Radian Measure and Dynamic Trigonometry

Transcription

1 cob980_ch0_089-.qd 0//09 7:0 Page 89 Debd MHDQ-New:MHDQ:MHDQ-.: CHAPTER CONNECTIONS Radian Measue and Dnamic Tigonomet CHAPTER OUTLINE. Angle Measue in Radians 90. Ac Length, Velocit, and the Aea of a Cicula Secto 97. The Unit Cicle 08. The Tigonomet of Real Numbes 7 The geat geen cicles poduced b the pocess of pivot iigation often aouse the cuiosit of ailine passenges fling ovehead. Hee, the tools of tigonomet can be used to investigate the spinkle s angula velocit (the numbe of evolutions pe hou), its linea velocit (how fast each spinkle along the adial am is moving), as well as the size of the aea being iigated. In calculations like these, the measue of an angle is often given in a new unit called adians, as this contibutes to the simplification of the fomulas involved, and man othe fomulas and pocedues. The use of adians also enables a cleae view of the tigonometic functions as functions of a eal numbe (athe than meel functions of an angle), thus etending thei influence on both pue and applied mathematics. This application appeas as Eecise 0 in Section. 89

2 . Angle Measue in Radians Leaning Objectives In Section. ou will lean how to: A. Use adians fo angle measue B. Find the adian measue of the standad angles C. Convet between degees and adians fo nonstandad angles WORTHY OF NOTE Using the popeties of atios, we note that since both (adius) and s (ac length) ae measued in like units, the units actuall cancel, making adians a unitless measue: s units. units s While angle measue based on a 0 cicle has been accepted fo centuies, its basic constuct is no bette than othe measues poposed and used ove time (stadia, gons, cis, points, mils, gadients, and so on). In the 870s, mathematicians Thomas Mui and James Thomson (the bothe of Lod Kelvin) advocated the need fo a new unit that stated the measue of an angle in tems of a cicle s inheent chaacteistics, athe than abitail declaed numbes like 0 (degees), 00 (gadients), o 000 (mils). A. The Radian Measue of an Angle To help develop this new unit of angle measue, we use a cental cicle, which is a cicle in the -plane with its cente at the oigin. A cental angle is an angle whose vete is at the cente of the cicle. Fo cental angle intesecting the cicle at points B and C, we sa cicula ac BC, denoted BC, subtends BAC, as shown in Figue.. The lette s is commonl used to epesent ac length, and if we define adian (abbeviated ad) to be the measue of an angle subtended b an ac equal in length to the adius, then ad when s (see Figue.). We can then find the adian measue of an cental angle b dividing the length of the subtended ac b : s BC adians. Figue. Figue. Cental angle BAC B B s A adian C A C Radians If cental angle is subtended b an ac that is equal in length to the adius, then adian. Radian Measue of an Angle If is a cental angle in a cicle of adius, and is subtended b an ac of length s, then the measue of in adians is s. 90 -

3 - Section. Angle Measue in Radians 9 EXAMPLE Finding the Radian Measue of an Angle If the cicle shown in the figue has adius 8 cm, what is the adian measue of angle? s 8 cm Solution Using the fomula s with s 8 and 8 gives 8 8 substitute 8 fo s and 8 fo 9 esult 9 The angle measues, o.5 ad. A. You ve just leaned how to find the adian measue of an angle Now t Eecises 7 though 8 The angle in Eample was fomed b a counte-clockwise otation. Recall that angles fomed b a clockwise otation ae consideed negative angles. The fomula fo adian measue still applies, but with 0. See Eecises 9 though. B. Radian Measue of the Standad Angles Recall the cicumfeence of a cicle is C. While ou ma not have consideed this befoe, note the fomula can be witten as C #, which implies that the adius, o an ac of length, can be wapped aound the cicumfeence of the cicle.8 times, as illustated in Figue.. This shows the adian measue of a full 0 otation is : ad 0. This can be veified using the elation adians s The elation ad 0 enables us to state the adian. measues of the standad angles using a simple division. Fo ad 80 we have WORTHY OF NOTE We will often use the convention that unless degee measue is eplicitl implied o noted with the smbol, adian measue is being used. In othe wods, and.7,, all indicate angles measued in adians. division b : 90 division b : 5 division b : 0 division b : 0. See Figue.. The adian measues of these standad angles pla a majo ole in this chapte, and ou ae encouaged to become ve familia with them. Additional convesions can quickl be found using multiples of these fou. Figue. A Figue. 90 o q.57 C 0 o u 5 o d 0 o k o.7

4 9 CHAPTER Radian Measue and Dnamic Tigonomet - EXAMPLE Finding the Radian Measue of Standad Angles Use multiples of a standad angle to find the eact adian measue of the angles given. a. 0 b. 5 c. 70 Solution a. Since 0 is a multiple of both 0 and 0, we can use eithe angle. If we use 0, we have 0 0 a b 0 is a multiple of 0 substitute 0 with ad WORTHY OF NOTE Since a full otation is adians, angles and k ae coteminal fo an intege k when is measued in adians. See Eecises 5 though 8. b c. 5 a b 5 simplif a b B. You ve just leaned how to find the adian measue of the standad angles Now t Eecises 9 though Note in Eample (c), we could have teated 70 as a multiple of 0 o 5, and obtained the same answe afte simplification. C. Conveting between Degees and Radians The elationship 80 also gives the factos needed fo conveting fom degees to adians o fom adians to degees, even if is a nonstandad angle. Dividing b we have 80 while division b shows Multipling a given angle, b the appopiate convesion facto gives the equivalent measue. Degees/Radians Convesion Factos To convet fom adians to degees: 80 multipl b. To convet fom degees to adians: multipl b 80. EXAMPLE Conveting fom Degees to Radians Convet each angle fom degees to adians. Wite each esult in eact fom and in appoimate fom, ounded to thousandths. a. 75 b. 50

5 -5 Section. Angle Measue in Radians 9 Solution a. Fo degees to adians, use the convesion facto # ewite epession 5.09 appl convesion facto eact fom appoimate fom (to thousandths) b. Begin b conveting fom DMS notation to decimal degees, then use the convesion facto a 80 b a50 0 b 50. appl convesion facto ewite epession eact fom appoimate fom (to thousandths) Now t Eecises 7 though 8 EXAMPLE Conveting fom Radians to Degees Convet each angle fom adians to degees. Wite each esult in eact fom and in appoimate fom, ounded to hundedths as needed. a. b Solution a. Fo adians to degees, use the convesion facto. appl convesion facto a80 b a80 b 7.5 ewite epession eact fom, b. appl convesion facto 5 5 a 80 b, C. You ve just leaned how to convet between degees and adians fo nonstandad angles 5 a80 b ewite epession eact fom appoimate fom (to hundedths) Now t Eecises 9 though As in Eample, appoimating the adian measue of an angle lends insight to its size. Figue. on page 9 is a useful tool fo detemining the quadant of the teminal side of an angle measued in adians and shows that the angle in Eample (a) teminates in QIV, while the angle in (b) teminates in QIII. See Eecises 5 though 7.

