\\ Chapter 4 Trigonometric Functions

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1 c0 0.qd // :7 PM Page 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 times as long, and >.. radians Quick Review.. C= #.= in.. C= #.=9. m.. r = # 8 = = ft. (a) s=7. ft (b) s=9.77 km. (a) v=. m/sec (b) v=8.0 ft/sec r = # = m 0 mi r # 80 ft mi # mi r # 80 ft mi # 8.8 ft sec # r = 88 ft/sec sec r = ft/sec sec mi # 00 sec 80 ft r = m 0. ft # mi # 00 sec = 90 m sec 80 ft r Section. Eercises. '= a + =. 0 b. '= a + =. 0 b. 8 '"= a8 + = b. 8 0'"= a8 + 0 = b.. = (0 # 0.)'= '. 9.7 =9 (0 # 0.7)'=9 ' =8 (0 # 0.)'=8 9.' =8 9'(0 # 0.)"=8 9'" =99 (0 # 0.7)'=99.' =99 '(0 # 0.)"=99 '" For #9, use te formula s=r, and te equivalent forms r=s/ and =s/r # rad 80 = # rad 80 =. 0 # rad 80 =. 0 # rad 80 =. 7.7 #.8 rad # 0.0 rad 80. '= a +.07 rad 0 b =. # '= =7. # a rad 0 b # 80 = 0 # 80 = # 80 0 = 8 # 80 = # 80 = 0 0 # 80 = 7 # 80 L.9.. # 80 L 7.8. s=0 in.. s=70 cm 7. r=/ ft 8. r=7./ cm 9. = radians 0. = radians 7 0. r= cm Corigt 0 Pearson Education, Inc.

2 c0 0.qd /8/ 0:08 AM Page Section. Angles and Teir Measures. s=( ft)(8 ) a ft 0 b =. = s /r = 9 rad and s = r u= cm. = s /r =. rad and r = s /u = km. Te angle is 0 #, so te curved side 80 = 8 rad measures Te two straigt sides measure in. 8 in. eac, so te erimeter is ++ L in. 8. Te angle is 00 #, so 80 = 9 rad 7 = 9 r. Ten r = L cm. 7. Five ieces of track form a semicircle, so eac arc as a central angle of / radians. Te inside arc lengt is r i (/) and te outside arc lengt is r o (/). Since r o (/) - r i (/) =. inces, we conclude tat r o - r i =.(/) L. inces. 8. Let te diameter of te inner (red) circle be d. Te inner circle s erimeter is 7.7 inces, wic equals d. Ten te net-largest (ellow) circle as a erimeter of (d + + ) = d + = L 7. inces. 9. (a) NE is. (b) NNE is.. (c) WSW is (a) SSW is 0.. (b) WNW is 9.. (c) NNW is 7... ESE is closest at... SW is closest at.. Te angle between tem is =9 '= radians, so te distance is about s=r =()(0.9). statute miles.. Since C = d, a tire travels a distance d wit eac revolution. (a) Eac tire travels at a seed of 800 d in. er min, or 800d in. min a ba0 min r ba mi b L.8d mi/r.,0 in. Veicle d Seed.8d Nissan Leaf SL. 0.7 m Cev Volt.. m Tesla S m d in. (b) a so eac mile rev ba mi,0 in. b = d,0 mi/rev,,0 requires L 0,8 revolutions. d d 0,8 Leaf: L revolutions. 0,8 Tesla: L revolutions 7.7 Te Leaf must make just over more revolutions. (c) In eac revolution, te tire would cover a distance of d new rater tan d old, so tat te car would travel (d new ) (d old ) = d new /d old = 7.7/. L.08 miles for ever mile te car s instruments would sow. Bot te odometer and seedometer readings would be low.. v= ft/sec and r=0 in., so =v/r= a ft # 0 sec sec min b, a0 in. # ft # rad in. rev b 7.0 rm. S. (a) mm. W = R 00 Q S = WR 00. mm= in., so S = WR # in. 00. = WR 0 9. WR (b) D+S=D+ a WR =D+ in. 0 b 70 (c) Leaf: D = 7 + # 0 L. in. 70 Volt: 7 8 mi 0 D = 7 + # 70 mi # 8 b. L. in. Tesla S: D = 9 + # L 7.7 in. 70 Escalade: D = 8 + # L. in =000 rm and r= in., so v=r = a in. # teet b # in. a000 rev # rad # min,.7 teet er min rev 0 sec b second. 8. c. a. 9 stat mi 0. 7 naut mi 9 statute miles 0,800 naut mi 0,800 naut mi. 89 stat mi # 778 nautical miles 9 stat mi. (a) Lane as inside radius 7 m, wile te inside radius of lane is 8 m, so over te wole semicircle, te difference is 8-7 =. m. (Tis would be te answer for an two adjacent lanes.) (b) 8 - =.708 m. Corigt 0 Pearson Education, Inc.

3 c0 0.qd // :7 PM Page Cater Trigonometric Functions. (a) s=r =()( )= 0. in., or.89 ft (b) r =.8 ft. s=r =() a = ft 80 b rev. (a) =0 # rad # min v = rad/sec min rev 0 sec (b) v= Rv =(7 cm) a rad =8 cm/sec sec b (c) v =v/r= a8 cm ( cm)=7 rad/sec sec b rev. (a) = # rad # min =. rad/sec min rev 0 sec (b) v= rv =(. m) a. rad =. m/sec sec b (c) Te radius to tis alfwa oint is r*= r=0. m, so v=r* =(0. m) a. rad =.7 m/sec. sec b 7. True. In te amount of time it takes for te merr-goround to comlete one revolution, orse B travels a distance of r, were r is B s distance from te center. In te same time, orse A travels a distance of (r)=( r) twice as far as B. 8. False. If all tree radian measures were integers, teir sum would be an integer. But te sum must equal, wic is not an integer. 9. = a rad Te answer is C. 80 b = If te erimeter is times te radius, te arc is two radii long, wic imlies an angle of radians. Te answer is A.. Let n be te number of revolutions er minute. in. rev min a ban ba0 rev min r ba mi,0 in. b L n m. Solving n=0 ields n L 9. Te answer is B.. Te size of te circle does not affect te size of te angle. Te radius and te subtended arc lengt bot double, so tat teir ratio stas te same. Te answer is C. In #, we need to borrow and cange it to 0' in order to comlete te subtraction.. '-8 '=8 0'. 7 09'-7 0'= 09'. 9 '-87 9'=9 7'-87 9'= 7'. 0'-80 '= 08' In #7 70, find te difference in te latitude. Convert tis difference to minutes; tis is te distance in nautical miles. Te Eart s diameter is not needed. 7. Te difference in latitude is 0'- '= 0' =80 minutes of arc, wic is 80 naut mi. 8. Te difference in latitude is 7 '-7 7'=9 9' =89 minutes of arc, wic is 89 naut mi. 9. Te difference in latitude is 9'-9 7'= 0' =90 minutes of arc, wic is 90 naut mi. 70. Te difference in latitude is 0'- '=8 ' = minutes of arc, wic is naut mi. 7. Te wole circle s area is r ; te sector wit central u angle makes u / of tat area, or # r = ur. 7. (a) A= (.9) a.8 =0.9 ft. b (b) A= (.) (.7)=.7 km. 7. B mi A 7. Bike weels: v = v /r=( ft/sec # in./ft) ( in.).7 rad/sec. Te weel srocket must ave te same angular velocit: v = v.7 rad/sec. For te edal srocket, we first need te velocit of te cain, using te weel srocket: v L A in.ba.7 rad/secb 8.87 in./sec. Ten te edal srocket s angular velocit is v = (8.87 in./sec) (. in.) 8.9 rad/sec. Section. Trigonometric Functions of Acute Angles Eloration. sin and csc, cos and sec, and tan and cot. tan. sec.. sin and cos Eloration. Let =0. Ten sin = 0.8 csc = =. cos = sec = tan =.7 cot = = Te values are te same, but for different functions. For eamle, sin 0 is te same as cos 0, cot 0 is te same as tan 0, etc.. Te value of a trig function at is te same as te value of its co-function at Corigt 0 Pearson Education, Inc.

