Section 4.8 Solving Problems with Trigonometry

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1 9 Cater Trigonometric Functions. (a) Te orizontal asymtote of te gra on te left is y =. (b) Te two orizontal asymtotes of te gra on te rigt are y= an y=. (c) Te gra of y = sin - a will look like te gra b on te left. () Te gra on te left is ecreasing on bot connecte intervals. (, π/) y (, π/) Section.8 Solving Problems wit Trigonometry Eloration. Te arametrization soul rouce te unit circle.. Te graer is actually graing te unit circle, but te y-winow is so large tat te oint never seems to get above or below te -ais. It is flattene vertically.. Since te graer is lotting oints along te unit circle, it covers te circle at a constant see. Towar te etremes its motion is mostly vertical, so not muc orizontal rogress (wic is all tat we see) occurs. Towar te mile, te motion is mostly orizontal, so it moves faster.. Te irecte istance of te oint from te origin at any T is eactly cos T, an =cos t moels simle armonic motion. Quick Review.8. b= cot.9, c= csc 9.. a= cos 8 9., b= sin 8.8. b=8 cot 8-8 cot., c=8 csc 8 9., a=8 csc.8. b= cot - cot 8., c= csc., a= csc comlement: 8, sulement: 8. comlement:, sulement: Amlitue: ; erio:. Amlitue: ; erio: / Section.8 Eercises All triangles in te sulie figures are rigt triangles.. tan =, so = tan = ft 9. ft. (, π/) y (, π/). tan =, so = tan ft 8. ft. Let be te lengt of te orizontal leg. Ten tan ft =, so = = cot 8. ft. tan 9 ft 9. tan =, so = =9 cot ft. tan 9 ft. Let / be te wire lengt (te yotenuse); ten ft cos 8 =, so /= = sec ft. O cos 8 Let be te tower eigt (te vertical leg); ten tan 8 =, so = tan 8 8. ft. ft ft. cos =, so /= = sec.8 ft. O cos tan =, so = tan.9 ft. ft l ft. tan 8 =, so =8 tan 8 8 ft ft. 8 8 ft 8. Let be te eigt of te smokestack; ten tan 8 =, so =8 tan 8. ft. 8 ft 9. tan 8 =, so = tan 8 89 ft. ft 8 ft

2 Section.8 Solving Problems wit Trigonometry 9. =sin ft.9 ft. tan =, so = tan. m m. tan 9 8 =, so = tan 9 8. ft ft 9 8 ft LP. tan =, so LP=. tan.98 mi.. mi. tan = an tan =, so = cot + an += cot. Ten cot = cot -, so = 9. ft. cot - cot ft ft m. Let be te elevation of te bottom of te eck, an be te eigt of te eck. Ten tan = an ft + tan =, so = tan ft an += ft tan ft. Terefore =(tan -tan ). ft.. Let be te istance travele, an let be car s ening istance from te base of te builing. Ten ft ft tan = an tan =, so += + cot ft an = cot ft. Terefore =(cot -cot ) 9 ft.. Te two legs of te rigt triangle are te same lengt ( knots # r= naut mi), so bot acute angles are. Te lengt of te yotenuse is te istance: L 8.8 naut mi. Te bearing is 9 + =. 8. Te two legs of te rigt triangle are knots # r= 8 naut mi an knots # r= naut mi. Te istance can be foun wit te Pytagorean Teorem: =8 L 8.88 naut mi. Te acute angle at Fort Lauerale as measure tan, so te bearing is +tan 8. (see figure below) Fort 8 Lauerale ft 9. Te ifference in elevations is 9 ft. If te wit of te 9 ft canyon is w, ten tan 9 =, so w w=9 cot 9 8 ft. 9 9 ft w. Te istance from te base of te tower satisfies ft tan =, so = cot ft. ft. Te acute angle in te triangle as measure O 8 - =, so tan =. Ten ft O= tan 8 ft.. tan =, so = tan. mi. mi. If is te eigt of te vertical san, tan =,so. ft =. tan 9.8 ft. ft. Te istance satisfies tan. =, so = cot. 8 ft.. ft. Let be te istance from te boat to te sore, an let be te sort leg of te smaller triangle. For te two triangles, te larger acute angles are an 8. Ten tan 8 = an tan =, or = cot 8 + an += cot. Terefore = 9 ft. cot - cot 8. If t is te time until te boats collie, te law enforcement boat travels t naut mi. During tat same time, te smugglers craft travels t naut mi, were is tat craft s see. Tese two istances are te legs of a rigt triangle t (sown); ten tan =, so = tan t =. knots. t t. (a) Frequency: =8 cycles/sec. = (b) = cos t inces. (c) Wen t=.8,.8; tis is about. in. left of te starting osition (wen t=, =). 8. (a) Frequency: cycle/sec. = = (b) =8 cos t cm. (c) Since cycle takes sec, tere are cycles/min.

