepare for Secion.7 PS. y sin ampliude, period PS. y cos ampliude, period 6 PS. y sin ampliude, period PS. maimum a PS. minimum a PS6. f () cos is symmeric o y ais. Secion.7. a, p, phase shif. a, p, phase shif / 8. a, p, phase shif / 6 7. a, p, phase shif / / 9. p, phase shif /. p 6, phase shif 8 / /

22 Secion.7 7. p, phase shif / 8. p, phase shif 6 / / 7. y sin 0 period, phase shif 9. y cos period, phase shif. y an + < + < < < period, phase shif. y co 8 0 < < 8 < < < < 9 period, phase shif. y sec period, phase shif 7. y csc 0 period 6, phase shif 9. y sin period 6, phase shif. y cos period, phase shif. y sin +, p. y cos, p 7. y sin, p

23 7 Chaper : Trigonomeric Funcions 9. y cos( ) +, p. y sin( + ), p. y sin, p phase shif phase shif phase shif. y an, p 7. y sec, p 9. y csc, p. y sin. y + sin. y sin + cos 7. sine curve, a,, b b phase shif c, c b 6 y sin 9. cosecan curve,, b b phase shif c, c b y csc 6. secan curve,, b, phase shif c, c, y sec b b 6. a. phase shif: c..(6) 7. monhs b /6 period: (6) monhs b /6 b. Firs graph y.cos. Because he phase shif 6 is 7. monhs, shif he graph of y 7. unis o he righ o produce he graph of y. Now shif he graph of y upward 7 unis o produc he graph of S. c. 7. monhs afer January is he middle of Augus.

24 Secion y.sin beween 990 and 006 y.sin ( 6) +.( 6) + y ppm difference 0 ppm 67. rpm a 7 p 0. min p b 0. b b 0 s 7cos Change 6 rpm o radians/second. 6 rpm 6 radians minue minue 60 sec radians/sec in seconds, θ increased by radians. an s 00 00an s for 0 <. or 7. < 0 7. ampliude A k 9 cycles cycle B B 6 y cos A 6:00 P.M.,. y cos y cos + 9 y + 9 y f 7. y sin cos 7. y cos+ sin 77. y sin 79. y sin 8.

25 76 Chaper : Trigonomeric Funcions... Connecing Conceps 8. sine funcion, a, p, b b phase shif c, c b y sin 8. angen funcion, p b b phase shif c c b y an 87. gh [ ( )] ( cos ) + cos sin as 0 9. y sin... Prepare for Secion.8 PS. PS. PS. cos (0) cos0 PS. 8 9 PS. cos 6 cos( ) cos8 PS6. y cos Secion.8. y sin. y cos. y cos ampliude p frequency p ampliude p / frequency p ampliude p frequency p

26 Secion y sin ampliude p / frequency p 9. a p frequency. b b y cos. p. ampliude. b b y cos. ampliude p b b y sin. ampliude p b b y sin 7. ampliude frequency p frequency b b y sin 9. ampliude frequency p b b y cos. ampliude. frequency 0. p b b y.cos. ampliude p b b y cos. ampliude p b b y cos 7. ampliude fee, a k 8 frequency f m period p f y acos f cos cos

27 78 Chaper : Trigonomeric Funcions 9. a. frequency f 9 96 cycles/s period p s 96 b. The ampliude needs o increase.. a. The pseudoperiod is. There are 0 complee oscillaions of lengh in 0 0. b. f ( ) < > ( ) 0.0 for all 9.8 neares enh.. ampliude a 78 7 period p. h 7cos ( ) +.. a. The pseudoperiod is. There are 0 complee oscillaions of lengh in 0 0. b. f ( ) < > ( ) 0.0 for all 7.0 neares enh. Xmin 6,Xma 6,Xscl, Ymin 0.0, Yma 0.0, Yscl a. The pseudoperiod is. There are 0 complee oscillaions of lengh in 0 0. b. f ( ) < > ( ) 0.0 for all 9. neares enh. Xmin 70, Xma 7, Xscl Ymin 0.0, Yma 0.0, Yscl a. The pseudoperiod is. There are 0 complee oscillaions of lengh in 0 0. b. f ( ) < > ( ) 0.0 for all 6. neares enh. Xmin 8, Xma 0, Xscl, Ymin 0.0, Yma 0.0, Yscl 0.00 Xmin,Xma 7,Xscl, Ymin 0.0, Yma.0, Yscl Connecing Conceps m 9m. p, p k k p Increasing he main mass o 9 m will riple he period.. yes Xmin 0, Xma, Xscl, Ymin,Yma.,Yscl 0.

28 Assessing Conceps 79. yes Xmin 0, Xma 0, Xscl, Ymin, Yma 9, Yscl... Eploring Conceps wih Technology Sinusoidal Families.. All hree sine graphs have, a period of, -inerceps a n, and no phase shif, bu heir ampliudes are,, and 6 respecively. All hree sine graphs have a period of and an ampliude of, bu heir phase shifs are /, / 6, and /, respecively.... Assessing Conceps. True. False; sec θ an θ is an ideniy.. False; rad 7... True. 6. (0, ) 7. The period is Shif he graph of y o he lef unis. / 9. All real numbers ecep muliples of. 0. The verical asympoes are and.