4 c0 0.qd // :7 PM Page 7 Section. Trigonometric Functions of Acute Angles 7 Quick Review. 8. Te adjacent side lengt is - 9 = 88 =,. = + = 0= 9 9 so sin =, cos =, tan = ; csc =,. = 8 + = 08= 9. = 0-8 = sec =, cot =.. = - 9 = =. 8. ft # in. 9. Using a rigt triangle wit otenuse 7 and legs = 00.8 in. (oosite) and 7 - = 0 = 0 (adjacent), ft ft # mi = 7 we ave sin =, cos =, tan = ; L mi ft a=(0.88)(0.)=7.9 km csc =, sec =, cot = b =.9 L.89 ft 0. Using a rigt triangle wit otenuse and legs.7 (oosite) and - = (adjacent), we ave 9. Å=. #..00 (no units) sin =, cos =, tan = ; csc =,. 0. ı=.9 #..899 (no units) sec =, cot =. 8.. Using a rigt triangle wit otenuse and legs Section. Eercises (adjacent) and - = 9 = (oosite),. sin =, cos =, tan =, csc =, sec =, we ave sin =, cos =, tan = ; cot =. csc =, sec =, cot = sin =, cos =, tan = ; csc =, 7 8. Using a rigt triangle wit otenuse 8 and legs (adjacent) and 8 - = 9 (oosite), we ave 7 sec =, cot = sin =, cos =, tan = ; csc =, sin =, cos =, tan = ; csc =, 8 sec =, cot =. 9 sec =, cot =.. Using a rigt triangle wit legs (oosite) and sin =, cos =, tan = ; csc =, 9 (adjacent) and otenuse + 9 = 0, we ave sec =, cot =. sin =, cos =, tan = ; csc =, Te otenuse lengt is 7 + = 70, so 0 9 sec =, cot = sin =, cos =, tan = ; csc =, Using a rigt triangle wit legs (oosite) and 70 sec =, cot =. (adjacent) and otenuse + =, 7. Te adjacent side lengt is 8 - we ave sin =, cos =, tan = ; = 8 = 7, so 7 sin =, cos =, tan = ; csc =, csc =, sec =, cot = Using a rigt triangle wit legs (oosite) and sec =, cot =. 7 (adjacent) and otenuse + = 0, 7. Te oosite side lengt is - 8 = 7, so we ave sin =, cos =, tan = ; sin =, cos =, tan = ; csc =, csc =, sec =, cot =. sec =, cot =. 8 7 Corigt 0 Pearson Education, Inc.

5 c0 0.qd // :7 PM Page 8 8 Cater Trigonometric Functions. Using a rigt triangle wit otenuse and legs (oosite) and - = 9 (adjacent), 9 we ave sin =, cos =, tan = ; 9 9 csc =, sec =, cot = Using a rigt triangle wit otenuse and legs 9 (oosite) and - 9 = 8 = 87 (adjacent), we ave sin =, cos =, tan = ; csc =, sec =, cot = Using a rigt triangle wit otenuse 7 and legs (adjacent) and 7 - = = (oosite), we ave sin =, cos =, tan = ; csc =, sec =, cot = = =. sec = /cos L.. Squaring tis result ields.0000, so sec =.. sin 0 L Squaring tis result ields 0.700=/, so sin 0 = / = /. 7. csc (/) = /sin (/) L.7. Squaring tis result ields. or essentiall /, so csc (/) = / = / = /. 8. tan (/) L.70. Squaring tis result ields.0000, so tan (/) =. For #9 0, te answers marked wit an asterisk (*) sould be found in DEGREE mode; te rest sould be found in RADI- AN mode. Since most calculators do not ave te secant, cosecant, and cotangent functions built in, te recirocal versions of tese functions are sown * 0. 0.*. 0.9*. 0.9* cos 9 L.* sin 9 L.07* tan 0.89 L 0.80 cos. L tan (/8) L.. =0 =. =0 =. =0 =. = =. =0 =. = = 7. =0 = 8. =0 = sin (/0) L. 9. = 0. z = sin L.8 cos 9 L 9.0. =. = sin L 9. tan 7 L =. = 0 cos L 0. sin L 0. For # 8, coose wicever of te following formulas is aroriate: b a = c - b =c sin Å=c cos ı=b tan Å= tan a a b = c - a =c cos Å=c sin ı=a tan ı= tan b c = a + b a = cos a = a sin b = b sin a = b cos b If one angle is given, subtract from 90 to find te oter angle. a. b = tan b =. tan 0 L.79, a c = sin b =. L.9, a = 90 - b = 70 sin 0. a=c sin Å=0 sin., b=c cos Å=0 cos 7., ı=90 -Å=9 7. b=a tan ı=.8 tan., c = a cos b =.8 cos L 7., b = 90 - a = 8. b=a tan ı= tan 9 8., a c = Å=90 -ı= cos a = cos 9 L 9.7, As gets smaller and smaller, te side oosite gets smaller and smaller, so its ratio to te otenuse aroaces 0 as a limit. 0.. As gets smaller and smaller, te side adjacent to aroaces te otenuse in lengt, so its ratio to te otenuse aroaces as a limit.. = tan 7 0. ft. =+0 tan 8.8 ft. A = # L 7. ft sin. =0 tan ft. AC=00 tan ft. Connect te tree oints on te arc to te center of te circle, forming tree triangles, eac wit otenuse 0 ft. Te orizontal legs of te tree triangles ave lengts 0 cos 7..87, 0 cos 7.07, and 0 cos Te widts of te four stris are, terefore,.87-0=.87 (stri A) =. (stri B) =.8 (stri C) 0-9.9=0.7 (stri D) Allen needs to correct is data for stris B and C. 7. False. Tis is onl true if is an acute angle in a rigt triangle. (Ten it is true b definition.) 8. False. Te larger te angle of a triangle, te smaller its cosine. Corigt 0 Pearson Education, Inc.