6 9 CHAPTER Radian Measue and Dnamic Tigonomet - TECHNOLOGY HIGHLIGHT Decimal Degee and Radian Convesions Most gaphing calculatos ae pogammed to compute convesions involving angle measue. On the TI-8 Plus, this is accomplished using the featue fo degee to adian convesions while in adian MODE, and the featue fo adian to degee convesions while in degee MODE. Both ae found on the ANGLE submenu located at nd APPS, as is the : DMS featue used fo convesion to the degees, minutes, seconds fomat (see Figue.5). We ll illustate b conveting both standad and nonstandad angles. To convet 80, 7, and 5 to adians, be sue ou ae in adian MODE then ente 80 nd APPS ENTER (the : featue is the default), then ENTER once again to eecute the opeation. The sceen shows a value of.595, which we epected since 80. Fo 7 and 5, we simpl ecall 80 ( nd ENTER ) and ovewite the desied value (see Figue.). Fo adian-to-degee convesions, be sue ou ae in degee MODE and complete the convesions in a simila manne, using nd APPS : instead of :. Figue.5 Figue. 5 Eecise : Use ou gaphing calculato to convet the adian measues, and.7 to,, degees, then veif each using standad angles o a convesion facto. Eecise : Epeiment with the nd APPS : DMS featue, and use it to convet 08.7 to the DMS fomat. Veif the esult manuall.. EXERCISES CONCEPTS AND VOCABULARY Fill in each blank with the appopiate wod o phase. Caefull eead the section if needed.. A cicle of adius will have -intecepts of ( 5, 0) and (5, 0). On this cicle, an ac of length 5 subtends an angle of adians.. The epession theta equals two degees is witten using the notation. The epession, theta equals two adians is simpl witten.. Fo the adian measue of a standad angle in simplified fom, the numeato will alwas be a multiple of. All such nonquadantal angles will have a denominato of,, o, while the quadantal angles will have a denominato of o.. If is not a special angle, multipl b to convet adians to degees. To convet degees to adians, multipl b. 5. Discuss/Eplain how the adian measue of the standad angles ( 0, 5, and 0 ) and thei multiples can be found without using a convesion facto.. Discuss/Eplain the diffeence between and adian. Eactl what is a adian? Without an convesions, eplain wh an angle of ad teminates in QIII.

7 -7 Section. Angle Measue in Radians 95 DEVELOPING YOUR SKILLS Use the diagam shown and infomation given to find the adian measue of each angle. 7. s in., in Use the diagam shown and infomation given to find the adian measue of each angle s m, m s 5 cm, 5 cm s 0 ft, 0 ft s dm, dm s 8 d, d s. mi,. mi s 5. mm,. mm s km, km s 0 km, 8 km s 8 ft, d s cm, 0 mm s cm, 7 cm s 5 ft, 5 ft s in.,.5 in. s. m, 7 m s 5 d, 5 d s 7 mm, 7 mm Find two positive angles and two negative angles that ae coteminal with the angle given. Answes ma va Convet the following degee measues to adians in eact fom, without the use of a calculato s s Convet each degee measue to adians. Leave in eact fom Convet each degee measue to adians. Round to the neaest ten-thousandth Convet each adian measue to degees, without the use of a calculato Convet each adian measue to degees. Round to the neaest tenth If the following angles ae dawn in standad position on the cental cicle shown, identif the point a f though which the teminal side passes. c d b e a f 5. a.. a. b. 5 b. c. c.

8 cob980_ch0_089-.qd 0/0/09 8:5PM Page 9 ntt MHDQ-New:MHDQ:MHDQ-.: 9-8 CHAPTER Radian Measue and Dnamic Tigonomet Without using a calculato, identif the teminal side quadant of the given angles in standad position WORKING WITH FORMULAS 7. Toque applied b a wench: F sin A wench used to fee a stuck nut applies a toque (measued in foot-pounds) when a foce F (in pounds) is applied feet fom the nut at an angle to the wench. Find how much toque is being applied to a paticulal stubbon nut if 5 lb of foce is being applied 8 in. fom the nut, at an angle of to the wench. How much foce should be applied at this angle and point if the nut will onl elease when the applied toque eaches 50 ft-lb? 7. Hosepowe fo an electic winch: H w 98,000 Ignoing the fact that the cable ma wind ove itself, an electic winch with dum of adius (in inches) tuning at pm is appling H units of hosepowe when it lifts a weight of w pounds. How much hosepowe is equied of a winch to lift 00 lb if the dum has adius 8 in. and is tuning at 90 pm? If the maimum hosepowe of the moto is 5 hp, what is the geatest weight this winch can lift? APPLICATIONS 75. Automobile acing: While pepaing the annual Bonneville Speedwa, the Utah State Highwa Depatment maks out a.75-mi-adius cicula tack in the Bonneville Salt Flats. If a mechanical failue ends And s ace afte onl 7 mi, though what cental angle (in adians) did he tavel? 7. Planeta obits: The moon Titon obits the planet Neptune in a nea pefect cicula path of adius 5,759 km. When the moon has taveled,9,8 km, what cental angle does its path subtend? 77. Automobile acing: An enduance ace on the cicula tack at the Bonneville Speedwa (see Eecise 75) consists of 5.5 laps. What is the adian measue of the cental angle swept out b a ca that completes the ace? 78. Planeta obits: Titon completes one full obit of Neptune eve das (see Eecise 7). What is the adian measue of the cental angle Titon makes in 5 das? 79. Automobile testing: A 00-ft-diamete cicle painted on a smooth, flat piece of pavement is known as a skid pad. A small ca gaduall acceleates ove 000 ft befoe it begins to slip off the pad. To the neaest whole degee, what is the measue of the cental angle swept out b the ca? 80. Amusement pak ides: The Tumble Bug tavels on a 00-ft-diamete cicula tack. What cental angle is subtended b the 70-ft-long tain of si cas? Round to the neaest whole degee. 8. The Pantheon: The tomb of the atist Raphael is in one of the eight niches of the cicula otunda of this ancient building. Epess in adians the cental angle of.5 fomed b the main entance and the commemoative bust of this geat painte. 8. Submaine design: The top-hatch of a eseach submaine is opened b otating the hatch wheel.75 counteclockwise. Epess this angle in adians.

