6 Trigonometric Functions of Angles

Transcription

1 57050_06_ch06_p qd 07/04/008 05:59 PM Pge Trigonometric Functions of ngles

2 57050_06_ch06_p qd 07/04/008 05:59 PM Pge ngle Mesure 6. Trigonometr of Right Tringles 6. Trigonometric Functions of ngles 6.4 The Lw of Sines 6.5 The Lw of osines hpter Overview The trigonometric functions cn be defined in two different but equivlent ws s functions of rel numbers (hpter 5) or s functions of ngles (hpter 6). The two pproches to trigonometr re independent of ech other, so either hpter 5 or hpter 6 m be studied first. We stud both pproches becuse different pplictions require tht we view these functions differentl. The pproch in this chpter lends itself to geometric problems involving finding ngles nd distnces. Suppose we wnt to find the distnce to the sun. Using tpe mesure is of course imprcticl, so we need something besides simple mesurement to tckle this problem. ngles re es to mesure for emple, we cn find the ngle formed b the sun, erth, nd moon b simpl pointing to the sun with one rm nd the moon with the other nd estimting the ngle between them. The ke ide then is to find reltionship between ngles nd distnces. So if we hd w to determine distnces from ngles, we d be ble to find the distnce to the sun without going there. The trigonometric functions provide us with just the tools we need. If is right tringle with cute ngle u s in the figure, then we define sin u to be the rtio /r. Tringle is similr to tringle, so r r lthough the distnces nd r re different from nd r, the given rtio is the sme. Thus, in n right tringle with cute ngle u, the rtio of the side opposite ngle u to the hpotenuse is the sme nd is clled sin u. The other trigonometric rtios re defined in similr fshion. ' Gregor D. Dimijin M.D. r ' In this chpter we lern how trigonometric functions cn be used to mesure distnces on the erth nd in spce. In Eercises 6 nd 6 on pge 487, we ctull de- r' ' ' ' 467 HPTER 6 Trigonometric Functions of ngles 467

3 57050_06_ch06_p qd 07/04/008 05:59 PM Pge HPTER 6 Trigonometric Functions of ngles termine the distnce to the sun using trigonometr. Right tringle trigonometr hs mn other pplictions, from determining the optiml cell structure in beehive (Eercise 67, pge 497) to eplining the shpe of rinbow (Eercise 69, pge 498). In the Focus on Modeling, pges 5 5, we see how surveor uses trigonometr to mp town. SUGGESTED TIME ND EMPHSIS clss. Essentil mteril. POINTS TO STRESS. Degree nd rdin mesurements of ngles.. oterminl ngles.. rc lengths. 4. Liner nd ngulr speed. 6. ngle Mesure n ngle O consists of two rs R nd R with common verte O (see Figure ). We often interpret n ngle s rottion of the r R onto R. In this cse, R is clled the initil side, nd R is clled the terminl side of the ngle. If the rottion is counterclockwise, the ngle is considered positive, nd if the rottion is clockwise, the ngle is considered negtive. terminl side O R initil side Positive ngle R R O terminl side initil side Negtive ngle R Figure ngle Mesure The mesure of n ngle is the mount of rottion bout the verte required to move R onto R. Intuitivel, this is how much the ngle opens. One unit of mesurement for ngles is the degree. n ngle of mesure degree is formed b rotting the initil side 60 of complete revolution. In clculus nd other brnches of mthemtics, more nturl method of mesuring ngles is used rdin mesure. The mount n ngle opens is mesured long the rc of circle of rdius with its center t the verte of the ngle. Rdin mesure of Definition of Rdin Mesure If circle of rdius is drwn with the verte of n ngle t its center, then the mesure of this ngle in rdins (bbrevited rd) is the length of the rc tht subtends the ngle (see Figure ). Figure The circumference of the circle of rdius is p nd so complete revolution hs mesure p rd, stright ngle hs mesure p rd, nd right ngle hs mesure SMPLE QUESTIONS Tet Questions () Which two of the following ngles (given in rdin mesure) re coterminl? 0 p p p p (b) Which two of the following ngles re coterminl? nswers () 0 nd p (b) 80 nd HPTER 6 Trigonometric Functions of ngles

4 57050_06_ch06_p qd 08/04/008 05:58 PM Pge 469 SETION 6. ngle Mesure 469 Figure Rdin mesure O rd p/ rd. n ngle tht is subtended b n rc of length long the unit circle hs rdin mesure (see Figure ). O π rd O π rd O rd DRILL QUESTION Find n ngle with mesure (in p rdins) between - p nd tht is coterminl with the ngle of p mesure in stndrd position. nswer Since complete revolution mesured in degrees is 60 nd mesured in rdins is p rd, we get the following simple reltionship between these two methods of ngle mesurement. - p Reltionship between Degrees nd Rdins 80 p rd rd 80 p b p. To convert degrees to rdins, multipl b. 80 p 80 rd 80. To convert rdins to degrees, multipl b. p Mesure of = rd Mesure of Å57.96* Figure 4 To get some ide of the size of rdin, notice tht rd nd rd n ngle u of mesure rd is shown in Figure 4. Emple onverting between Rdins nd Degrees p () Epress 60 in rdins. (b) Epress rd in degrees. 6 Solution The reltionship between degrees nd rdins gives p p () (b) 6 rd p 6 b80 b 0 80 b rd p rd p note on terminolog: We often use phrse such s 0 ngle to men n ngle whose mesure is 0. lso, for n ngle u, we write u 0 or u p/6 to men the mesure of u is 0 or p/6 rd. When no unit is given, the ngle is ssumed to be mesured in rdins. LTERNTE EXMPLE Epress 0 in rdins. NSWER p 9 LTERNTE EXMPLE b p Epress rd in degrees. 9 NSWER 0 IN-LSS MTERILS Mke n nlog between ngle mesurement nd clock mesurement. In 48 hours 5 minutes, how fr will the minute hnd hve moved? One nswer is tht it will go round the clock fce 484 times. nother nswer is tht it moved one qurter of the w round the clock fce, reltive to where it strted. HPTER 6 Trigonometric Functions of ngles 469