6 c0 0.qd // :7 PM Page 9 Section. Trigonometr Etended: Te Circular Functions 9 9. sec 90 = is undefined. Te answer is E. cos 90 = 0 o 70. sin = Te answer is A. =. 7. If te unknown sloe is m, ten m sin =,so m = - = -csc u. Te answer is D. sin u 7. For all, cos. Te answer is B. 7. For angles in te first quadrant, sine values will be increasing, cosine values will be decreasing, and onl tangent values can be greater tan. Terefore, te first column is tangent, te second column is sine, and te tird column is cosine. 7. For angles in te first quadrant, secant values will be increasing, and cosecant and cotangent values will be decreasing. We recognize tat csc (0 )=. Terefore, te first column is secant, te second column is cotangent, and te tird column is cosecant. 7. Te distance d A from A to te mirror is cos 0 ; te distance from B to te mirror is d B =d A -. Ten PB = = - L.9 m. 7. Let P be te oint at wic we sould aim; let Å and ı be te angles as labeled in #7. Since Å=ı, tan Å=tan ı. P sould be inces to te rigt of C, were is cosen 0 - so tat tan Å= tan ı=. Ten = 0 0=(0-), so =0, wic gives =8. Aim 8 in. to te rigt of C (or in. to te left of D). 77. One ossible roof: (sin u) + (cos u) = a a c b + a b c b = a c + b c = a + b c = c c d B cos a = d A - cos 0 = - cos 0 (Ptagorean teorem: a +b =c.) = 78. Let be te lengt of te altitude to base b and denote te area of te triangle b A. Ten a = sin u =a sin. Since A=, we can substitute =a sin to get b A = ab sin. Section. Trigonometr Etended: Te Circular Functions Eloration. Te side oosite in te triangle as lengt and te otenuse as lengt r. Terefore sin u = o. = r. cos u = adj = r. tan u = o adj = r r. cot = ; sec = ; csc = Eloration. Te -coordinates on te unit circle lie between and, and cos t is alwas an -coordinate on te unit circle.. Te -coordinates on te unit circle lie between and, and sin t is alwas a -coordinate on te unit circle.. Te oints corresonding to t and t on te number line are wraed to oints above and below te -ais wit te same -coordinates. Terefore cos t and cos ( t) are equal.. Te oints corresonding to t and t on te number line are wraed to oints above and below te -ais wit eactl oosite -coordinates. Terefore sin t and sin ( t) are oosites.. Since is te distance around te unit circle, bot t and t+ get wraed to te same oint.. Te oints corresonding to t and t+ get wraed to oints on eiter end of a diameter on te unit circle. Tese oints are smmetric wit resect to te origin and terefore ave coordinates (, ) and (, ). Terefore sin t and sin (t+ ) are oosites, as are cos t and cos (t+ ). 7. B te observation in (), tan t and tan(t+ ) are ratios - of te form and, wic are eiter equal to eac - oter or bot undefined. 8. Te sum is alwas of te form + for some (, ) on te unit circle. Since te equation of te unit circle is + =, te sum is alwas. 9. Answers will var. For eamle, tere are similar statements tat can be made about te functions cot, sec, and csc. Quick Review tan = =. cot = Corigt 0 Pearson Education, Inc.

7 c0 0.qd // :7 PM Page Cater Trigonometric Functions 7. csc = 8. sec = 9. Using a rigt triangle wit otenuse and legs (oosite) and - = (adjacent), we ave sin =, cos =, tan = ; csc =, sec =, cot =. 0. Using a rigt triangle wit otenuse 7 and legs (adjacent) and 7 - = 8 (oosite), we ave sin =, cos =, tan = ; csc =, sec =, cot =. 8 Section. Eercises. Te 0 angle lies on te ositive -ais (0-0 =90 ), wile te oters are all coterminal in Quadrant II.. Te - angle lies in Quadrant I a -, + = b wile te oters are all coterminal in Quadrant IV. In #, recall tat te distance from te origin is r= +.. sin =, cos =, tan = ; csc =, sec =, cot =.. sin =, cos =, tan = ; csc =, sec =, cot =.. sin =, cos =, tan =; csc =, sec =, cot =.. sin =, cos =, tan = ; csc =, sec =, cot =. 7. sin =, cos =, tan = ; csc =, sec =, cot =. 8. sin =, cos =, tan = ; csc =, sec =, cot =. 9. sin =, cos =0, tan undefined; csc =, sec undefined, cot =0. 0. sin =0, cos =, tan =0; csc undefined, sec =, cot undefined.. sin =, cos =, tan = ; csc =, sec =, cot =.. sin =, cos =, tan = ; csc =, sec =, cot =. For #, determine te quadrant(s) of angles wit te given measures, and ten use te fact tat sin t is ositive wen te terminal side of te angle is above te -ais (in Quadrants I and II) and cos t is ositive wen te terminal side of te angle is to te rigt of te -ais (in quadrants I and IV). Note tat since tan t= sin t/cos t, te sign of tan t can be determined from te signs of sin t and cos t: lf sin t and cos t ave te same sign, te answer to (c) will be + ; oterwise it will be.tus tan t is ositive in Quadrants I and III.. Tese angles are in Quadrant I. (a)+(i.e., sin t 7 0). (b)+(i.e., cos t 7 0). (c)+(i.e., tan t 7 0).. Tese angles are in Quadrant II. (a) +. (b). (c).. Tese angles are in Quadrant III. (a). (b). (c) +.. Tese angles are in Quadrant IV. (a). (b) +. (c). For #7 0, use strategies similar to tose for te revious roblem set. 7. is in Quadrant II, so cos is negative is in Quadrant III, so tan 9 is ositive rad is in Quadrant II, so cos is negative rad is in Quadrant II, so tan is negative.. (a) (, ); tan = = Q =.. (b) (, ); tan is in Quadrant II, = = -. so is negative. 7. (a) (, ); is in Quadrant III, so and are bot negative. tan 7 =.. (b) (, ); 0º is in Quadrant IV, so is ositive wile is negative. tan (-0 ) = -. For #, recall tat te reference angle is te acute angle formed b te terminal side of te angle in standard osition and te -ais.. Te reference angle is 0. A rigt triangle wit a 0 angle at te origin as te oint P(, ) as one verte, wit otenuse lengt r=, so cos 0 = =. r. Te reference angle is 0. A rigt triangle wit a 0 angle at te origin as te oint P(, ) as one verte, so tan 00 = = -. Corigt 0 Pearson Education, Inc.

8 c0 0.qd // :7 PM Page 7 Section. Trigonometr Etended: Te Circular Functions 7 7. Te reference angle is te given angle,. A rigt triangle wit a radian angle at te origin as te oint P(, ) as one verte, wit otenuse lengt r=, so r sec = =. 8. Te reference angle is. A rigt triangle wit a radian angle at te origin as te oint P(, ) as one verte, r wit otenuse lengt r=, so csc = =. 9. Te reference angle is (in fact, te given angle is coterminal wit ). A rigt triangle wit a radian angle at te origin as te oint P(,) as one verte, wit otenuse lengt r=, so sin = =. r 0. Te reference angle is (in fact, te given angle is coterminal wit ). A rigt triangle wit a radian angle at te origin as te oint P(, ) as one verte, 7 wit otenuse lengt r=, so cos = =. r. Te reference angle is (in fact, te given angle is coterminal wit ). A rigt triangle wit a radian angle at te origin as te oint P(, ) as one verte, - so tan = =.. Te reference angle is. A rigt triangle wit a radian angle at te origin as te oint P(, ) as one verte, so cot = =.. cos =cos = 7. cos =cos =. sin =sin = 9 7. cot =cot = 7. 0 is coterminal wit 70, on te negative -ais. (a) (b) 0 (c) Undefined is coterminal wit 90, on te ositive -ais. (a) (b) 0 (c) Undefined 9. 7 radians is coterminal wit radians, on te negative -ais. (a) 0 (b) (c) 0 0. radians is coterminal wit radians, on te negative -ais. (a) (b) 0 (c) Undefined -7. radians is coterminal wit radians, on te ositive -ais. (a) (b) 0 (c) Undefined. radians is coterminal wit 0 radians, on te ositive -ais. (a) 0 (b) (c) 0. Since cot 7 0, sin and cos ave te same sign, so sin u sin = + - cos u =, and tan =. cos u =. Since tan 0, sin and cos ave oosite signs, so cos = - sin u=, and cos u cot = =. sin u sin u. cos = + - sin u=, so tan = cos u = and sec = =. cos u. sec as te same sign as cos, and since cot 7 0, sin must also be negative. Wit =, = 7, and 7 r= + 7 = 8, we ave sin = and 8 cos = Since cos 0 and cot 0, sin must be ositive. Wit =, =, and r= + =, we ave sec = and csc =. 8. Since sin 7 0 and tan 0, cos must be negative. Wit =, =, and r= + =, we ave csc = and cot =. 9. sin a =sin a b = + 9,000b 0. tan (,,7 )-tan (7,, ) =tan ( )-tan ( )=0. cos a,, b = cos a b = 0-70,000. tan a b=tan a =undefined b. Te calculator s value of te irrational number is necessaril an aroimation. Wen multilied b a ver large number, te sligt error of te original aroimation is magnified sufficientl to trow te trigonometric functions off.. sin t is te -coordinate of te oint on te unit circle after measuring counterclockwise t units from (, 0). Tis will reeat ever units (and not before), since te distance around te circle is. sin 8. Â=.9 sin sin. sin = (a) Wen t=0, d=0. in. (b) Wen t=, d=0.e 0. cos 0.8 in. Corigt 0 Pearson Education, Inc.