9 -9 Section. Ac Length, Velocit, and the Aea of a Cicula Secto 97 EXTENDING THE CONCEPT 8. Detemine the adian measue of both minute and second. Use ou esults to detemine the adian measue of 7 8. Check ou esults b fist conveting the angle to decimal fom. 8. Titon completes one obit of Neptune eve si Eath das (see Eecise 78) in the opposite diection of the planet s -h-long otation. Fom the point of view of an astonaut on the suface of Neptune, find the cental angle (in adians) Titon appeas to sweep out in h. MAINTAINING YOUR SKILLS 85. (.) Find the eact value of sin 0, cos 0, 88. (.) Find cot given the teminal side of in and tan 0 using efeence angles. 8. (.) Wite sec in tems of csc. 87. (.) Find the acute angle A such that standad position is coincident with., (.) Find the efeence angle of the following: cos A Round to the neaest tenth a. 5 b. 7 of a degee. 90. (.) Fo tiangle ABC, find the measues of all thee angles if A, B 7, and C.. Ac Length, Velocit, and the Aea of a Cicula Secto Leaning Objectives In Section. ou will lean how to: A. Use adians to compute the length of a subtended ac B. Solve applications involving angula velocit and linea velocit C. Calculate the aea of a cicula secto Pio to the widespead use of adians, the length of an ac was found using a popotional pat of 0 :, o Solving fo s amount of otation length of ac 0 0 s. ields s a b, o s a afte egouping. While this fomula is cetainl usable, it s ve unwield and has limited value in man pactical applications. 0 0 b With in adians we know 0, giving b division. Afte substituting 0 fo, the fomula quickl simplifies to s, a much moe elegant esult but 0 once again, must be epessed in adians. Figue.7 s A. The Length of a Subtended Ac s Recall that in a cicle of adius, the adian measue of a cental angle is, whee s is the length of the subtended ac (see Figue.7). In pactice, measuing the length of an ac is often a moe difficult task than measuing the angle. Note that b multipling both sides of s b, we obtain a fomula fo the length of an ac: s povided is in adians. Ac Length If is a cental angle in a cicle of adius, then the length of the subtended ac s is s, povided is epessed in adians.

10 98 CHAPTER Radian Measue and Dnamic Tigonomet -0 EXAMPLE Using the Fomula fo Ac Length If the cicle in Figue.7 has adius 0 cm, what is the length of the ac subtended b an angle of.5 ad? Solution Using the fomula s with 0 and.5 gives s substitute 0 fo and.5 fo esult The subtended ac has a length of 5 cm. Now t Eecises 7 though 8 Note that if the angle is epessed in degees, we fist convet it to adians befoe appling the fomula. One aea whee these convesions ae necessa is applications involving longitude and latitude (see Figue.8). The latitude of a fied point on the Eath s suface tells how man degees noth o south of the equato the point is, as measued fom the cente of the Eath. The longitude of a fied point on the Eath s suface tells how man degees east o west of the Pime Meidian (though Geenwich, England) the point is, as measued along the equato to the longitude line going though the point. Fo eample, the cit of New Oleans, Louisiana, is located at 0 N latitude, 90 W longitude (see Figue.8) New Oleans 0 Figue.8 Pime meidian Equato WORTHY OF NOTE EXAMPLE Appling the Ac Length Fomula: Distances Between Cities Note that 90 mi was used because Quito and Macapá ae both on the equato. Fo othe cities shaing the same longitude but not on the equato, the adius of the Eath at that longitude must be used. See Eecise. The cities of Quito, Ecuado, and Macapá, Bazil, both lie ve nea the equato, at a latitude of 0. Howeve, Quito is at appoimatel 78 west longitude, while Macapá is at 5 west longitude (see Figue.8). Assuming the Eath has a adius of 90 mi, how fa apat ae these cities? Solution Fist we note that of longitude sepaate the two cities. Using the convesion facto we find the equivalent adian measue 80 is 7 a The ac length fomula gives 80 b 0. s 90a 0 b 59 ac length fomula; in adians substitute 90 fo and esult fo 0 Quito and Macapá ae appoimatel 8 mi apat (see Woth of Note in the magin). Now t Eecises 9 and 50

11 - Section. Ac Length, Velocit, and the Aea of a Cicula Secto 99 A. You ve just leaned how to use adians to compute the length of a subtended ac WORTHY OF NOTE Geneall speaking, the velocit of an object is its change in position pe unit time, and can be eithe positive o negative. The ate o speed of an object is the magnitude of the velocit, egadless of diection. B. Angula and Linea Velocit The angula velocit of an object is defined as the amount of otation pe unit time. Hee, we often use the smbol (omega) to epesent the angula velocit, and to epesent the angle though which the teminal side has otated, measued in adians:. Fo instance, a Feis wheel tuning at 0 evolutions pe minute has an angula t velocit of substitute fo evolution The linea velocit of an object is defined as a change of position o distance taveled pe unit time. In the contet of angula motion, we conside the distance taveled b a point on the cicumfeence of the Feis wheel, which is equivalent to the length of the esulting ac s. This elationship is epessed as V s, a fomula that can be t witten diectl in tems of the angula velocit since s : V a. t t b Angula and Linea Velocit 0 evolutions min 0 min 0 ad min t 0 0 Given a cicle of adius with point P on the cicumfeence, and cental angle in adians with P on the teminal side. If P moves along the cicumfeence at a unifom ate:. The ate at which changes is called the angula velocit,. The ate at which the position of P changes is called the linea velocit V, V t t. V. EXAMPLE Using the Velocit Fomulas A point P is otating aound the cicumfeence of a cicle with adius ft at a constant ate. If it takes 5 sec fo the point to otate though an angle of 50, a. What is the angula velocit of P? b. What is the linea velocit of P? Solution a. Since ou fomulas equie to be in adians, we fist convet fom degees to adians a 80 b appl convesion facto ewite epession esult P 50 ft