3 9 Cater Trigonometric Functions 9. Te frequency is cycles/sec, so = # = raians/sec. Assuming te initial osition is = cm: = cos t.. =8, so = raians/sec.. (a) Te amlitue is a= ft, te raius of te weel. (b) k= ft, te eigt of te center of te weel. (c) rotations/sec, so = / raians/sec. =. (a) = rm= rotations/sec, so = /. t One ossibility is = 8 cos +9 m. (b) [, ] by [, ] (c) (). m; ()= m.. (a) Given a erio of, we ave =. b b = so b = We select te ositive =. value so b =. (b) Using te ig temerature of 8 an a low temerature of 8, we fin a = = so a = 8-8 an we will select te ositive value k = = (c) is alfway between te times of te minimum an maimum. Using te maimum at time t= an te - minimum at time t=, we ave =. So, = + =. () Te fit is very goo for y = sin a t - b +. [, ] by [, 88] Algebraic solution: Solve for t. sin a t - b + = sin a t - b + = sin a t - b = t - = sin- a b t - L.99,.8 Note: sin =sin ( - ) t., 9. Using eiter meto to fin t, fin te ay of te year. 9. as follows: # L 9 an # L 8. Tese reresent May 9 an October.. () All ave te correct erio, but te oters are incorrect in various ways. Equation (a) oscillates between an ±, wile equation (b) oscillates between an ±. Equation (c) is te closest among te incorrect formulas: it as te rigt maimum an minimum values, but it oes not ave te roerty tat ()=8. Tis is accomlise by te orizontal sift in ().. (a) Solve tis graically by fining te zero of te function P = t - sin a t. Te zero occurs at aroimately b.. Te function is ositive to te rigt of te zero. So, te so began to make a rofit in Marc. [, ] by [, 88] (e) Tere are several ways to fin wen te mean temerature will be. Graical solution: Gra te line t= wit te curve sown above, an fin te intersection of te two curves. Te two intersections are at t. an t 9.. [, ] by [, ] (b) Solve tis graically by fining te maimum of te function P = t - sin a t. Te maimum occurs b at aroimately., so te so enjoye its greatest rofit in November. [, ] by [, ]

4 Section.8 Solving Problems wit Trigonometry 9. (a) Using te function W = -.t sin a t b, were t is measure in monts after January of te first year an W is measure in ouns, we ave t= at te beginning. Tis gives 9.8 sin a # W = -. + b = ouns. At te en of two years, t=, wic gives sin a # W = -. b = 8 ouns. (b) Solve tis graically by fining te maimum of te function W = -.t sin a t. Te maimum occurs at t=., were W L. b [, ] by [, ] (c) Solve tis graically by fining te minimum of te function W = -.t sin a t. Te minimum occurs at t L., were W L. b [, ] by [, ]. True. Te frequency an te erio are recirocals: f=/t. So te iger te frequency, te sorter te erio. 8. False. One nautical mile equals about. statute miles, an one knot is one nautical mile er our. So, in te time tat te car travels statute mile, te si travels about. statute miles. Terefore te si is traveling faster. 9. If te builing eigt in feet is, ten tan 8 =/. So = tan 8 L 8. Te answer is D.. By te Law of Cosines, te istance is c= a + b - ab cos u = + + cos naut. mi. Te answer is B.. Moel te tie level as a sinusoial function of time, t. r, min= min is a alf-erio, an te amlitue is alf of -9=. So use te moel f(t)= cos ( t/)- wit t= at 8: PM. Tis takes on a value of at t=. Te answer is D.. Te answer is A.. (a) [,.] by [., ] (b) Te first is te best. Tis can be confirme by graing all tree equations. (c) About oscillations/sec. = L 9. (a) Newborn: about ours. Four-year-ol: about ours. Ault: about ours. (b) Te ault slee cycle is eras most like a sinusoi, toug one migt also ick te newborn cycle. At least one can eras say tat te four-year-ol slee cycles is least like a sinusoi.. Te -gon can be slit into congruent rigt triangles wit a common verte at te center. Te legs of tese triangles measure a an.. Te angle at te center is, so a=. cot cm = L.. Te -gon can be slit into congruent rigt triangles wit a common verte at te center. Te legs of tese triangles measure a an., wile te yotenuse as lengt r. Te angle at te center is, so = r=. csc cm L.8. Coosing oint E in te center of te rombus, we ave AEB wit rigt angle at E, an mjeab=. Ten AE=8 cos in., BE=8 sin in., so tat AC=AE. in. an BD= BE.9 in. 8. (a) BE= tan.8 ft. (b) CD=BE+ tan. ft. (c) AE+ED= sec + sec 9 ft, so te total istance across te to of te roof is about 8 ft. 9. =tan.. mi. Observe tat tere are two (congruent) rigt triangles wit yotenuse mi (see figure below). Te acute angle ajacent to te mi leg as measure cos =cos, so = cos.. raians. Te arc lengt is s=r ( mi)(.) mi. mi mi (.)() =. mi