9 c0 0.qd // :7 PM Page 7 7 Cater Trigonometric Functions 8. Wen t=0, =0. (rad). Wen t=., =0. cos rad. 9. Te difference in te elevations is 00 ft, so d=00/sin. Ten: (a) d= ft. (b) d=00 ft. (c) d 9. ft. 0. Januar (t=): sin =0.. Aril (t=): sin.8. June (t=): sin =7.. October (t=0): sin December (t=): sin =7.. June and December are te same; eras b June most eole ave suits for te summer, and b December te are beginning to urcase tem for net summer (or as Cristmas resents, or for mid-winter vacations).. True. An angle in a triangle measures between 0 and 80. Acute angles (<90 ) determine reference triangles in Quadrant I, were te cosine is ositive, wile obtuse angles (>90 ) determine reference triangles in Quadrant II, were te cosine is negative.. True. Te oint determines a reference triangle in Quadrant IV, wit r = 8 + (-) = 0. Tus sin =/= /0 = 0... If sin =0., ten sin ( )+csc = sin + sin u = 0.+ =.. Te answer is E. 0.. If cos =0., ten cos ( + )= cos = 0.. Te answer is B.. (sin t) +(cos t) = for all t. Te answer is A.. sin = - - cos u, because tan =(sin )/(cos )>0. So sin = - - = -. B 9 Te answer is A. 7. Since sin 7 0 and tan 0, te terminal side must be in Quadrant II, so =. 8. Since cos 7 0 and sin 0, te terminal side must be in Quadrant IV, so =. 9. Since tan 0 and sin 0, te terminal side must be 7 in Quadrant IV, so =. 70. Since sin 0 and tan 7 0, te terminal side must be in Quadrant III, so =. 7. Te two triangles are congruent: Bot ave otenuse, and te corresonding angles are congruent te smaller acute angle as measure t in bot triangles, and te two acute angles in a rigt triangle add u to /. 7. Tese coordinates give te lengts of te legs of te triangles from Eercise 7, and tese triangles are congruent. For eamle, te lengt of te orizontal leg of te triangle wit verte P is given b te (absolute value of te) - coordinate of P; tis must be te same as te (absolute value of te) -coordinate of Q. Q( b, a) 7. One ossible answer: Starting from te oint (a, b) on te unit circle at an angle of t, so tat cos t=a ten measuring a quarter of te wa around te circle (wic corresonds to adding / to te angle), we end at ( b, a), so tat sin (t + /) = a. For (a, b) in Quadrant I, tis is sown in te figure above; similar illustrations can be drawn for te oter quadrants. 7. One ossible answer: Starting from te oint (a, b) on te unit circle at an angle of t, so tat sin t=b ten measuring a quarter of te wa around te circle (wic corresonds to adding / to te angle), we end at ( b, a), so tat cos (t + /) = b= sin t. For (a, b) in Quadrant I, tis is sown in te figure above; similar illustrations can be drawn for te oter quadrants. 7. Starting from te oint (a, b) on te unit circle at an angle of t, so tat cos t=a ten measuring a quarter of te wa around te circle (wic corresonds to adding / to te angle), we end at ( b, a), so tat sin (t + /) = a. Tis olds true wen (a, b) is in Quadrant II, just as it did for Quadrant I. P(a, b) Q( b, a) t + π t + π P(a, b) t t t (, 0) (, 0) 7. (a) Bot triangles are rigt triangles wit otenuse, and te angles at te origin are bot t (for te triangle on te left, te angle is te sulement of -t). Terefore te vertical legs are also congruent; teir lengts corresond to te sines of t and -t. (b) Te oints P and Q are reflections of eac oter across te -ais, so te are te same distance (but oosite directions) from te -ais. Alternativel, use te congruent triangles argument from (a). Corigt 0 Pearson Education, Inc.

10 c0 0.qd // :7 PM Page 7 ` Section. Gras of Sine and Cosine: Sinusoids Seven decimal laces are sown so tat te sligt differences can be seen. Te magnitude of te relative error is less tan % wen œ œ 0. (aroimatel). Tis can be seen b etending te table to larger values of, or b graing ƒ sin u -u ƒ ƒ sin u ƒ ` sin sin - sin - sin u Let (, ) be te coordinates of te oint tat corresonds to t under te wraing. Ten + +(tan t) =+ a = b = =(sec t). (Note tat + = because (, ) is on te unit circle.) 79. Tis Talor olnomial is generall a ver good aroimation for sin in fact, te relative error (see Eercise sin - sin -a - b 77) is less tan % for œ œ (aro.). It is better for close to 0; it is sligtl larger tan sin wen 0 and sligtl smaller wen Tis Talor olnomial is generall a ver good aroimation for cos in fact, te relative error (see #77) is less tan % for œ œ. (aro.). It is better for close to 0; it is sligtl larger tan cos wen Z 0. cos - + cos u - a - u + bu Section. Gras of Sine and Cosine: Sinusoids Eloration. / (at te oint (0, )). / (at te oint (0, )). Bot gras cross te -ais wen te -coordinate on te unit circle is 0.. (Calculator eloration). Te sine function tracks te -coordinate of te oint as it moves around te unit circle. After te oint as gone comletel around te unit circle (a distance of ), te same attern of -coordinates starts over again.. Leave all te settings as te are sown at te start of te eloration, ecet cange Y T to cos(t). Quick Review.. In order: +,+,-,-. In order: +,-,-,+. In order: +,-,+,-. # 80 =. -0 # 80 = -. 0 # 80 = 7. Starting wit te gra of, verticall stretc b to obtain te gra of. 8. Starting wit te gra of, reflect across te -ais to obtain te gra of. 9. Starting wit te gra of, verticall srink b 0. to obtain te gra of. Corigt 0 Pearson Education, Inc.