12 00 CHAPTER Radian Measue and Dnamic Tigonomet - Appling the fomula fo angula velocit gives: t angula velocit fomula 7 substitute fo and 5 fo t invet and multipl esult (eact fom) 7 P is moving at an angula velocit of adians pe second b. Using the eact esult fom pat (a) and the fomula fo linea velocit, we have V a 7 b P is moving at a linea velocit of 5 a 7 b linea velocit fomula 7 substitute fo and fo 0 esult (eact fom) feet pe second. Now t Eecises 9 though Revolutions pe unit time is a common unit of measue fo angula velocit, as in evolutions pe second (ps) fo moden dental dills and evolutions pe minute (pm) fo automotive engines. Fo this eason, man applications ae stated using this unit. But in computations involving the fomulas fo angula and linea velocit, the units must be epessed in adians pe unit time. EXAMPLE Using Angula Velocit to Detemine Linea Velocit The wheels on a acing biccle have a adius of in. How fast is the cclist taveling in miles pe hou, if the wheels ae tuning at 00 pm? 00 ev Solution Note that. min 00 min 00 min Using the fomula V gives a linea velocit of V in. 00 min To convet this to miles pe hou we convet minutes to hous h 0 min and inches to miles mi 580 in.:,50. in. 0 min mi a b a b a b. mph. min h,0 in. The biccle is taveling about. mph.,50. in.. min Now t Eecises 5 though 5

13 - Section. Ac Length, Velocit, and the Aea of a Cicula Secto 0 To help undestand the elationship between angula velocit and linea velocit, conside two lage olles with a adius of. ft, used to move an industial conveo belt. The olles have a cicumfeence of C. ft 0.05 ft, meaning that fo each evolution of the olles, an object on the belt will move 0.05 ft (fom P to P ). P P. ft About 0 ft fo each evolution B. You ve just leaned how to solve applications involving angula velocit and linea velocit If the olles ae otating at 0 pm, an object on the belt (o a point on the cicumfeence of a olle), will be moving at a ate of 0 # ft/min (about. miles pe hou). In othe wods, V 0 evolutions min 0 # min 0 min. ft 0 min 0 ft pe min t substitute fo evolution fomula fo velocit substitute. ft fo, 0 fo esult C. The Aea of a Cicula Secto Using a cental angle measued in adians, we can also develop a fomula fo the aea of a cicula secto (a pie slice) using a popotion. Recall the total aea of a cicle is and the adian measue of a 0 otation is. The atio of the aea of a secto to the total aea will be identical to the atio of the subtended angle to one full otation. See Figue.9. Using A to A epesent the aea of the secto, we have and solving fo A gives A. Figue.9 A Aea of a Secto If is a cental angle in a cicle of adius, the aea of the secto fomed is A, povided is epessed in adians. EXAMPLE 5 Using the Fomula fo the Aea of a Secto What is the aea of the cicula secto fomed b a cental angle of of the cicle is 7 ft? Round to tenths., if the adius

14 0 CHAPTER Radian Measue and Dnamic Tigonomet - Solution Using the fomula A we have A a b7 a b 9 ft substitute 7 fo, esult The aea of this secto is appoimatel 07. ft. fo Now t Eecises 5 though In man applications, the computations of cicula aeas and sectos ae combined with the concept of angula velocit, as in Eample. EXAMPLE Finding the Aea of a Cicula Secto The second hand of a wistwatch measues mm fom the cental ais to its tip. Find the aea of the watch face the second hand passes ove in a 0-sec time peiod. Solution To begin, we find the angle the second hand passes though in 0 sec. Since the second hand moves one full evolution ( ) in 0 sec, its angula velocit is t 0 0 angula velocit fomula substitute fo and 0 fo t simplif Multipling both sides of the angula velocit fomula b t ields a fomula t fo the angle: t. Using this fomula and the value of, we can find the angle the second hand passes though in 0 sec. t angle fomula substitute fo and 0 fo t 0 ewite epession esult With adians, we now compute the aea as befoe. A aea of a secto fomula C. You ve just leaned how to calculate the aea of a cicula secto a b 88 8 mm substitute fo and fo multipl esult (eact fom) The second hand of the watch passes ove an aea of mm in 0 sec. Now t Eecises 55 and 5

15 -5 Section. Ac Length, Velocit, and the Aea of a Cicula Secto 0. EXERCISES CONCEPTS AND VOCABULARY Fill in each blank with the appopiate wod o phase. Caefull eead the section if needed.. Fo an fied point on the Eath s suface, gives its location as the numbe of degees noth o south of the, while gives the numbe of degees east o west of the.. The fomula fo ac length is. The fomula fo the aea of a secto is A. Fo both fomulas, must be in.. The fomula fo angula velocit is. The fomula fo the linea velocit of a point P on the cicumfeence of a cicle moving with an angula velocit is, as long as is in pe unit time. DEVELOPING YOUR SKILLS. The angle can be found b both sides of the fomula fo velocit b. If the fomula fo velocit is multiplied b, the ac length is the esult. 5. Discuss/Eplain the diffeence between angula velocit and linea velocit. In paticula, wh does one depend on the adius while the othe does not? Include an eample fom ou own epeience.. Develop a fomula fo the aea of a secto, if the angle is in degees. Discuss what ou discove. Use the fomula fo ac length to find the value of the unknown quantit: s. 7..5; 80 m 8..; 9 cm 9. s 007 mi; 7 mi 0. s 5. km;,0 km. ; s.9 d. ; s 8.8 nautical miles. ; mi. ; in. 5. s 5.5 ft; 980 ft. s 9. mm; 800 mm 7. 0 ; s 5.5 km ; s 77 m Point P passes though a cental angle in time t as it tavels aound a cicle. Find its eact angula velocit in adians pe unit time ; t 8 sec ; t ; t 0 h. 70 ; t min. 0 ; t 7 das. 90 ; t 5 sec 5. 0 ; t min. 00 ; t 5 h Point P tavels aound a cicle of adius as descibed. Find its linea velocit, ounded to the neaest hundedth as necessa Use the fomula fo aea of a cicula secto to find the value of the unknown quantit: A ad/sec; 8 in. ad/min; 5 ft ad/h;. mi ad/h; 0.0 km 8 ; t 0.8 sec; mm 8 ; t. min;. d 0; t h;. km 5; t h; mi 5;.8 km ; 5 mi A 080 mi ; 0 mi A 7.5 cm ;.5 cm 7 ; A.5 m 0. 9 ; A 75 cm