5 98 Cater Trigonometric Functions. (a). 9 or raians [,.9] by [.,.] (b) One retty goo matc is y=.9 sin[(t-.)] (tat is, a=.9, b=, =.). Answers will vary but soul be close to tese values. A goo estimate of a can be foun by noting te igest an lowest values of Pressure from te ata. For te value of b, note te time between maima (aro..8-.8=. sec); tis is te erio, so b L. Finally, since. L.8 is te location of te first eak after t=, coose so tat (.8-). Tis gives L.. (c) Frequency: about 9 Hz. L. L It aears to be a G. () Eercise a b, so te frequency is again about 9 Hz; it also aears to be a G. Cater Review. On te ositive y-ais (between quarants I an II); 8 # =.. Quarant II; # 8 =.. Quarant III; # =. 8. Quarant IV; # =. 8. Quarant I; 8 # = Quarant II; # = Quarant I; # = Quarant II; # =. 9. or raians For #, it may be useful to lot te given oints an raw te terminal sie to etermine te angle. Be sure to make your sketc on a square viewing winow.. =tan a = = raians b. = = raians. = = raians. = = raians. = +tan ( ) 9.. raians. =tan.. raians. sin = 8. cos = 9. tan( )=. sec( )=. sin. csc = =. sec a - =. tan a - = b b. csc =. sec 8 =. cot( 9 )= 8. tan = 9. Reference angle: ; use a rigt triangle wit = sie lengts, ( ), an (yotenuse). sin a - =, cos a - =, tan a - = ; b b b csc a - =, sec a - = ; cot a - =. b b b. Reference angle: ; use a rigt triangle wit = sie lengts ( ),, an (yotenuse) sin =, cos =, tan = ; csc =, sec = ; cot =.. Reference angle: ; use a rigt triangle wit sie lengts ( ), ( ), an (yotenuse). sin( )=, cos( )=, tan( )=; csc( )=, sec( )=, cot( )=.. Reference angle: ; use a rigt triangle wit sie lengts,, an (yotenuse). sin =, cos =, tan = ; csc =, sec =, cot =.

6 Cater Review 99. Te yotenuse lengt is cm, so sin Å=, cos Å=, tan Å=, csc Å=, sec Å=, cot Å=. For #, since we are using a rigt triangle, we assume tat is acute.. Draw a rigt triangle wit legs (ajacent) an - = =, an yotenuse. sin =, cos =, tan = ; csc =, sec =, cot =.. Draw a rigt triangle wit legs 8 (ajacent) an, an yotenuse 8 + = 89=. 8 sin =, cos =, tan = ; csc =, 8 8 sec =, cot = raians 8.. or.9 raians For #9, coose wicever of te following formulas is aroriate: b a= c - b =c sin Å=c cos ı=b tan Å= tan ı a b= c - a =c cos Å=c sin ı=a tan ı= tan Å a a b b c= a + b = = = = cos ı sin Å sin ı cos Å If one angle is given, subtract from 9 to fin te oter angle. If neiter Å nor ı is given, fin te value of one of te trigonometric functions, ten use a calculator to aroimate te value of one angle, ten subtract from 9 to fin te oter. 9. a=c sin Å= sin 8., b=c cos Å = cos.8, ı=9 -Å=. a= c - b = - 8 =. For te angles, we 8 know cos Å= ; using a calculator, we fin = Å.8, so tat ı=9 -Å.. a. b=a tan ı= tan 8., c= cos ı =., Å=9 -ı= cos 8. a=c sin Å=8 sin 8., b=c cos Å 8 cos 8 =., ı=9 -Å=. a= c - b = - = =.9. For te angles, we know cos Å= ; using a calculator, we fin Å., so tat ı=9 -Å.8.. c= a + b =. +. = 9... For. te angles, we know tan Å= ; using a calculator, we. fin Å 8.9, so tat ı=9 -Å... sin an cos : Quarant III. cos an : Quarant II sin. sin an cos : Quarant II 8. an : Quarant II cos sin 9. Te istance OP=, so sin =, cos =, tan = ; csc =, sec =, cot =.. OP= 9, so sin =, cos =, tan = ; csc =, sec =, cot =.. OP=, so sin =, cos =, tan = ; csc =, sec =, cot =. 9. OP= 9, so sin =, cos =, tan = ; csc =, sec =, cot = Starting from y=sin, translate left units. [, ] by [.,.]. Starting from y=cos, vertically stretc by ten translate u units. [, ] by [, ]. Starting from y=cos, translate left units, reflect across -ais, an translate u units. [, ] by [, ]