11 c0 0.qd // :7 PM Page 7 7 Cater Trigonometric Functions 0. Starting wit te gra of, translate down units to obtain te gra of. Section. Eercises In #, for =a sin, te amlitude is a. If a >, tere is a vertical stretc b a factor of a, and if a <, tere is a vertical srink b a factor of a. Wen a<0, tere is also a reflection across te -ais.. Amlitude ; vertical stretc b a factor of.. Amlitude /; vertical srink b a factor of /.. Amlitude ; vertical stretc b a factor of, reflection across te -ais.. Amlitude 7/; vertical stretc b a factor of 7/, reflection across te -ais.. Amlitude 0.7; vertical srink b a factor of Amlitude.; vertical stretc b a factor of., reflection across te -ais. In #7, for =cos b, te eriod is / b. If b >, tere is a orizontal srink b a factor of / b, and if b <, tere is a orizontal stretc b a factor of / b. Wen b<0, tere is also a reflection across te -ais. For =a cos b, a as te same effects as in #. 7. Period /; orizontal srink b a factor of /. 8. Period /(/)=0 ; orizontal stretc b a factor of /(/)=. 9. Period /7; orizontal srink b a factor of /7, reflection across te -ais. 0. Period /0.= ; orizontal stretc b a factor of /0.=., reflection across te -ais.. Period /= ; orizontal srink b a factor of /. Also a vertical stretc b a factor of.. Period /(/)= ; orizontal stretc b a factor of /(/)=/. Also a vertical srink b a factor of /. In #, te amlitudes of te gras for =a sin b and =a cos b are governed b a, wile te eriod is governed b b, just as in #. Te frequenc is /eriod.. For = sin (/), te amlitude is, te eriod is /(/)=, and te frequenc is /( ).. For = (/) sin, te amlitude is /, te eriod is /=, and te frequenc is /.. For = sin (/), te amlitude is, te eriod is /(/)=, and te frequenc is /( ). Note: Te frequenc for eac gra in #7 is /( ). 7. Period, amlitude= 8. Period, amlitude=. 9. Period, amlitude= 0. Period, amlitude= [, ] b [, ] [, ] b [, ]... Period, amlitude=0.. Period, amlitude= 0. [, ] b [, ]. For = cos (/), te amlitude is, te eriod is /(/)=, and te frequenc is /( ). 0. [, ] b [, ] Corigt 0 Pearson Education, Inc.

12 c0 0.qd // :7 PM Page 7 Section. Gras of Sine and Cosine: Sinusoids 7. Period,. Period, amlitude=, amlitude=, frequenc=/ frequenc = /( )... Period /,. Period /, amlitude=0., amlitude=0, frequenc=/ frequenc = / Period 8, 8. Period /, amlitude=, amlitude=8, frequenc=/(8 ) frequenc = /( ) 9. Maimum: aat, ; minimum: (at 0,, ). b 7 Zeros:,,,. 0. Maimum: aat -, ; minimum: aat,. b b Zeros: 0,,.. =sin as to be translated left or rigt b an odd multile of. One ossibilit is =sin (+ ).. =sin as to be translated rigt b lus an even multile of. One ossibilit is =sin (- /).. Starting from =sin, orizontall srink b and verticall srink b 0.. Te eriod is /. Possible window: c - b c -.,, d d. Starting from =cos, orizontall srink b and verticall stretc b.. Te eriod is /. Possible window: c - b [, ]., d Period ; amlitude.; [, ] b [, ]. 0. Period /; amlitude ; c - b [, ]., d. Period ; amlitude ; [, ] b [, ].. Period ; amlitude ; [, ] b [ 0, 0].. Period ; amlitude ; [, ] b [, ].. Period ; amlitude ; [, ] b [, ].. Maimum: aat and ; b minimum: aat and. b Zeros: 0,,.. Maimum: (at 0); minimum: (at ). Zeros:. 7. Maimum: (at 0,, ); minimum: 7 aat and. Zeros:,,,. b 8. Maimum: aat and ; b minimum: - a at - and. Zeros: 0,,. b [, ] b [ 0.7, 0.7] [, ] b [ 0.7, 0.7] For # For #. Starting from =cos, orizontall stretc b, verticall srink b, reflect across te -ais. Te eriod is. Possible window: [, ] b [, ].. Starting from =sin, orizontall stretc b and verticall srink b. Te eriod is 0. Possible window: [ 0, 0 ] b [, ]. [, ] b [, ] [ 0, 0 ] b [, ] For # For # 7. Starting from =cos, orizontall srink b and verticall stretc b. Te eriod is. Possible window: [, ] b [.,.]. Corigt 0 Pearson Education, Inc.

13 c0 0.qd // :7 PM Page 7 7 Cater Trigonometric Functions 8. Starting from =sin, orizontall stretc b, verticall stretc b, and reflect across te -ais. Te eriod is 8. Possible window: [ 8, 8] b [, ]. [, ] b [.,.] [ 8, 8] b [, ] For Eercise 7 For Eercise 8 9. Starting wit, verticall stretc b. 0. Starting wit, translate rigt units and verticall srink b.. Starting wit, orizontall srink b.. Starting wit, orizontall stretc b and verticall srink b. For #, gra te functions or use facts about sine and cosine learned to tis oint.. (a) and (b). (a) and (b). (a) and (b) bot functions equal cos. (a) and (c) sina + b =sin ca - =cosa - b + d b In #7 0, for =a sin (b(-)), te amlitude is a, te eriod is / b, and te ase sift is. 7. One ossibilit is = sin. 8. One ossibilit is = sin (/). 9. One ossibilit is =. sin (-). 0. One ossibilit is =. sin (-).. Amlitude, eriod, ase sift, vertical translation unit u.. Rewrite as =. sin c a -. bd - Amlitude., eriod, ase sift, vertical translation unit down.. Rewrite as = cos c a - bd Amlitude, eriod, ase sift, vertical 8 translation units u.. Amlitude, eriod, ase sift, vertical translation units down.. Amlitude, eriod, ase sift 0, vertical translation unit u.. Amlitude, eriod, ase sift 0, vertical translation units down Amlitude, eriod, ase sift -, vertical translation unit down. 8. Amlitude, eriod 8, ase sift, vertical translation unit u. 9. = sin (a=, b=, =0, k=0). 70. = sin[(+0.)] (a=, b=, =0., k=0). 7. (a) Tere are two oints of intersection in tat interval. (b) Te coordinates are (0, ) and (,. ) (.8, 0.9). In general, two functions intersect were cos =, i.e., =n, n an integer. 7. a= and b =.. = 7 7. Te eigt of te rider is modeled b =0- cos a, were t=0 corresonds 0 tb to te time wen te rider is at te low oint. =0 - wen =cos a. Ten, so t 0 tb 0 t L sec. 7. Te lengt L must be te distance traveled in 0 min b an object traveling at 0 ft/sec: L=800 sec # 0 ft 97,000 ft, or about 8 mi sec = 7. (a) A model of te det of te tide is d= cos c (t - 7.) d + 9, were t is ours since. midnigt. Te first low tide is at :00 A.M. (t=). (b) At :00 A.M. (t=): about 8.90 ft. At 9:00 P.M. (t=): about 0. ft. (c) :0 A.M. (t=. alfwa between :00 A.M. and 7: A.M.). 7. (a) second. (b) Eac eak corresonds to a eartbeat tere are 0 er minute. (c) [0, 0] b [80, 0] Corigt 0 Pearson Education, Inc.