16 0 CHAPTER Radian Measue and Dnamic Tigonomet - Find the angle, adius, ac length, and/o aea as needed, until all values ae known.... s 9 in. m.5 5 cm d A 8 mm A ft WORKING WITH FORMULAS 7. Havesine fomula, fist pat: h sin a b cos cos sin a b The geat cicles of a sphee have the same cente and adius as the sphee itself. As a esult, the shotest path between two points on a sphee will alwas be an ac of a geat cicle. In navigation, the havesine fomula is often used to detemine the shotest distance between two points on the suface of the Eath, and uses the value of h detemined b the fomula shown. Hee, the espective latitudes of each point ae given b and, and thei longitudes b and, with all angles in adians. Find the value of h fo the cities of Houston, Teas ( 0 N, 95 W), and San Fancisco, Califonia ( 8 N, W). 8. Havesine fomula, second pat: d R sin ( h) The distance d between two points on a geat cicle of adius R is found using the fomula shown, whee h is the value detemined b the havesine fomula in Eecise 7. Given the adius of the Eath is 90 mi, use ou calculato in adian mode and the esult fom Eecise 7 to find the distance between Houston and San Fancisco. Check ou esult using an Intenet distance calculato o GPS device (which will most likel be using this fomula anwa). APPLICATIONS 9. Ac length: The cit of Pittsbugh, Pennslvania, is diectl noth of West Palm Beach, Floida. Pittsbug is at 0. noth latitude, while West Palm Beach is at. noth latitude. Assuming the Eath has a adius of 90 mi, how fa apat ae these cities? 50. Ac length: Both Libeville, Gabon, and Jamame, Somalia, lie nea the equato, but on opposite ends of the Afican continent. If Libeville is at 9. east longitude and Jamame is.5 east longitude, how wide is the continent of Afica at the equato? 5. Riding a ound-a-bout: At the pak two blocks fom ou home, the kids ound-a-bout has a adius of 5 in. About the time the kids stop sceaming, Faste, Dadd, faste! I estimate the ound-a-bout is tuning at evolutions pe second. (a) What is the elated angula velocit? (b) What is the linea velocit (in miles pe hou) of Eli and Reno, who ae hanging on fo dea life at the im of the ound-a-bout?

17 -7 Section. Ac Length, Velocit, and the Aea of a Cicula Secto Canival ides: At canivals and fais, the Gavit Dum is a popula ide. People stand along the wall of a cicula dum with adius ft, which begins spinning ve fast, pinning them against the wall. The dum is then tuned on its side b an amatue, with the ides sceaming and squealing with delight. As the dum is aised to a nea-vetical position, it is spinning at a ate of 5 pm. (a) What is the angula velocit in adians? (b) What is the linea velocit (in miles pe hou) of a peson on this ide? 5. Speed of a winch: A winch is being used to lift a tubine off the gound so that a tactotaile can back unde it and load it up fo tanspot. The winch dum has a adius of in. and is tuning at 0 pm. Find (a) the angula velocit of the dum in adians, (b) the linea velocit of the tubine in feet pe second as it is being aised, and (c) how long it will take to get the load to the desied height of ft (ignoe the fact that the cable ma wind ove itself on the dum). 5. Speed of a cuent: An instument called a flowmete is used to measue the speed of flowing wate, like that in a ive o steam. A cude method involves placing a paddle wheel in the cuent, and using the wheel s adius and angula velocit to calculate the speed of wate flow. If the paddle wheel has a adius of 5. ft and is tuning at 0 pm, find (a) the angula velocit of the wheel in adians and (b) the linea velocit of the wate cuent in miles pe hou. 55. Aea of a secto: A wate spinkle is set to shoot a steam of wate a distance of m and otate though an angle of 0. (a) What is the aea of the lawn it wates? (b) Fo m, what angle is equied to wate twice as much aea? (c) Fo 0, what ange fo the wate steam is equied to wate twice as much aea? 5. Aea of a secto: A motion detecto can detect movement up to 5 m awa though an angle of 75. (a) What aea can the motion detecto monito? (b) Fo 5 m, what angle is equied to monito 50% moe aea? (c) Fo 75, what ange is equied fo the detecto to monito 50% moe aea? 57. Angula and linea velocit: The planet Jupite s lagest moon, Ganmede, otates aound the planet at a distance of about 5,000 mi, in an obit that is pefectl cicula. If the moon completes one otation about Jupite in 7.5 das, (a) find the angle that the moon moves though in da, in both degees and adians, (b) find the angula velocit of the moon in adians pe hou, and (c) find the moon s linea velocit in miles pe second as it obits Jupite. 58. Angula and linea velocit: The planet Neptune has an obit that is neal cicula. It obits the Sun at a distance of 97 million km and completes one evolution eve 5. (a) Find the angle that the planet moves though in in both degees and adians and (b) find the linea velocit (km/h) as it obits the Sun. 59. Cente-pivot iigation: A standad /-mi iigation sstem has a adius of 00 m and otates once eve das. Find the linea velocit of the outside set of wheels to the neaest tenth of a mete pe hou. 0. Aea of cop field: If a centepivot iigation sstem with adius of 500 m otates once eve 00 h, find the aea of a cop field iigated in da. Round to the neaest squae mete. EXTENDING THE CONCEPT. The adius of the Eath at the equato ( 0 N latitude) is appoimatel 90 mi. Beijing, China, is located at 9.5 N latitude, E longitude. Philadelphia, Pennslvania, is located at the same latitude, but at 75 W longitude. (a) Use the diagam given and a cofunction elationship to find the adius of the Eath (paallel to the equato) at this latitude; (b) use the ac length fomula to compute the shotest distance between these two cities along this latitude; and (c) if the supesonic Concode flew a diect flight between Beijing and Philadelphia along this latitude, appoimate the flight time assuming a cuising speed of 50 mph. Note: The shotest distance is actuall tavesed b heading nothwad, using the ac of a geat cicle that goes though these two cities. Noth Pole h R h South Pole R R N