5 57050_06_ch06_p qd 07/04/008 05:59 PM Pge HPTER 6 Trigonometric Functions of ngles ngles in Stndrd Position n ngle is in stndrd position if it is drwn in the -plne with its verte t the origin nd its initil side on the positive -is. Figure 5 gives emples of ngles in stndrd position. EXMPLE Find n ngle between 0 nd p tht is coterminl with 00. NSWER 00-0p L () Figure 5 ngles in stndrd position 0 (b) 0 (c) Two ngles in stndrd position re coterminl if their sides coincide. In Figure 5 the ngles in () nd (c) re coterminl. 0 (d) LTERNTE EXMPLE () Find ngles tht re coterminl with the ngle u = 6 in stndrd position. (b) Find ngles tht re coterminl with the ngle u = 5p in 6 stndrd position. NSWERS () We dd n positive or negtive multiple of 60 to 6. nswers include 4 nd 98. (b) We dd n positive or negtive multiple of to 5p 6. nswers include - 7p 6. 7p 6 nd EXMPLE Find n ngle between 0 nd 60 tht is coterminl with -64. Emple oterminl ngles () Find ngles tht re coterminl with the ngle u 0 in stndrd position. (b) Find ngles tht re coterminl with the ngle u p in stndrd position. Solution () To find positive ngles tht re coterminl with u, we dd n multiple of 60. Thus Figure nd re coterminl with u 0. To find negtive ngles tht re coterminl with u, we subtrct n multiple of 60. Thus nd re coterminl with u. (See Figure 6.) 0 0* (b) To find positive ngles tht re coterminl with u, we dd n multiple of p. Thus p p 7p 0 nd 90* p p 4p 0 _0* NSWER 6 EXMPLE We cn pproimte the shpe of the Erth b sphere of rdius 960 miles. If we wnted to wlk fr enough to trverse ectl one degree in ltitude, how fr trip would we hve to tke? NSWER 960 # # p 80 L 69 miles 470 HPTER 6 Trigonometric Functions of ngles

6 57050_06_ch06_p qd 07/04/008 05:59 PM Pge 47 SETION 6. ngle Mesure 47 re coterminl with u p/. To find negtive ngles tht re coterminl with u, we subtrct n multiple of p. Thus p p 5p nd re coterminl with u. (See Figure 7.) p 4p p Figure 7 0 π 0 7π 5π _ 0 Emple oterminl ngles Find n ngle with mesure between 0 nd 60 tht is coterminl with the ngle of mesure 90 in stndrd position. Solution We cn subtrct 60 s mn times s we wish from 90, nd the resulting ngle will be coterminl with 90. Thus, is coterminl with 90, nd so is the ngle 90 (60) 570. To find the ngle we wnt between 0 nd 60, we subtrct 60 from 90 s mn times s necessr. n efficient w to do this is to determine how mn times 60 goes into 90, tht is, divide 90 b 60, nd the reminder will be the ngle we re looking for. We see tht 60 goes into 90 three times with reminder of 0. Thus, 0 is the desired ngle (see Figure 8). LTERNTE EXMPLE Find n ngle with mesure between 0 nd 60 tht is coterminl with the ngle of mesure 65 in stndrd position. NSWER 95 90* 0 0* 0 EXMPLE Find n ngle between 0 nd p 88p tht is coterminl with. r s Figure 8 Length of irculr rc n ngle whose rdin mesure is u is subtended b n rc tht is the frction u/p of the circumference of circle. Thus, in circle of rdius r, the length s of n rc tht subtends the ngle u (see Figure 9) is s u circumference of circle p NSWER 4p Figure 9 s ur u pr ur p IN-LSS MTERILS Students often get the flse impression tht ngles with integer mesure re lws in degrees, nd tht ngles with mesures tht re rtionl multiples of p re in rdins. sk students to sketch ngle nd p p then rdin ngle. Similrl, sk students to sketch ngles with mesures rdins nd degrees. HPTER 6 Trigonometric Functions of ngles 47

7 57050_06_ch06_p qd 07/04/008 05:59 PM Pge HPTER 6 Trigonometric Functions of ngles IN-LSS MTERILS Show tht the fmilir formuls = pr nd = pr re ctull specil cses of formuls in this section. Length of irculr rc In circle of rdius r, the length s of n rc tht subtends centrl ngle of u rdins is s r u Solving for u, we get the importnt formul u s r This formul llows us to define rdin mesure using circle of n rdius r: The rdin mesure of n ngle u is s/r, where s is the length of the circulr rc tht subtends u in circle of rdius r (see Figure 0). r Figure 0 The rdin mesure of u is the number of rdiuses tht cn fit in the rc tht subtends u; hence the term rdin. rd r r rd r r LTERNTE EXMPLE 4 Find the length of n rc of circle with rdius m tht subtends centrl ngle of 5. NSWER 7p 4 LTERNTE EXMPLE 4b centrl ngle u in circle of rdius 9 m is subtended b n rc of length m. Find the mesure of u in rdins. NSWER 4 The formul s ru is true onl when u is mesured in rdins. Figure r u r Emple 4 rc Length nd ngle Mesure () Find the length of n rc of circle with rdius 0 m tht subtends centrl ngle of 0. (b) centrl ngle u in circle of rdius 4 m is subtended b n rc of length 6 m. Find the mesure of u in rdins. Solution () From Emple (b) we see tht 0 p/6 rd. So the length of the rc is (b) the formul u s/r, we hve re of irculr Sector s r u 0 p 6 5p m u s r 6 4 rd The re of circle of rdius r is pr. sector of this circle with centrl ngle u hs n re tht is the frction u/p of the re of the entire circle (see Figure ). So the re of this sector is u re of circle p u p pr r u IN-LSS MTERILS The formul for the length of circulr rc cn be verified eperimentll. Hve students either drw circles or use trsh cn lids, soup cns, plstic cups, pie tins, bking pns, or wht hve ou. Mesure rc length nd rdius for 90 ngles, 45 ngles, nd so on, verifing tht s = ru. 47 HPTER 6 Trigonometric Functions of ngles