14 c0 0.qd // :7 PM Page 77 Section. Gras of Sine and Cosine: Sinusoids (a) Te maimum d is aroimatel.. Te amlitude is (.-7.)/=7.. Scatterlot: [0, ] b [0, 90] [0,.] b [7, ] (b) Te eriod aears to be sligtl greater tan 0.8, sa 0.8. (c) Since te function as a minimum at t=0, we use an inverted cosine model: d(t)= 7. cos ( t/0.8)+.. (d) [0,.] b [7, ] 78. (a) Te amlitude is.7, alf te diameter of te turntable. (b) Te eriod is.8, as can be seen b measuring from minimum to minimum. (c) Since te function as a minimum at t=0, we use an inverted cosine model: d(t)=.7 cos ( t/.8)+7.7. (d) [0,.] b [9, 8] 79. One ossible answer is T =. cos a ( - 7)b +.. Start wit te general form sinusoidal function = a cos (b( - )) + k, and find te variables a, b,, and k as follows: 79 - Te amlitude is ƒ a ƒ = =.. We can arbitraril coose to use te ositive value, so a=.. Te eriod is monts. = Q ƒ b ƒ =. ƒ b ƒ = Again, we can arbitraril coose to use te ositive value, so b =. Te maimum is at mont 7, so te ase sift = Te vertical sift k = = One ossible answer is =. cos a ( - 7)b +.. Start wit te general form sinusoidal function = a cos (b( - )) + k, and find te variables a, b,, and k as follows: 9-0 Te amlitude is ƒ a ƒ = =.. We can arbitraril coose to use te ositive value, so a=.. Te eriod is monts. = Q ƒ b ƒ =. ƒ b ƒ = Again, we can arbitraril coose to use te ositive value, so b =. Te maimum is at mont 7, so te ase sift = Te vertical sift k = =.. [0, ] b [0, 80] 8. False. Since =sin is a orizontal stretc of =sin b a factor of, =sin as twice te eriod, not alf. Remember, te eriod of =sin b is / b. 8. True. An cosine curve can be converted to a sine curve of te same amlitude and frequenc b a ase sift, wic can be accomlised b an aroriate coice of C (a multile of /). 8. Te minimum and maimum values differ b twice te amlitude. Te answer is D. 8. Because te gra asses troug (, 0), f()=0. And lus eactl two eriods equals 9, so f(9)=0 also. But f(0) deends on ase and amlitude, wic are unknown. Te answer is D. 8. For f()=a sin (b+c), te eriod is / b, wic ere equals /0= /0. Te answer is C. 8. Tere are solutions er ccle, and 000 ccles in te interval. Te answer is C. 87. (a) [, ] b [.,.] Corigt 0 Pearson Education, Inc.

15 c0 0.qd // :7 PM Page Cater Trigonometric Functions (b) cos Te coefficients given as 0 ere ma sow u as ver small numbers (e.g.,.*0 ) on some calculators. Note tat cos is an even function, and onl te even owers of ave nonzero (or a least non-small ) coefficients. (c) Te Talor olnomial is - ; te + = coefficients are fairl similar. 88. (a) [, ] b [.,.] (b) sin Te coefficients given as 0 ere ma sow u as ver small numbers (e.g.,.*0 ) on some calculators. Note tat sin is an odd function, and onl te odd owers of ave nonzero (or a least non-small ) coefficients. (c) Te Talor olnomial is - ; = te coefficients are somewat similar. 89. (a) = sec. = (b) f = ( ccles er sec ), or Hertz (Hz). sec (c) [0, 0.0] b [, ] 90. Since te cursor moves at a constant rate, its distance from te center must be made u of linear ieces as sown (te sloe of te line is te rate of motion). So, it is not sinusoidal. A gra of a sinusoid is included for comarison. [0, ] b [.,.] 9. (a) a-b must equal. (b) a-b must equal. (c) a-b must equal k. 9. (a) a-b must equal. (b) a-b must equal. (c) a-b must equal k. For #9 9, note tat A and C are one eriod aart. Meanwile, B is located one-fourt of a eriod to te rigt of A, and te -coordinate of B is te amlitude of te sinusoid. 9. Te eriod of tis function is and te amlitude is. B and C are located (resectivel) units and units to te rigt of A. Terefore, B=(0, ) and C= a., 0b 9. Te eriod of tis function is and te amlitude is.. B and C are located (resectivel) units and units to te rigt of A. Terefore B= a and,.b C = a 9., 0b 9. Te eriod of tis function is and te amlitude is. B and C are located (resectivel) units and units to te rigt of A. Terefore B= a and C= a., b, 0b 9. Te first coordinate of A is te smallest ositive suc n + tat - =n, n and integer, so = must equal. Te eriod of tis function is and te amlitude is. B and C are located (resectivel) units and units to te rigt of A. Terefore A= a,, 0b B = a., b, and C = a, 0b 97. (a) Since sin ( )= sin (because sine is an odd function) a sin [ B(-)]+k= a sin[b(-)]+k. Ten an eression wit a negative value of b can be rewritten as an eression of te same general form but wit a ositive coefficient in lace of b. (b) A sine gra can be translated a quarter of a eriod to te left to become a cosine gra of te same sinusoid. Tus = a sin cba( - ) + # b bd + k = a sin cba - a - as te same b bbd + k gra as = a cos [b( - )] + k. We terefore coose H = -. b (c) Te angles + and determine diametricall oosite oints on te unit circle, so te ave oint smmetr wit resect to te origin. Te -coordinates are terefore oosites, so sin( + )= sin. (d) B te identit in (c). = a sin [b( - ) + ] + k = -a sin [b( - )] + k. We terefore coose H = -. b (e) Part (b) sows ow to convert = a cos [b( - )] + k to = a sin [b( - H)] + k, and arts (a) and (d) sow ow to ensure tat a and b are ositive. Corigt 0 Pearson Education, Inc.

16 c0 0.qd // :7 PM Page 79 Section. Gras of Tangent, Cotangent, Secant, and Cosecant Section. Gras of Tangent, Cotangent, Secant, and Cosecant 79 Eloration. Te gras do not seem to intersect.. Set te eressions equal and solve for : k cos =sec k cos =/cos k(cos ) = (cos ) = /k Since k 7 0, tis requires tat te square of cos be negative, wic is imossible. Tis roves tat tere is no value of for wic te two functions are equal, so te gras do not intersect. Quick Review.. Period. Period. Period. Period For # 8, recall tat zeros of rational functions are zeros of te numerator, and vertical asmtotes are found at zeros of te denominator (rovided te numerator and denominator ave no common zeros).. Zero:. Asmtote: =. Zero:. Asmtote: = 7. Zero:. Asmtotes: = and = 8. Zero:. Asmtotes: =0 and = For #9 0, eamine gras to suggest te answer. Confirm b cecking f( )=f() for even functions and f( )= f() for odd functions. 9. Even: ( ) += + 0. Odd: = ( ) Section. Eercises. Te gra of = csc must be verticall stretced b comared to =csc, so = csc and =csc.. Te gra of = tan must be verticall stretced b 0 comared to =0. tan, so = tan and =0. tan.. Te gra of = csc must be verticall stretced b and orizontall srunk b comared to =csc,so = csc and =csc.. Te gra of =cot(-0.)+ must be translated units u and 0. units rigt comared to =cot,so =cot(-0.)+ and =cot.. Te gra of =tan results from srinking te gra of = tan orizontall b a factor of. Tere are vertical asmtotes at = -.,,, [, ] b [, ]. Te gra of = cot results from srinking te gra of =cot orizontall b a factor of and reflecting it across te -ais. Tere are vertical asmtotes at = -., -, 0,, b [, ], 7. Te gra of =sec results from srinking te gra of =sec orizontall b a factor of. Tere are vertical asmtotes at odd multiles of., b [, ] 8. Te gra of =csc results from srinking te gra of =csc orizontall b a factor of. Tere are vertical asmtotes at = -, -., 0,, [, ] b [, ] Corigt 0 Pearson Education, Inc.