18 0 CHAPTER Radian Measue and Dnamic Tigonomet -8. The Duvall famil is out on a famil biccle ide aound Ceve Coue Lake. The adult bikes have a pedal spocket with a -in. adius, wheel spocket with -in. adius, and ties with a -in. adius. The kids bikes have pedal spockets with a.5-in. adius, wheel spockets with.5-in. adius, and ties with a 9-in. adius. (a) If adults and kids both pedal at 50 pm, how fa ahead (in ads) ae the adults afte min? (b) If adults pedal at 50 pm, how fast do the kids have to pedal to keep up? MAINTAINING YOUR SKILLS. (.) Convet to degees.. (.) Convet to adians. Leave the esult in tems of (.) Find the eact length of the legs of a tiangle whose hpotenuse measues 0 in.. (.) A small boat is 0 ft awa fom the base of one of the towes of the Golden Gate Bidge. The angle of elevation fom its position to the top of the towe is 8. To the neaest foot, how tall is the towe? Given a 0, b 9, and c 7 use the tiangle shown to find the following. 7. (.) csc 8. (.) cot b Eecises 7 and 8 a c MID-CHAPTER CHECK. The cit of Las Vegas, Nevada, is located at 0 noth latitude, west longitude. (a) Convet both measues to decimal degees. (b) If the adius of Eecise the Eath is 90 mi, how fa noth of the equato is Las 8 cm Vegas?. Find the angle subtended b the ac shown in the figue, then detemine the aea of the secto.. Find the eact adian measue of the following angles. a. 0 b. 5 c cm. Convet the following angles to degee measue. Round to the neaest hundedth if necessa. a. b.. c. 5. Name one positive and one negative angle that ae 5 coteminal with. Answes ma va.. At a kid s canival, the me-go-ound makes 0 evolutions pe minute, with the outemost ides a distance of 5 ft fom the cente. (a) What is the angula velocit of these ides? (b) What is the linea velocit of these ides in miles pe hou? 7. A windshield wipe has a adius of 5 in. and sweeps though an angle of 05 as it wipes the windshield. (a) To the neaest whole, what is the aea of the windshield it wipes? (b) To the neaest whole, what angle (in degees) would be needed fo the wipe to wipe an aea of A in? 8. Identif the quadant of the teminal side of the given angles when in standad position. a..9 b. c. 5. d.. 9. In a 9 naval battle, a fault topedo launched b the HMS Tinidad followed a cicula path and stuck its own ship. Given the velocit of the topedo was 5 m/s duing its 8-sec tip, find the adius of the cicula path it took. 0. Befoe it closed in 97, the Langhone Speedwa was bette known as The Big Left Tun because of its -mi-long cicula tack. On Apil, 90, a oung A. J. Fot captued the checkeed flag in a 50-lap championship ace at this speedwa. Though what cental angle had he diven duing the fist 9. mi of the 50-mi ace?

19 -9 Reinfocing Basic Concepts 07 Moe on Radians REINFORCING BASIC CONCEPTS To incease ou undestanding and appeciation of adian measue, conside the potacto shown in Figue.0, which is maked in both degees and adians. Besides the obvious convesions it illustates, 5, 0 and so on, we, note that eithe sstem is adequate fo measuing the amount of otation fo an angle in standad position. As dawn, this would be paticulal tue fo multiples of 5 Howeve, angles measued in adians have a twofold advantage ove. Ee cm those measued in degees. Fist, the fomula s in adians) enables us to find the length of the elated cicula ac diectl, without having to detemine a popotional pat of the 0 in a full cicle. Fo the potacto shown, 7 cm and using an angle of 0 we have s 7a cm. To veif calculations, b 7. Figue. of this kind, take an semicicula potacto ou have available and detemine its adians adian adius. On a sheet of pape, stand the potacto on the 0 end, mak whee the 7 0 meets the pape and oll the potacto along a staight line to 0 and make anothe mak. Then daw a line segment between these two maks and measue its length the esult will be ve close to s a Repeat this pocess fo othe b. angles measued in adians. Second, if we also mak the potacto in unit adians, as shown in Figue. anote o just moe than.07 adb, the measue of the ac will be numeicall equal to the measue of the angle. If ou epeat the pevious epeiment and oll the potacto to t ad, the esulting line segment will be eactl twice as long as the adius. T it! This again shows that we can view t as eithe an angle in adians o as the length of the elated ac. Moe impotantl, when this unit potacto is seen as the uppe half of a unit cicle (see Figue.), we ae also eminded of wh we can view the tig functions as functions of a eal numbe. Specificall, this is because the ac length t, t, acts as a cicula numbe line that associates an eal numbe t with a unique point (, ) on the unit cicle. Fo cos t and sin t, the tig functions ae now indeed functions of an eal numbe, since the eal numbe line, cicula o oth- Figue. adians.5 ewise, is infinite in length and etends in both a positive and negative diection (note that this view of tigonomet is independent of 0.75 (0, ) adian the ight tiangle view). Using the gid povided in Figue., we 0.5 estimate that an ac length of t.5 units coesponds to the point 5 (0., 0.95) on the unit cicle. Afte veifing , 0.5 adians we use a calculato to suppot ou findings and sue enough, cos and sin Use this infomation to complete the following eecises. (, 0) (, 0) Eecise : Estimate the point (, ) on the unit cicle associated with the values of t indicated, then veif that. Finall, use a calculato to show cos t and sin t fo (a) t 0.5, (b) t 0.5, (c) t 0.75, and (d) t. Eecise : Estimate the point (, ) on the unit cicle associated with the values of t indicated, then veif that. Finall, use standad values to show cos t and sin t fo (a) t (b) t (c) t 5 and (d) t.,,, adians Figue Ee unit