8 57050_06_ch06_p qd 07/04/008 05:59 PM Pge 47 SETION 6. ngle Mesure 47 re of irculr Sector In circle of rdius r,there of sector with centrl ngle of u rdins is r u The formul r u is true onl when u is mesured in rdins. s r Figure Emple 5 re of Sector Find the re of sector of circle with centrl ngle 60 if the rdius of the circle is m. Solution To use the formul for the re of circulr sector, we must find the centrl ngle of the sector in rdins: 60 60p/80 rd p/ rd. Thus, the re of the sector is r u p b p m irculr Motion Suppose point moves long circle s shown in Figure. There re two ws to describe the motion of the point liner speed nd ngulr speed. Liner speed is the rte t which the distnce trveled is chnging, so liner speed is the distnce trveled divided b the time elpsed. ngulr speed is the rte t which the centrl ngle u is chnging, so ngulr speed is the number of rdins this ngle chnges divided b the time elpsed. Liner Speed nd ngulr Speed Suppose point moves long circle of rdius r nd the r from the center of the circle to the point trverses u rdins in time t. Let s ru be the distnce the point trvels in time t. Then the speed of the object is given b LTERNTE EXMPLE 5 Find the re of sector of circle with centrl ngle 4 if the rdius of the circle is 45 m. NSWER 45p The smbol v is the Greek letter omeg. ngulr speed Liner speed v u t s t Emple 6 Finding Liner nd ngulr Speed bo rottes stone in -ft-long sling t the rte of 5 revolutions ever 0 seconds. Find the ngulr nd liner velocities of the stone. Solution In 0 s, the ngle u chnges b 5 p 0p rdins. So the ngulr speed of the stone is v u t 0p rd 0 s p rd/s LTERNTE EXMPLE 6 disk with -inch dimeter spins t the rte of 45 revolutions per minute. Find the ngulr nd liner velocities of point t the edge of the disk in rdins per second nd inches per second, respectivel. NSWER IN-LSS MTERILS One cn demonstrte the reltionship between liner nd ngulr speed this w: The techer sits in rotting office chir, nd spins t rte of, s, p rdins per five seconds. Now hve one student stnd close to the professor, nd one stnd frther w, nd hve them both tr to run in circles t tht rte. Note tht the one frther w hs to trvel much fster, demonstrting tht if the ngulr speed is held constnt, nd r is incresed, then v is incresed s well. ngulr speed = rd/s p liner speed = 8p in/s HPTER 6 Trigonometric Functions of ngles 47

9 57050_06_ch06_p qd 07/04/008 05:59 PM Pge HPTER 6 Trigonometric Functions of ngles The distnce trveled b the stone in 0 s is s 5 pr 5 p 90p ft. So the liner speed of the stone is s t 90p ft 0 s 9p ft/s Notice tht ngulr speed does not depend on the rdius of the circle, but onl on the ngle u. However, if we know the ngulr speed v nd the rdius r, we cn find liner speed s follows: s/t ru/t ru/t rv. Reltionship between Liner nd ngulr Speed If point moves long circle of rdius r with ngulr speed v, then its liner speed is given b rv LTERNTE EXMPLE 7 womn is riding biccle whose wheels re 0 inches in dimeter. If the wheels rotte t 50 revolutions per minute, find the speed t which she is trveling in miles per hour. NSWER.4 mph Emple 7 Finding Liner Speed from ngulr Speed womn is riding biccle whose wheels re 6 inches in dimeter. If the wheels rotte t 5 revolutions per minute (rpm), find the speed t which she is trveling, in mi/h. Solution The ngulr speed of the wheels is p 5 50p rd/min. Since the wheels hve rdius in. (hlf the dimeter), the liner speed is rv # 50p 0,0. in./min Since there re inches per foot, 580 feet per mile, nd 60 minutes per hour, her speed in miles per hour is 0,0. in./min 60 min/h in./ft 580 ft/mi 6,6 in./h 6,60 in./mi 9.7 mi/h 6. Eercises Find the rdin mesure of the ngle with the given degree mesure Find the degree mesure of the ngle with the given rdin mesure. 7p p 5p p p p p The mesure of n ngle in stndrd position is given. Find two positive ngles nd two negtive ngles tht re coterminl with the given ngle p p 4 IN-LSS MTERILS If the students hve tken some phsics, present them with this prdo: We know tht Einstein sid tht the speed of n object cnnot eceed 86,000 miles per second, regrdless of reference frme. Let the reference frme be bsketbll. Hve them clculte how fst the sun is moving if the bsketbll is spinning on finger, reltive to the frme of reference of the bsketbll. Then hve them clculte how fst the strs re moving. These numbers will be fster thn the speed of light! This prdo is ctull misunderstnding of reltivit. The rules bout the speed of light being mimum, nd most of reltivit in generl, pplies to wht is clled n inertil reference frme one where objects in motion remin in motion, nd objects t rest remin t rest. The reference frmes of spinning bsketbll or n ccelerting trin re not inertil reference frmes. 474 HPTER 6 Trigonometric Functions of ngles