7 Cater Trigonometric Functions. Starting from y=sin, translate rigt units, vertically stretc by, reflect across -ais, an translate own units. [, ] by [, ]. Starting from y=tan, orizontally srink by. [.,. ] by [, ] 8. Starting from y=cot, orizontally srink by, vertically stretc by, an reflect across -ais (in any orer). by [, ], 9. Starting from y=sec, orizontally stretc by, vertically stretc by, an reflect across -ais (in any orer). [, ] by [ 8, 8]. Starting from y=csc, orizontally srink by. [, ] by [, ] For #, recall tat for y=a sin[b(-)] or y=a cos[b(-)], te amlitue is a, te erio is, b an te ase sift is. Te omain is always ( q, q), an te range is [ a, a].. f()= sin. Amlitue: ; erio: ; ase sift: ; omain: ( q, q); range: [, ].. g()= cos. Amlitue: ; erio: ; ase sift: ; omain: ( q, q); range: [, ].. f()=. sin c a -. Amlitue:.; erio: 8 b ; ase sift: ; omain: ( q, q); range: [.,.]. 8. g()= sin c a -. Amlitue: ; erio: ; 9 b ase sift: ; omain: ( q, q); range: [, ]. 9. y= cos c a -. Amlitue: ; erio: ; b ase sift: ; omain: ( q, q); range: [, ].. g()= cos c a +. Amlitue: ; erio: ; b ase sift: ; omain: ( q, q); range: [, ]. For # 8, gra te function. Estimate a as te amlitue of te gra (i.e., te eigt of te maimum). Notice tat te value of b is always te coefficient of in te original functions. Finally, note tat a sin[b(-)]= wen =, so estimate using a zero of f() were f() canges from negative to ositive.. a., b=, an., so f(). sin(-.). 8. a., b=, an.8, so f(). sin[(+.8)]. 9. L 9.99 L.8 raians. L.8 L. raians. = raians. = raians. Starting from y=sin, orizontally srink by. Domain: c -. Range: c -.,,. Starting from y=tan, orizontally srink by. Domain: ( q, q ). Range: a -., b. Starting from y=sin, translate rigt unit, orizontally srink by, translate u units. Domain: c,. Range: c -., +. Starting from y=cos, translate left unit, orizontally srink by, translate own units. Domain: [, ]. Range: [, -].

8 Cater Review. = 8. = 9. = sin 8. As Sq, S. 8. As Sq, e / sin(-) S ; as S-q, te function oscillates from ositive to negative, an tens to q in absolute value. 8. tan(tan )=tan = 8. cos =cos a cos = b 8. tan(sin sin )=, were is an angle in c -, cos wit sin =. Ten cos = - sin =. =.8 an tan =. 88. cos cos a - = - b 89. Perioic; erio. Domain Z, n an integer. + n Range: [, q). 9. Not erioic. Domain: ( q, q). Range: [, ]. 9. Not erioic. Domain: Z, n an integer. + n Range: [ q, q). 9. Perioic; erio. Domain: ( q, q). Range: aroimately [,.]. 9. s=r =() a b = 9. Draw a rigt triangle wit orizontal leg (if, raw te orizontal leg rigt; if, raw it left), vertically leg -, an yotenuse. If, let be te acute angle ajacent to te orizontal leg; if, let be te sulement of tis angle. Ten =cos, so - tan =tan(cos )=. 9. tan =, so = tan.8 ft. ft ft ft 9. tan = an tan 8 =, so + = cot an += cot 8. Ten = cot 8 - cot 9 ft. ft PQ 98. tan =, so PQ= tan. mi. 99. See figure below tan = an tan =, so = tan an +8= tan. Ten tan +8= 8 tan, so = ft. tan - tan. tan '=, so = tan ' 9. ft ft ft 8 8 nort tower sout tower 8 ft > < 9. tan 8 =, so = tan 8 m. m ft. Let be te angle of elevation. Note tat sin =, ft so = sin. (a) If =, ten = sin 8 ft. (b) If =, ten = sin ft. m 8 ft

\\ Chapter 4 Trigonometric Functions

$\\ Chapter 4 Trigonometric Functions$ 9 Cater Trigonometric Functions \\ Cater Trigonometric Functions Section. Angles and Teir Measures Eloration. r. radians ( lengts of tread). No, not quite, since te distance r would require a iece of tread