20 . The Unit Cicle Leaning Objectives In Section. ou will lean how to: A. Locate points on a unit cicle and use smmet to locate othe points B. Use special tiangles to find points on a unit cicle and locate othe points using smmet C. Define the si tig functions in tems of a point on the unit cicle In Section., we noted that using adians as a unit of angle measue simplified calculations involving ac length, angula and linea velocit, and the aea of a cicula secto. In much the same wa, the definitions of the tig functions seen in Chapte can be simplified b consideing points on the cicumfeence of a unit cicle ( ). Fo eample, ecall that cos. If, we have cos, with each of the othe tig functions likewise moe simple. In pactice, the impotance of using a unit cicle goes fa beond these algebaic simplifications and is tul the beginning of a moe moden (and moe useful) view of tigonomet. A. The Unit Cicle A cicle is defined as the set of all points in a plane that ae a fied distance called the adius, fom a fied point called the cente. Since the definition involves distance, we can constuct the geneal equation of a cicle using the distance fomula. Assume the cente has coodinates (h, k) and let (, ) epesent an point on the gaph. Since the distance between these points is the adius, the distance fomula ields h k. Figue. Squaing both sides gives h k. Fo cental cicles both h and k ae zeo, and the (0, ) esult is the equation fo a cental cicle of adius : 7 0. The unit cicle is defined as a (, ) cental cicle with adius unit:. As such, the figue can easil be gaphed b dawing a cicle though the fou quadantal points (, 0),, 0, (0, ), and 0, as in Figue.. To find othe points on the cicle, we simpl select an value of, whee, then substitute and solve fo ; o an value of, whee, then solve fo. (, 0) (0, ) (0, 0) (, 0) EXAMPLE Finding Points on a Unit Cicle Find a point on the unit cicle given with (, ) in QII. Solution Using the equation of a unit cicle, we have unit cicle equation a b substitute fo a b subtact esult With (, ) in QII, we choose The point is. a, b. Now t Eecises 7 though

21 - Section. The Unit Cicle 09 Additional points on the unit cicle can be found using smmet. The simplest eamples come fom the quadantal points, whee (, 0) and, 0 ae on opposite sides of the -ais, and (0, ) and 0, ae on opposite sides of the -ais. In geneal, if a and Figue. b ae positive eal numbes and (a, b) is on the unit cicle, then a, b, a, b, and a, b ae also on the cicle because a cicle is smmetic to both aes and the oigin! Fo the point,, a fom Eample, thee othe points, b ae a in QIII, a in QIV, and, b, b,, a in QI. See Figue.., b EXAMPLE Using Smmet to Locate Points on a Unit Cicle Name the quadant containing 5, 5 and veif it s on a unit cicle. Then use smmet to find thee othe points on the cicle. Solution Since both coodinates ae negative, 5, 5 is in QIII. Substituting into the equation fo a unit cicle ields unit cicle equation a substitute 5 fo and 5 fo 5 b a 5 b?,, simplif 5 5? 5 esult checks 5 Since is on the unit cicle, 5, 5, 5, 5, 5 5,, 5 5, 5 5 and 5, 5 ae also on the cicle due to smmet (see figue). A. You ve just leaned how to locate points on a unit cicle and use smmet to locate othe points B. Special Tiangles and the Unit Cicle Now t Eecises 9 though The special tiangles fom Section. can also be used to find points on a unit cicle. As usuall witten, the tiangles state a popotional elationship between thei sides afte assigning a value of to the shotest side. Howeve, pecisel due to this popotional elationship, we can divide all sides b the length of the hpotenuse, giving it a length of unit (see Figues.5 and.). Figue.5 Figue. k u divide b u k q w d d divide b d d

22 0 CHAPTER Radian Measue and Dnamic Tigonomet - We then place the tiangle within the unit cicle, and eflect it fom quadant to quadant to find additional points. We use the sides of the tiangle to detemine the absolute value of each coodinate, and the quadant to give each coodinate the appopiate sign. Note the angles in these special tiangles ae now epessed in adians. EXAMPLE Using a Special Tiangle and Smmet to Locate Points on a Unit Cicle Use the : : tiangle fom Figue. to find fou points on the unit cicle. Solution Begin b supeimposing the tiangle in QI, noting it gives the point a, b shown in Figue.7. B eflecting the tiangle into QII, we find the additional point a on this cicle. Realizing we can simpl appl the cicle s, b emaining smmeties, we obtain the two additional points and a shown in Figue.8., b Figue.7 Figue.8 a, b d, d, d d,,, Now t Eecises 7 and 8 Appling the same idea to a : : tiangle would give the points a and a which includes the same point we found in, b, a, b, b, Eample. When a cental angle is viewed as a otation, Figue.9 each otation can be associated with a unique point (, ) on the teminal side, whee it intesects the unit (, ) cicle (see Figue.9). Fo the quadantal angles and, we associate the points (0, ),,,,, 0, 0,, and (, 0), espectivel. When this otation esults in a special angle, the association can be found using a special tiangle in a manne a, b,

23 - Section. The Unit Cicle simila to Eample. Figue.0 shows we associate the point a with, b, a with and b eoienting the tiangle, a is asso-, : :, b, b ciated with a otation of. k, Figue.0 d, u, Fo standad otations fom 0 to we have the following: Rotation 0 Associated point (, ) (0, 0) a a a (0, ),,, b b b Each of these points give ise to thee othes using the smmet of the cicle. With this smmet and a efeence angle, we can associate additional points on a unit cicle fo 7 Seveal eamples of the efeence angle concept ae shown in. Figue. fo 7 0 in adians. Figue. q q q q (, ) (, ) 0 q q (, ) (, ) Due to the smmeties of the cicle, efeence angles of and seve to fi,, the absolute value of the coodinates fo and, and we simpl use the appopiate sign fo each coodinate ( is alwas positive). As befoe, this depends solel on the quadant of the teminal side.