17 c0 0.qd // :7 PM Page Cater Trigonometric Functions 9. Te gra of = cot results from srinking te gra of =cot orizontall b a factor of and stretcing it verticall b a factor of. Tere are vertical asmtotes at = -, -., 0,, [, ] b [, ] 0. Te gra of = tan a results from stretcing te b gra of =tan orizontall b a factor of and stretcing it verticall b a factor of. Tere are vertical asmtotes at = -,,,. [, ] b [, ]. Te gra of =csc a results from orizontall b stretcing te gra of =csc b a factor of. Tere are vertical asmtotes at = -, -, 0,,. [, ] b [, ]. Te gra of = sec results from orizontall srinking te gra of =sec b a factor of and stretcing it verticall b a factor of. Tere are vertical asmtotes at odd multiles of. 8 [, ] b [, ]. Gra (a); Xmin= and Xma=. Gra (d); Xmin= and Xma=. Gra (c); Xmin= and Xma=. Gra (b); Xmin= and Xma= 7. Domain: All reals ecet integer multiles of Range: ( q, q) Continuous on its domain Decreasing on eac interval in its domain Smmetric wit resect to te origin (odd) Not bounded above or below No local etrema No orizontal asmtotes Vertical asmtotes = k for all integers k End beavior: lim cot and lim cot do not eist. : q : -q 8. Domain: All reals ecet odd multiles of Range: ( q, ] [, q) Continuous on its domain On eac interval centered at an even multile of : decreasing on te left alf of te interval and increasing on te rigt alf On eac interval centered at an odd multile of : increasing on te left alf of te interval and decreasing on te rigt alf Smmetric wit resect to te -ais (even) Not bounded above or below Local minimum at eac even multile of, local maimum at eac odd multile of No orizontal asmtotes Vertical asmtotes = k / for all odd integers k End beavior: lim sec and lim sec do not eist. : q : -q 9. Domain: All reals ecet integer multiles of Range: ( q, ] [, q) Continuous on its domain On eac interval centered at = (k an integer): + k decreasing on te left alf of te interval and increasing on te rigt alf On eac interval centered at : increasing on te + k left alf of te interval and decreasing on te rigt alf Smmetric wit resect to te origin (odd) Not bounded above or below Local minimum at eac =, local maimum + k at eac =, were k is an even integer in + k bot cases No orizontal asmtotes Vertical asmtotes: = k for all integers k End beavior: lim csc and lim csc do not eist. : q : -q 0. Domain: All reals ecet odd multiles of Range: ( q, q) Continuous on its domain Increasing on eac interval in its domain Smmetric wit resect to te origin (odd) Not bounded above or below No local etrema No orizontal asmtotes Vertical asmtotes =k for all odd integers k End beavior: lim tan (/) and lim tan (/) : q : -q do not eist. Corigt 0 Pearson Education, Inc.

18 c0 0.qd // :7 PM Page 8 Section. Gras of Tangent, Cotangent, Secant, and Cosecant 8. Starting wit =tan, verticall stretc b.. Starting wit =tan, reflect across te -ais.. Starting wit =csc, verticall stretc b.. Starting wit =tan, verticall stretc b.. Starting wit =cot, orizontall stretc b, verticall stretc b, and reflect across te -ais.. Starting wit =sec, orizontall stretc b, verticall stretc b, and reflect across te -ais. 7. Starting wit =tan, orizontall srink b and reflect across te -ais and sift u b units. 8. Starting wit =tan, orizontall srink b and verticall stretc b and sift down b units. 9. sec = 0. csc =.. sec = -. cos = = sin = = cot = - tan = - = cos = - = csc = sin = =. cot = tan = = -. tan =. L 0.9. sec =. cos =. L. 7. cot = 0. tan = csc =. sin = -. -( 0.7) csc = sin = 0. or tan = or (a) One elanation: If O is te origin, te rigt triangles wit otenuses OP and OP, and one leg (eac) on te -ais, are congruent, so te legs ave te same lengts. Tese lengts give te magnitudes of te coordinates of P and P ; terefore, tese coordinates differ onl in sign. Anoter elanation: Te reflection of oint (a, b) across te origin is ( a, b). sin t (b) tan t=. cos t = b a sin (t - ) -b (c) tan(t- )= =. -a = b cos (t - ) a = tan t (d) Since oints on oosite sides of te unit circle determine te same tangent ratio, tan(t_ )=tan t for all numbers t in te domain. Oter oints on te unit circle ield triangles wit different tangent ratios, so no smaller eriod is ossible. (e) Te tangent function reeats ever units; terefore, so does its recirocal, te cotangent (see also #).. Te terminal side asses troug (0, 0) and (cos, sin ); sin - 0 sin te sloe is terefore m= = =tan. cos - 0 cos. For an, a (+)= = = a (). f b f( + ) f() f b Tis is not true for an smaller value of, since tis is te smallest value tat works for f.. (a), (b) Te angles t and t+ determine oints (cos t, sin t) and (cos(t+ ), sin(t+ )), resectivel. Tese oints are on oosite sides of te unit circle, so te are reflections of eac oter about te origin. Te reflection of an oint (a, b) about te origin is ( a, b), so cos(t+ )= cos t and sin(t+ )= sin t. sin (t + ) -sin t sin t (c) tan(t+ )= = = =tan t. cos (t + ) -cos t cos t In order to determine tat te eriod of tan t is,we would need to sow tat no satisfies tan(t+)= tan t for all t. 0. (a) d=0 sec = cos (b) d,8 ft Corigt 0 Pearson Education, Inc.

19 c0 0.qd // :7 PM Page 8 8 Cater Trigonometric Functions 800. (a) =800 cot = ft tan (b),0 ft (c) # 80 0 = 9 For #7 0, te equations can be rewritten (as sown), but generall are easiest to solve graicall. 7. sin =cos ; ; cos =sin ; 0. or.7 9. cos = ; ;.07 or ;.0 0. cos =sin ;.08 or.00. False. f()=tan is increasing onl over intervals on wic it is defined, tat is, intervals bounded b consecutive asmtotes.. True. Asmtotes of te secant function, sec =/cos, occur at all odd multiles of / (were cos =0), and tese are eactl te zeros of te cotangent function, cot =cos /sin.. Te cotangent curves are saed like te tangent curves, but te are mirror images. Te reflection of tan in te -ais is tan. Te answer is A.. sec just barel intersects its inverse, cos, and wen cos is sifted to roduce sin, tat curve and te curve of sec do not intersect at all. Te answer is E.. =k/sin and te range of sin is [, ]. Te answer is D.. =csc =/sin as te same asmtotes as =cos /sin =cot. Te answer is C. 7. On te interval [, ], f 7 g on about ( 0., 0) (0., ). [, ] b [ 0, 0] 0. Te look similar on tis window, but te are noticeabl different at te edges (near 0 and ). Also, if f were equal to g, ten it would follow tat = cos = =on tis interval, wic we know to be f g false. [0, ] b [ 0, 0]. csc =sec a - (or csc =sec a - a b + nbb for an integer n) Tis is a translation to te rigt of or a units. + nb. cot = tan a - (or cot = b tan a -a for an integer n). + nbb Tis is a translation to te rigt of or a units, + nb and a reflection in te -ais, in eiter order. 0. d=0 sec = cos [, ] b [ 0, 0] 8. On te interval [, ], f 7 g on about (,.) a - a., 0b,.b [, ] b [ 0, 0] 9. cot is not defined at 0; te definition of increasing on (a, b) requires tat te function be defined everwere in (a, b). Also, coosing a= / and b= /, we ave a b but f(a)= 7 f(b)=. [ 0., 0. ] b [0, 00]. (a) For an acute angle t, cos t=sin a te sine - tb of te comlement of t. Tis can be seen from te rigt-triangle definition of sine and cosine: if one of te acute angles is, ten te oter acute angle is, since all tree angles in a triangle must add - t to. Te side oosite te angle t is te side adjacent to te oter acute angle. (b) (cos t, sin t) Corigt 0 Pearson Education, Inc.