10 57050_06_ch06_p qd 07/04/008 05:59 PM Pge 475 SETION 6. ngle Mesure 475 p p The mesures of two ngles in stndrd position re given. Determine whether the ngles re coterminl.. 70, 40. 0, 0 5p p. 4., p 6, 7p , , Find n ngle between 0 nd 60 tht is coterminl with the given ngle Find n ngle between 0 nd p tht is coterminl with the given ngle. 7p 7p p 6 7p Find the length of the rc s s in the figure. 5p 5. Find the length of n rc tht subtends centrl ngle of 45 in circle of rdius 0 m. 5. Find the length of n rc tht subtends centrl ngle of rd in circle of rdius mi. 54. centrl ngle u in circle of rdius 5 m is subtended b n rc of length 6 m. Find the mesure of u in degrees nd in rdins. 55. n rc of length 00 m subtends centrl ngle u in circle of rdius 50 m. Find the mesure of u in degrees nd in rdins. 56. circulr rc of length ft subtends centrl ngle of 5. Find the rdius of the circle. 57. Find the rdius of the circle if n rc of length 6 m on the circle subtends centrl ngle of p/6 rd. 58. Find the rdius of the circle if n rc of length 4 ft on the circle subtends centrl ngle of Find the re of the sector shown in ech figure. () (b) 0.5 rd 80* Find the rdius of ech circle if the re of the sector is. () (b) 0 40* rd 50* 50. Find the ngle u in the figure. 5. Find the rdius r of the circle in the figure rd r 6. Find the re of sector with centrl ngle rd in circle of rdius 0 m. 6. sector of circle hs centrl ngle of 60. Find the re of the sector if the rdius of the circle is mi. 6. The re of sector of circle with centrl ngle of rd is 6 m. Find the rdius of the circle. 64. sector of circle of rdius 4 mi hs n re of 88 mi. Find the centrl ngle of the sector. 65. The re of circle is 7 cm. Find the re of sector of this circle tht subtends centrl ngle of p/6 rd. 66. Three circles with rdii,, nd ft re eternll tngent to one nother, s shown in the figure on the net pge. Find the re of the sector of the circle of rdius tht is cut off HPTER 6 Trigonometric Functions of ngles 475

11 57050_06_ch06_p qd 07/04/008 05:59 PM Pge HPTER 6 Trigonometric Functions of ngles b the line segments joining the center of tht circle to the centers of the other two circles. of the erth from the following observtions. He noticed tht on certin d the sun shone directl down deep well in Sene (modern swn). t the sme time in lendri, 500 miles north (on the sme meridin), the rs of the sun shone t n ngle of 7. to the zenith. Use this informtion nd the figure to find the rdius nd circumference of the erth. 500 mi lendri 7.* Rs of sun pplictions 67. Trvel Distnce cr s wheels re 8 in. in dimeter. How fr (in miles) will the cr trvel if its wheels revolve 0,000 times without slipping? 68. Wheel Revolutions How mn revolutions will cr wheel of dimeter 0 in. mke s the cr trvels distnce of one mile? 69. Ltitudes Pittsburgh, Pennslvni, nd Mimi, Pittsburgh Florid, lie pproimtel on Mimi the sme meridin. Pittsburgh hs ltitude of 40.5 N nd Mimi, 5.5 N. Find the distnce between these two cities. (The rdius of the erth is 960 mi.) 70. Ltitudes Memphis, Tennessee, nd New Orlens, Louisin, lie pproimtel on the sme meridin. Memphis hs ltitude 5 N nd New Orlens, 0 N. Find the distnce between these two cities. (The rdius of the erth is 960 mi.) 7. Orbit of the Erth Find the distnce tht the erth trvels in one d in its pth round the sun. ssume tht er hs 65 ds nd tht the pth of the erth round the sun is circle of rdius 9 million miles. [The pth of the erth round the sun is ctull n ellipse with the sun t one focus (see Section 0.). This ellipse, however, hs ver smll eccentricit, so it is nerl circulr.] 7. Nuticl Miles Find the distnce long n rc on the surfce of the erth tht subtends centrl ngle of minute minute 60 degree. This distnce is clled nuticl mile. (The rdius of the erth is 960 mi.) 74. Irrigtion n irrigtion sstem uses stright sprinkler pipe 00 ft long tht pivots round centrl point s shown. Due to n obstcle the pipe is llowed to pivot through 80 onl. Find the re irrigted b this sstem. 00 ft 80* Sene 75. Windshield Wipers The top nd bottom ends of windshield wiper blde re 4 in. nd 4 in. from the pivot point, respectivel. While in opertion the wiper sweeps through 5. Find the re swept b the blde. sun 7. ircumference of the Erth The Greek mthemticin Ertosthenes (c ) mesured the circumference 4 in. 5* 4 in. 476 HPTER 6 Trigonometric Functions of ngles