24 CHAPTER Radian Measue and Dnamic Tigonomet - EXAMPLE Finding Points on a Unit Cicle Associated with a Rotation B. You ve just leaned how to use special tiangles to find points on a unit cicle and locate othe points using smmet Detemine the efeence angle fo each otation given, then find the associated point (, ) on the unit cicle. a. 5 b. c. 7 Figue. 5 Solution a. A otation of teminates in QII: 5 The associated point is. a since 0 in QII. See, b Figue.. b. A otation of teminates in QIII: The associated point is. a since 0 and 0 in QIII., b 7 c. A otation of teminates in QIV: 7 The associated point is. a since 0 in QIV. See, b Figue.. C. Tigonometic Functions and Points on the Unit Cicle k Now t Eecises 9 though We can now define the si tigonometic functions in tems of a point (, ) on the unit cicle, with the use of ight tiangles fading fom view. Fo this eason the ae sometimes called the cicula functions. q 5 Figue. 7 q d The Cicula Functions Fo an otation and point P(, ) on the unit cicle associated with, cos sec ; 0 sin csc ; 0 tan ; 0 cot ; 0

25 -5 Section. The Unit Cicle Note that once sin, cos, and tan ae known, the values of csc, sec, and cot follow automaticall since a numbe and its ecipocal alwas have the same sign. See Figue.. QII < 0, > 0 (onl is positive) sin is positive tan is positive QIII < 0, < 0 (both and ae negative) Figue. QI > 0, > 0 (both and ae positive) All functions ae positive cos is positive QIV > 0, < 0 (onl is positive) EXAMPLE 5 Evaluating Tig Functions fo a Rotation Evaluate the si tig functions fo 5. q 5 Solution A otation of teminates in QIII, so 5 The associated point is. 5 d ` a since 0 and 0 in QIII., b This ields, C. You ve just leaned how to define the si tig functions in tems of a point on the unit cicle cosa 5 b sina 5 b Noting the ecipocal of is afte ationalizing, we have tana 5 b seca 5 b csca5 b cota5 b Now t Eecises 7 though 0 TECHNOLOGY HIGHLIGHT Gaphing the Unit Cicle When using a gaphing calculato to stud the unit cicle, it s impotant to keep two things in mind. Fist, most gaphing calculatos ae onl capable of gaphing functions, which means we must modif the equation of the cicle (and elations like ellipses, hpebolas, hoizontal paabolas, and so on) befoe it can be gaphed. Second, most standad viewing windows have the - and -values peset at [ 0, 0] even though the calculato sceen is 9 piels wide and piels high. This tends to compess the -values and give a skewed image of the gaph. Conside the equation. Fom ou wok in this section, we know this is the equation of a cicle centeed at (0, 0) with adius. Fo the calculato to gaph this elation, we must define it in two pieces, each of which is a function, b solving fo : oiginal equation isolate solve fo continued

26 CHAPTER Radian Measue and Dnamic Tigonomet - Figue.5 Y Note that we can sepaate this esult into two pats, each of which is a function. The gaph of Y gives the 0 uppe half of the cicle, while Y gives the lowe half. We can ente these on the Y sceen as shown, using the epession Y instead of eenteing the entie epession. The function vaiables Y, Y, Y, and so on, can be accessed 0 using VARS (Y-VARS) ENTER (:Function). Gaphing and Y on the standad sceen, the esult appeas ve small and moe elliptical than cicula (Figue.5). One wa to fi this (thee ae man othes), is to use the ZOOM :ZDecimal option, which places the tic maks equall spaced on both aes, instead of ting to foce both to displa points fom 0 to 0. 0 Figue.. Using this option gives the sceen shown in Figue.. An even bette gaph can be obtained using the ZOOM :Zoom In option (o b manuall esetting the window size). Using the TRACE featue enables us to view points on the unit cicle, but.7 ecall that this image is actuall the union of two gaphs and we ma need to jump between the uppe and lowe halves using the up o down aows Eecise : Use the TRACE featue to veif the point (0., 0.8) is on the unit cicle, as well as the othe thee elated points given b smmet (as shown in Eample ). Eecise : Use the nd TRACE (CALC) featue to evaluate the function at What do ou notice. about the output? Fo cos o sin, what value of can we associate with this point? Eecise : What othe standad values can ou identif as ou TRACE aound the cicle?. EXERCISES CONCEPTS AND VOCABULARY Fill in each blank with the appopiate wod o phase. Caefull eead the section if needed.. A cental cicle is smmetic to the ais, the ais and to the.. On a unit cicle, cos, sin, and tan ; while,, and. 5. Discuss/Eplain how knowing onl one point on the unit cicle, actuall gives the location of fou points. Wh is this helpful to a stud of the cicula functions?. Since, 5 is on the unit cicle, the point in QII is also on the cicle.. Refeence angles ae the angles fomed b the teminal side of angles in standad position and the. The must measue between and adians.. Discuss/Eplain how the unit cicle simplifies the definitions of the tig functions and wh the points on this cicle ae sufficient to descibe an otation.

27 -7 Section. The Unit Cicle 5 DEVELOPING YOUR SKILLS Given the point is on a unit cicle, complete the odeed pai (, ) fo the quadant indicated. Fo Eecises 7 to, answe in adical fom as needed. Fo Eecises 5 to 8, ound esults to fou decimal places. 7., 0.8; QIII 8. 0., ; QII 9. a 5 QIV 0. a, 8 ; QIV, b; 7 b. a ; QI. a, ; QIII, b 7 b. a ; QII. a, ; QI, b 5 b 5., 0.7; QIII. (0.9909, ); QIV 7. (, 0.98); QII 8. (0.59, ); QI Veif the point given is on a unit cicle, then use smmet to find thee moe points on the cicle. Results fo Eecises 9 to ae eact, esults fo Eecises to ae appoimate. 9. a 0., b. a., 5 b. (0.5, 0.9). a 7, b a, b , , , Use a tiangle with a hpotenuse of length : : to veif that a is a point on the unit cicle., b 8. Use the esults fom Eecise 7 to find thee additional points on the cicle and name the quadant of each point. Find the efeence angle associated with each otation, then find the associated point (, ) on the unit cicle Without the use of a calculato, state the eact value of the tig functions fo the given angle. A diagam ma help. 7. a. sina b. sina b b c. sina5 d. sina 7 b b e. sina9 f. sina b b g. sina5 h. b 8. a. tana b. tana b b c. tana d. tana 5 b b e. tana7 f. tana b b g. tana h. b 9 9. a. cos b. cos 0 sina b tana 0 b c. cosa d. cosa b b 0. a. sin b. sin 0 c. sina d. sina b b WORKING WITH FORMULAS. Unit sphee: z The unit cicle is actuall the intesection of the unit sphee with the -plane ( z 0). An odeed tiple (,, z) in thee dimensional space that satisfies this equation is on the suface of the unit sphee. Find two possible values of each missing vaiable fo the following points on the unit sphee. a. a, b. a c. a 7,,, 9 9, 9 b b 7, zb