29 80 Chaper : Trigonomeric Funcions... Chaper Review. complemen: 90 6 [.] supplemen:80 6. θ [.] θ is coerminal wih α 60 and θ' α'. Since 80 < α < 70, 80 + α ' α 80 + a 60 α ' 80 θ [.]. [.]. s rθ 7 ( ) [.] m 6. s θ [.] r 0 0. V w r rad/sec [.] r For eercises 8 o, csc θ, r, y, and y. 8. cosθ [.] 9. coθ [.] r y 0. y sin θ [.]. sec θ r [.] r., y, r + 0 ( ) sinθ 0 0 cosθ anθ cscθ 0 secθ 0 coθ

30 Chaper Review 8. a. b. sec 0 an c. co ( ) d. cos [.]. a. cos 0.6 b. co. 0.6 c. sec6.6 d. an.0777 [.] cos φ,, r, y [.] r. ( ) a. b. y sinφ r y anφ y an φ, y,, r + [.] 6. ( ) ( ) a. b. r secφ r cscφ y sin φ, y, r, [.] 7. ( ) a. cosφ r b. coφ y 8. a. W ( ) (,0) [.] b. c. W, W, 9. f ( ) sin( ) an ( ) f ( ) sin( ) an ( ) ( sin )( an ) sin an f ( ) [.] The funcion defined by f ( ) sin( ) an( ) is an even funcion.

31 8 Chaper : Trigonomeric Funcions 0.. cos cos ( ) + cos ( ) + cos y an( ) y an an an ( ). sin φ + + an φ [.]. cos φ sec φ sinφ + anφ + cosφ coφ + cosφ + sinφ sinφ+ cosφ cosφ cosφ+ sinφ sinφ ( + ) ( + ) sinφ sinφ cosφ cosφ cosφ sinφ anφ [.]. cos φ + sin φ cscφ cscφ sinφ [.]. sin φ(an φ + ) sin φsec φ [.] sin φ cos φ an φ 6. an φ + + [.] an an φ φ sec φ an φ 7. cos φ sin φ sin φ sin φ 0 [.] cos sin cos φ φ φ sin φ csc φ 8. y cos( ) [.] a ; period b c phase shif b 9. y an [.6] no ampliude; period b phase shif 0

32 Chaper Review 8 0. y sin + [.] a ; period b c / phase shif b 9. y cos + [.] a ; period b c / phase shif b. y sec [.6] no ampliude; period b c / phase shif b 8. y csc [.6] no ampliude; period b c / phase shif b. y cos, p. y sin, p 6. / y sin, p / 7. y cos, p phase shif 8. y sin, p + phase shif 8 9. y cos ( ), p phase shif 0. y an, p. y co, p. y an, p / phase shif. y co, p + phase shif 8. y csc, p. y sec +, p phase shif phase shif 6

33 8 Chaper : Trigonomeric Funcions 6. y sin 7. y cos+ 8. y cos y sin 0. sin y. y sin cos.. h sin.. h.sin mi [.]. Speed for inner ring: Speed for ouer ring: v s v s (.) () f/s f/s The ouer swing has a greaer speed of f/s. [.]. h an. 8. h 8.an.. fee [.] () co8 + h h h () co7 h Subsiue for in equaion (). h 80 co8 + co 7 h 80 Solve for h. co8 co 7 h h 80 co8 co 7 80 h 6 f co8 co 7 [.]

34 Chaper Tes 8 6. y.sin0 [.8] ampliude. p b 0 frequency p 7. ampliude 0. [.8] k 0 f m p y 0.cos f 0.cos y 0.cos 8. f ( ) 0.0 for all 7. < > [.8] Xmin, Xma 0, Xcsl Ymin.0, Yma.0, Yscl Quaniaive Reasoning QR. a. m n m n () () period 6 c. / m n m / n period () () e. / m n m / n period 7. () () b. / m n m n period (6) () d. / m n 8 m 8 / n period (6) 8 (9) f. / m n m n period () () QR.. m.n. m period. s. n.(9).()... Chaper Tes. 0 0 [.] [.]. s rθ [.] s 0( 7 ) 80 s. cm. rev w 6 [.] sec rev rad w 6 sec rev w rad/sec. v rw [.] cm/sec 6. r 7 + r 8 8 secθ [.] 7

35 86 Chaper : Trigonomeric Funcions 7. csc [.] 8. an cos sin [.] 9. [.] 6 cos y sin W(, y) W, 0. sec cos sec cos cos cos cos cos sin [.]. period [.6]. b a ; period [.7] b phase shif. period [.7] /. y cos, p phase shif c /6 b /. y sec, p 6. Shif he graph [of y sin( )] [.7] unis o he righ and down uni. 7. y sin 8. y sin cos

36 Cumulaive Review p a, b b y cos or y sin [.8] an. h h an. h co. an 7. h. h. + h co. Solve for h. h an hco. an7. (. + hco. ) h.an7. + han7. co. h h han7. co..an7. h( an7. co. ).an7..an7. h an7. co. h. meers The heigh of he ree is approimaely. meers. [.]... Cumulaive Review. d ( ( )) + ( ) [.]. c a + b + b b b [.]. Inerceps: (, 0), (, 0), ( 9, 0) [.]. f ( ) f( ) ( ) + + Odd funcion [.]

37 88 Chaper : Trigonomeric Funcions. f( ) y y (y ) y y y y y( ) y f ( ) [.] 6. Domain: (,) (, ) [.] ( + )( ) The soluions are and. [.] 8. Shif he graph of y f () horizonally unis o he righ. [.] 9. Reflec he graph of y f () across he y-ais. [.[ [.]. 80 [.]. f sin sin + 6 [.]. f sin + sin [.]. cos sin [.]. negaive [.] 6. θ 0 [.] Since 80 < θ < 70, θ + 80 θ θ 0 7. θ [.] Since < θ <, θ + θ θ 8. Domain: (, ) [.] 9. Range: [, ] [.] 0. opp anθ adj hypoenuse opp sinθ [.] hyp