12 57050_06_ch06_p qd 07/04/008 05:59 PM Pge 477 SETION 6. ngle Mesure The Tethered ow cow is tethered b 00-ft rope to the inside corner of n L-shped building, s shown in the figure. Find the re tht the cow cn grze. 8. Speed of urrent To mesure the speed of current, scientists plce pddle wheel in the strem nd observe the rte t which it rottes. If the pddle wheel hs rdius 0.0 m nd rottes t 00 rpm, find the speed of the current in m/s. 0 ft 50 ft 00 ft 60 ft 50 ft 77. Winch winch of rdius ft is used to lift hev lods. If the winch mkes 8 revolutions ever 5 s, find the speed t which the lod is rising. 84. iccle Wheel The sprockets nd chin of biccle re shown in the figure. The pedl sprocket hs rdius of 4 in., the wheel sprocket rdius of in., nd the wheel rdius of in. The cclist pedls t 40 rpm. () Find the ngulr speed of the wheel sprocket. (b) Find the speed of the biccle. (ssume tht the wheel turns t the sme rte s the wheel sprocket.) in. in. 4 in. 78. Fn ceiling fn with 6-in. bldes rottes t 45 rpm. () Find the ngulr speed of the fn in rd/min. (b) Find the liner speed of the tips of the bldes in in./min. 79. Rdil Sw rdil sw hs blde with 6-in. rdius. Suppose tht the blde spins t 000 rpm. () Find the ngulr speed of the blde in rd/min. (b) Find the liner speed of the swteeth in ft/s. 80. Speed t Equtor The erth rottes bout its is once ever h 56 min 4 s, nd the rdius of the erth is 960 mi. Find the liner speed of point on the equtor in mi/h. 8. Speed of r The wheels of cr hve rdius in. nd re rotting t 600 rpm. Find the speed of the cr in mi/h. 8. Truck Wheels truck with 48-in.-dimeter wheels is trveling t 50 mi/h. () Find the ngulr speed of the wheels in rd/min. (b) How mn revolutions per minute do the wheels mke? 85. onicl up conicl cup is mde from circulr piece of pper with rdius 6 cm b cutting out sector nd joining the edges s shown. Suppose u 5p/. () Find the circumference of the opening of the cup. (b) Find the rdius r of the opening of the cup. [Hint:Use pr.] (c) Find the height h of the cup. [Hint: Use the Pthgoren Theorem.] (d) Find the volume of the cup. 6 cm 6 cm h r 6 cm HPTER 6 Trigonometric Functions of ngles 477

13 57050_06_ch06_p qd 07/04/008 05:59 PM Pge HPTER 6 Trigonometric Functions of ngles 86. onicl up In this eercise we find the volume of the conicl cup in Eercise 85 for n ngle u. () Follow the steps in Eercise 85 to show tht the volume of the cup s function of u is Vu 9 p u 4p u, (b) Grph the function V. (c) For wht ngle u is the volume of the cup mimum? Discover Discussion 0 u p 87. Different Ws of Mesuring ngles The custom of mesuring ngles using degrees, with 60 in circle, dtes bck to the ncient blonins, who used number sstem bsed on groups of 60. nother sstem of mesuring ngles divides the circle into 400 units, clled grds. In this sstem right ngle is 00 grd, so this fits in with our bse 0 number sstem. Write short ess compring the dvntges nd disdvntges of these two sstems nd the rdin sstem of mesuring ngles. Which sstem do ou prefer? 88. locks nd ngles In one hour, the minute hnd on clock moves through complete circle, nd the hour hnd moves through of circle. Through how mn rdins do the minute nd the hour hnd move between :00 P.M. nd 6:45 P.M. (on the sme d)? SUGGESTED TIME ND EMPHSIS clss. Essentil mteril. 6. Trigonometr of Right Tringles In this section we stud certin rtios of the sides of right tringles, clled trigonometric rtios, nd give severl pplictions. POINTS TO STRESS. Definition of the si trigonometric functions s rtios of sides of right tringles.. Specil tringles: nd pplictions tht involve solving right tringles. hpotenuse opposite Trigonometric Rtios onsider right tringle with u s one of its cute ngles. The trigonometric rtios re defined s follows (see Figure ). The Trigonometric Rtios sin u opposite hpotenuse cos u djcent hpotenuse tn u opposite djcent djcent csc u hpotenuse opposite sec u hpotenuse djcent cot u djcent opposite Figure The smbols we use for these rtios re bbrevitions for their full nmes: sine, cosine, tngent, cosecnt, secnt, cotngent. Since n two right tringles with IN-LSS MTERILS Note: If hpter 5 ws covered, this point m hve lred been discussed. The etmolog of the word sine is firl interesting nd not tht well-known. It strts with the Indin word j, mening cord of bowstring. The rbs trnslted this word s jib. The written lnguge hd no vowels, so it looked like this: jb. In 45, the Spnish trnsltor Robert of heste hd to figure out wht vowels to put in. Due to the shpe of the curve, he thought the word ws jib, which ment the opening of grment tht shows womn s clevge. So he used the Ltin word for the cvit formed b curve: sinus. (This word eists in the English lnguge tod right behind our nose re the sinus cvities.) 478 HPTER 6 Trigonometric Functions of ngles