20 c0 0.qd // :7 PM Page 8 Section. Gras of Comosite Trigonometric Functions 8 (c) Using ^ODA ~ ^OCB (recall ~ means similar DA BC to ), =tan t=, so BC=tan t. OC = BC OD OD OC (d) Using ^ODA ~ ^OCB, =cot t= OB = OA OB, so OB= =sec t. cos t (e) BC is a tangent segment (art of te tangent line); OB is a secant segment (art of a secant line, wic crosses te circle at two oints). Te names cotangent and cosecant arise in te same wa as cosine te are te tangent and secant (resectivel) of te comlement. Tat is, just as BC and OB go wit jboc (wic as measure t), te also go wit jobc (te comlement of jboc, wit measure ). - t N kg (. m) a00 a9.8 m m = m b sec b kg (.7* 0 - m)sec Ï 0.07 sec Ï, so sec sec Ï.990, and Ï 0.89 radians.9.. (a) = cos(b)= = a sec (b) = # a sec (b) a sin(b+ /) a (b) =0. sina + b (c) a=/0.= and b=/ (d) = sec a. Te scatter lot is sown below, and b te fit is ver good so good tat ou sould realize tat we made te data u! = sin - cos [, ] b [, ] Not a Sinusoid = cos a 7 - b + sin a 7 b [, ] b [, ] Sinusoid Sinusoid Not a Sinusoid Quick Review.. Domain: ( q, q); range: [, ]. Domain: ( q, q), range: [, ]. Domain: [, q); range: [0, q). Domain: [0, q); range: [0, q). Domain: ( q, q); range: [, q). Domain: ( q, q); range: [, q) 7. As : - q, f() : q; as : q, f() : As : - q, f() : - q; as : q, f() : f g()= () -=-, domain: [0, q). g f()= -, domain: ( q, ] [, q). 0. f g()=(cos ) =cos, domain: ( q, q). g f()=cos( ), domain: ( q, q). Section. Eercises. Periodic. = sin ( + ) - cos [, ] b [, ] = cos + sin 7 [, ] b [, ] [ 0., 8.7] b [., ] Section. Gras of Comosite Trigonometric Functions Eloration = sin + cos = sin - cos. Periodic. [, ] b [.,.] [, ] b [, ] Sinusoid [, ] b [, ] Sinusoid [, ] b [.,.] Corigt 0 Pearson Education, Inc.

21 c0 0.qd // :7 PM Page 8 8 Cater Trigonometric Functions. Not eriodic. 9. Since te eriod of cos is, we ave cos (+ )=(cos(+ )) =(cos ) =cos. Te eriod is terefore an eact divisor of, and we see graicall tat it is. A gra for is sown: [, ] b [, 0]. Not eriodic. [, ] b [, ] [, ] b [, 0]. Not eriodic. 0. Since te eriod of cos is, we ave cos (+ )=(cos(+ )) =(cos ) =cos. Te eriod is terefore an eact divisor of, and we see graicall tat it is. A gra for is sown: [, ] b [, ]. Not eriodic. [, ] b [.,.]. Since te eriod of cos is, we ave cos ( + ) = (cos ( + )) = (cos ) = cos. Te eriod is terefore an eact divisor of, and we see graicall tat it is. A gra for is sown: 7. Periodic. [, ] b [, ] [, ] b [, ] 8. Periodic. [, ] b [ 0, 0]. Since te eriod of cos is, we ave cos (+ ) = (cos(+ )) = (cos ) = cos. Te eriod is terefore an eact divisor of, and we see graicall tat it is. A gra for is sown: [, ] b [ 0, 0] [, ] b [, ]. Domain: ( q, q). Range: [0, ]. [, ] b [ 0.,.] Corigt 0 Pearson Education, Inc.

22 c0 0.qd // :8 PM Page 8 Section. Gras of Comosite Trigonometric Functions 8. Domain: ( q, q). Range: [0, ]. 0. = -0.; = Domain: all Z n, n an integer. Range: [0, q). [ 0, 0] b [ 0, 0]. = - 0.; = Domain: ( q, q). Range: [, ]. [ 0, 0] b [, 8]. = ; = + [ 0, 0] b [ 0, 0] 7. Domain: all Z an integer. Range: ( q, 0]. + n, n 8. Domain: ( q, q). Range: [, 0]. In #9, te linear equations are found b setting te cosine term equal to ;. 9. = - ; = + For # 8, te function + is a sinusoid if bot and are sine or cosine functions wit te same eriod.. Yes (eriod ). Yes (eriod ). Yes (eriod ). No 7. No 8. No For #9, gra te function. Estimate a as te amlitude of te gra (i.e., te eigt of te maimum). Notice tat te value of b is alwas te coefficient of in te original functions. Finall, note tat a sin[b(-)]=0 wen =, so estimate using a zero of f() were f() canges from negative to ositive. 9. A., b=, and 0.9, so f(). sin[(-0.9)]. 0. A., b=, and 0., so f(). sin[(+0.)].. A., b=, and 0., so f(). sin[ (-0.)].. A., b=, and 0.0, so f(). sin[ ( +0.0)].. A., b=, and., so f(). sin(+.).. A., b=, and 0., so f(). sin[(-0.)]. [ 0, 0] b [ 0, 0] Corigt 0 Pearson Education, Inc.

23 c0 0.qd // :8 PM Page 8 8 Cater Trigonometric Functions. Te eriod is. [0, ] b [, ] [, ] b [.,.] 0. f oscillates u and down between and.. Te eriod is. As : q, f() : 0. [, ] b [, ] 7. Te eriod is. [0, ] b [, ]. f oscillates u and down between and -. As : q, f() : 0. [, ] b [, ] 8. Te eriod is. [0, ] b [.,.]. f oscillates u and down between e and e. As : q, f() : 0. [, ] b [, ] 9. (a) 0. (d). (c). (b). Te daming factor is e, wic goes to zero as gets large. So daming occurs as : q.. Te daming factor is, wic goes to zero as goes to zero (obviousl). So daming occurs as : 0.. Te amlitude,, is constant. So tere is no daming.. Te amlitude,, is constant. So tere is no daming. 7. Te daming factor is, wic goes to zero as goes to zero. So daming occurs as : Te daming factor is (/), wic goes to zero as gets large. So daming occurs as : q. 9. f oscillates u and down between. and.. As : q, f() : 0. [0,. ] b [, ]. Period : sin[(+ )]+cos[(+ )]= sin(+ )+ cos(+ )=sin + cos. Te gra sows tat no < could be te eriod. [, ] b [.,.8] Corigt 0 Pearson Education, Inc.