14 57050_06_ch06_p qd 07/04/008 05:59 PM Pge 479 SETION 6. Trigonometr of Right Tringles 479 Hipprchus (circ 40..) is considered the founder of trigonometr. He constructed tbles for function closel relted to the modern sine function nd evluted for ngles t hlf-degree intervls. These re considered the first trigonometric tbles. He used his tbles minl to clculte the pths of the plnets through the hevens. œ 5 Figure Figure ngle u re similr, these rtios re the sme, regrdless of the size of the tringle; the trigonometric rtios depend onl on the ngle u (see Figure ). 5 4 ß = 5 4 ç = 5 Emple Finding Trigonometric Rtios Find the si trigonometric rtios of the ngle u in Figure. Solution sin u cos u 5 tn u 5 csc u sec u ß = = ç = = 50 5 cot u 5 0 DRILL QUESTION onsider this tringle: Find sin u, sec u, nd tn u. nswer,, LTERNTE EXMPLE Find the si trigonometric rtios of the ngle u in the figure below. 4 å Figure 4 œ 7 Emple Finding Trigonometric Rtios If cos 4, sketch right tringle with cute ngle, nd find the other five trigonometric rtios of. Solution Since cos is defined s the rtio of the djcent side to the hpotenuse, we sketch tringle with hpotenuse of length 4 nd side of length djcent to.ifthe opposite side is, then b the Pthgoren Theorem, 4 or 7, so 7. We then use the tringle in Figure 4 to find the rtios. sin 7 4 csc 4 7 Specil Tringles cos 4 sec 4 tn 7 cot 7 ertin right tringles hve rtios tht cn be clculted esil from the Pthgoren Theorem. Since the re used frequentl, we mention them here. The first tringle is obtined b drwing digonl in squre of side (see Figure 5 on pge 480). the Pthgoren Theorem this digonl hs length. The œ 5 NSWER sin(u) =, tn(u) = 5, cos(u) = 5, cot(u) = 5, sec(u) = csc(u) = 5, LTERNTE EXMPLE onsider right tringle with s one of its cute ngles. If cos = 7 8, find the other five trigonometric rtios of. IN-LSS MTERILS Note: If hpter 5 ws covered, this point m hve lred been discussed. There is n inconsistenc in mthemticl nottion tht cn be mde eplicit t this time. We s tht Similrl, we s tht ut is written s csc. Unfortuntel, there is smbol, p = p -. = -. sin sin -, tht mens something entirel different the rcsine of. Even worse, s noted in the section, sin = (sin ) nd sin = (sin ). The eponent - is nottionl noml. NSWER sin() = 5 5, tn() = 8 7, 8 csc() = 5, sec() = 8 7, 7 cot() = 5 HPTER 6 Trigonometric Functions of ngles 479

15 57050_06_ch06_p qd 07/04/008 05:59 PM Pge HPTER 6 Trigonometric Functions of ngles ristrchus of Smos (0 0..) ws fmous Greek scientist, musicin, stronomer, nd geometer. In his book On the Sizes nd Distnces of the Sun nd the Moon, he estimted the distnce to the sun b observing tht when the moon is ectl hlf full, the tringle formed b the sun, moon, nd the erth hs right ngle t the moon. His method ws similr to the one described in Eercise 6 in this section. ristrchus ws the first to dvnce the theor tht the erth nd plnets move round the sun, n ide tht did not gin full cceptnce until fter the time of opernicus, 800 ers lter. For this reson he is often clled the opernicus of ntiquit. resulting tringle hs ngles 45,45, nd 90 (or p/4, p/4, nd p/). To get the second tringle, we strt with n equilterl tringle of side nd drw the perpendiculr bisector D of the bse, s in Figure 6. the Pthgoren Theorem the length of D is. Since D bisects ngle, we obtin tringle with ngles 0, 60, nd 90 (or p/6, p/, nd p/). œ 45* Figure 5 45* 60* Figure 6 0* œ We cn now use the specil tringles in Figures 5 nd 6 to clculte the trigonometric rtios for ngles with mesures 0,45, nd 60 (or p/6, p/4, nd p/). These re listed in Tble. D Tble Vlues of the trigonometric rtios for specil ngles u in degrees u in rdins sin u cos u tn u csc u sec u cot u p 0 6 p 45 4 p 60 For n eplntion of numericl methods, see the mrgin note on pge 46. It s useful to remember these specil trigonometric rtios becuse the occur often. Of course, the cn be reclled esil if we remember the tringles from which the re obtined. To find the vlues of the trigonometric rtios for other ngles, we use clcultor. Mthemticl methods (clled numericl methods) used in finding the trigonometric rtios re progrmmed directl into scientific clcultors. For instnce, when the SIN ke is pressed, the clcultor computes n pproimtion to the vlue of the sine of the given ngle. lcultors give the vlues of sine, cosine, nd tngent; the other rtios cn be esil clculted from these using the following reciprocl reltions: csc t sin t sec t cos t cot t tn t You should check tht these reltions follow immeditel from the definitions of the trigonometric rtios. We follow the convention tht when we write sin t, we men the sine of the ngle whose rdin mesure is t. For instnce, sin mens the sine of the ngle whose r- EXMPLES Tringles to solve:. 5* 00 NSWERS * 7 75* * * 40* HPTER 6 Trigonometric Functions of ngles

16 57050_06_ch06_p qd 07/04/008 05:59 PM Pge 48 SETION 6. Trigonometr of Right Tringles 48 din mesure is. When using clcultor to find n pproimte vlue for this number, set our clcultor to rdin mode; ou will find tht If ou wnt to find the sine of the ngle whose mesure is, set our clcultor to degree mode; ou will find tht Emple Using lcultor to Find Trigonometric Rtios With our clcultor in degree mode, nd writing the results correct to five deciml plces, we find With our clcultor in rdin mode, nd writing the results correct to five deciml plces, we find cos sin sin sin sec cos 88 cot tn.54 LTERNTE EXMPLE Use clcultor to find the trigonometric rtio: cot.8. Plese give the nswer to five deciml plces. NSWER pplictions of Trigonometr of Right Tringles tringle hs si prts: three ngles nd three sides. To solve tringle mens to determine ll of its prts from the informtion known bout the tringle, tht is, to determine the lengths of the three sides nd the mesures of the three ngles. Figure 7 0* Figure 8 r sin u b r cos u r b b Emple 4 Solving Right Tringle Solve tringle, shown in Figure 7. Solution It s cler tht 60. To find, we look for n eqution tht reltes to the lengths nd ngles we lred know. In this cse, we hve sin 0 /, so Similrl, cos 0 b/, so sin 0 6 b cos 0 b 6 It s ver useful to know tht, using the informtion given in Figure 8, the lengths of the legs of right tringle re r sin u nd b r cos u The bilit to solve right tringles using the trigonometric rtios is fundmentl to mn problems in nvigtion, surveing, stronom, nd the mesurement of distnces. The pplictions we consider in this section lws involve right tringles but, s we will see in the net three sections, trigonometr is lso useful in solving tringles tht re not right tringles. To discuss the net emples, we need some terminolog. If n observer is looking t n object, then the line from the ee of the observer to the object is clled LTERNTE EXMPLE 4 Solve tringle, shown in the figure below (find the sides nd b, nd the third unknown ngle). 0* NSWER 7, 7, 60 4 b IN-LSS MTERILS It is reltivel es to mke device to mesure ngle of elevtion. Tpe sod strw to the bottom of protrctor, nd tie string with weight to the center. When ou sight n object through the strw, ou cn figure out the ngle b noticing where the string flls. strw protrctor string weight HPTER 6 Trigonometric Functions of ngles 48

17 57050_06_ch06_p qd 07/04/008 05:59 PM Pge HPTER 6 Trigonometric Functions of ngles SMPLE QUESTION Tet Question Define ngle of elevtion. nswer nswers will vr. LTERNTE EXMPLE 5 gint redwood tree csts shdow 45 ft long. Find the height of the tree if the ngle of elevtion of the sun is.. NSWER 76 ft Thles of Miletus (circ ) is the legendr founder of Greek geometr. It is sid tht he clculted the height of Greek column b compring the length of the shdow of his stff with tht of the column. Using properties of similr tringles, he rgued tht the rtio of the height h of the column to the height h of his stff ws equl to the rtio of the length s of the column s shdow to the length s of the stff s shdow: h h s s Since three of these quntities re known, Thles ws ble to clculte the height of the column. ccording to legend, Thles used similr method to find the height of the Gret Prmid in Egpt, fet tht impressed Egpt s king. Plutrch wrote tht lthough he [the king of Egpt] dmired ou [Thles] for other things, et he prticulrl liked the mnner b which ou mesured the height of the prmid without n trouble or instrument. The principle Thles used, the fct tht rtios of corresponding sides of similr tringles re equl, is the foundtion of the subject of trigonometr. the line of sight (Figure 9). If the object being observed is bove the horizontl, then the ngle between the line of sight nd the horizontl is clled the ngle of elevtion. If the object is below the horizontl, then the ngle between the line of sight nd the horizontl is clled the ngle of depression. In mn of the emples nd eercises in this chpter, ngles of elevtion nd depression will be given for hpotheticl observer t ground level. If the line of sight follows phsicl object, such s n inclined plne or hillside, we use the term ngle of inclintion. Figure 9 The net emple gives n importnt ppliction of trigonometr to the problem of mesurement: We mesure the height of tll tree without hving to climb it! lthough the emple is simple, the result is fundmentl to understnding how the trigonometric rtios re pplied to such problems. Emple 5 Line of sight ngle of elevtion Horizontl Finding the Height of Tree gint redwood tree csts shdow 5 ft long. Find the height of the tree if the ngle of elevtion of the sun is 5.7. Solution Let the height of the tree be h. From Figure 0 we see tht h tn h 5 tn Therefore, the height of the tree is bout 56 ft. Line of sight Definition of tngent Multipl b 5 Use clcultor ngle of depression Horizontl h Figure 0 5.7* 5 ft IN-LSS MTERILS Emple 5 is good outline for n eperiment tht cn be done in rel life. Students cn estimte the height of their school, their fvorite roller coster, flgpole, or nthing else. If students mesure the distnce of ten of their pces, the cn pce off the distnce of shdows, which mkes it esier to mesure them. 48 HPTER 6 Trigonometric Functions of ngles

18 57050_06_ch06_p qd 07/04/008 05:59 PM Pge 48 SETION 6. Trigonometr of Right Tringles 48 Figure 4* 7* 500 ft The ke lbels SIN or INV SIN stnd for inverse sine. We stud the inverse trigonometric functions in Section 7.4. h k Emple 6 Problem Involving Right Tringles From point on the ground 500 ft from the bse of building, n observer finds tht the ngle of elevtion to the top of the building is 4 nd tht the ngle of elevtion to the top of flgpole top the building is 7. Find the height of the building nd the length of the flgpole. Solution Figure illustrtes the sitution. The height of the building is found in the sme w tht we found the height of the tree in Emple 5. h tn 4 Definition of tngent 500 h 500 tn 4 Multipl b Use clcultor The height of the building is pproimtel ft. To find the length of the flgpole, let s first find the height from the ground to the top of the pole: k tn k 500 tn To find the length of the flgpole, we subtrct h from k. So the length of the pole is pproimtel 55 ft. In some problems we need to find n ngle in right tringle whose sides re given. To do this, we use Tble (pge 480) bckwrd ; tht is, we find the ngle with the specified trigonometric rtio. For emple, if sin u, wht is the ngle u? From Tble we cn tell tht u 0.Tofind n ngle whose sine is not given in the tble, we use the SIN or INV SIN or RSIN kes on clcultor. For emple, if sin u 0.8, we ppl the SIN ke to 0.8 to get u 5. or 0.97 rd. The clcultor lso gives ngles whose cosine or tngent re known, using the OS or TN ke. LTERNTE EXMPLE 6 From point on the ground 700 ft from the bse of building, it is observed tht the ngle of elevtion to the top of the building is nd the ngle of elevtion to the top of flgpole top the building is. Find the height of the building nd the length of the flgpole. NSWER 69 ft, 8 ft 40 ft 6 ft Figure Emple 7 Solving for n ngle in Right Tringle 40-ft ldder lens ginst building. If the bse of the ldder is 6 ft from the bse of the building, wht is the ngle formed b the ldder nd the building? Solution First we sketch digrm s in Figure. If u is the ngle between the ldder nd the building, then sin u So u is the ngle whose sine is 0.5. To find the ngle u, we use the clcultor. With our clcultor in degree mode, we get u 8.6 SIN ke on LTERNTE EXMPLE 7 50-ft ldder lens ginst building. If the bse of the ldder is 7 ft from the bse of the building, wht is the ngle formed b the ldder nd the building? NSWER 8 IN-LSS MTERILS In ddition to figuring out the height of n object, we cn lso figure out distnces. If ou know building is, for emple, 700 ft high, ou cn tell how fr w ou re b mesuring the ngle of elevtion from where ou re stnding. HPTER 6 Trigonometric Functions of ngles 48

19 57050_06_ch06_p qd 07/04/008 05:59 PM Pge HPTER 6 Trigonometric Functions of ngles 6. Eercises 6 Find the ect vlues of the si trigonometric rtios of the ngle u in the tringle * 5 6* 5 6 Epress nd in terms of trigonometric rtios of u Sketch tringle tht hs cute ngle u, nd find the other five trigonometric rtios of u. 7. sin u 8. cos u cot u 0. tn u 7 8 Find () sin nd cos b, (b) tn nd cot b, nd (c) sec nd csc b å 7 å 5 4. sec u 7. 8 Evlute the epression without using clcultor.. sin p 6 cos p 6 4. sin 0 csc 0 5. sin 0 cos 60 sin 60 cos 0 csc u 9 4 Find the side lbeled. In Eercises nd 4 stte our nswer correct to five deciml plces * 0* 6. sin 60 cos cos 0 sin 0 8. sin p cos p 4 sin p 4 cos p b Solve the right tringle * 0* 45* 6 75* HPTER 6 Trigonometric Functions of ngles

20 57050_06_ch06_p qd 07/04/008 05:59 PM Pge 485 SETION 6. Trigonometr of Right Tringles * π π π π * 65* 4. Epress the length in terms of the trigonometric rtios of u. 0 0* 5 7. Use ruler to crefull mesure the sides of the tringle, nd then use our mesurements to estimte the si trigonometric rtios of u. 44. Epress the length, b, c, nd d in the figure in terms of the trigonometric rtios of u. 8. Using protrctor, sketch right tringle tht hs the cute ngle 40. Mesure the sides crefull, nd use our results to estimte the si trigonometric rtios of Find correct to one deciml plce. 9. d c b 00 60* 0* * 0* pplictions 45. Height of uilding The ngle of elevtion to the top of the Empire Stte uilding in New York is found to be from the ground t distnce of mi from the bse of the building. Using this informtion, find the height of the Empire Stte uilding. HPTER 6 Trigonometric Functions of ngles 485

21 57050_06_ch06_p qd 07/04/008 06:00 PM Pge HPTER 6 Trigonometric Functions of ngles 46. Gtew rch plne is fling within sight of the Gtew rch in St. Louis, Missouri, t n elevtion of 5,000 ft. The pilot would like to estimte her distnce from the Gtew rch. She finds tht the ngle of depression to point on the ground below the rch is. () Wht is the distnce between the plne nd the rch? (b) Wht is the distnce between point on the ground directl below the plne nd the rch? 47. Devition of Lser em lser bem is to be directed towrd the center of the moon, but the bem strs 0.5 from its intended pth. () How fr hs the bem diverged from its ssigned trget when it reches the moon? (The distnce from the erth to the moon is 40,000 mi.) (b) The rdius of the moon is bout 000 mi. Will the bem strike the moon? 48. Distnce t Se From the top of 00-ft lighthouse, the ngle of depression to ship in the ocen is. How fr is the ship from the bse of the lighthouse? 49. Lening Ldder 0-ft ldder lens ginst building so tht the ngle between the ground nd the ldder is 7. How high does the ldder rech on the building? 50. Lening Ldder 0-ft ldder is lening ginst building. If the bse of the ldder is 6 ft from the bse of the building, wht is the ngle of elevtion of the ldder? How high does the ldder rech on the building? 5. ngle of the Sun 96-ft tree csts shdow tht is 0 ft long. Wht is the ngle of elevtion of the sun? 5. Height of Tower 600-ft gu wire is ttched to the top of communictions tower. If the wire mkes n ngle of 65 with the ground, how tll is the communictions tower? 5. Elevtion of Kite mn is ling on the bech, fling kite. He holds the end of the kite string t ground level, nd estimtes the ngle of elevtion of the kite to be 50. If the string is 450 ft long, how high is the kite bove the ground? 54. Determining Distnce womn stnding on hill sees flgpole tht she knows is 60 ft tll. The ngle of depression to the bottom of the pole is 4, nd the ngle of elevtion to the top of the pole is 8. Find her distnce from the pole. 55. Height of Tower wter tower is locted 5 ft from building (see the figure). From window in the building, n observer notes tht the ngle of elevtion to the top of the tower is 9 nd tht the ngle of depression to the bottom of the tower is 5. How tll is the tower? How high is the window? 9* 5* 5 ft 56. Determining Distnce n irplne is fling t n elevtion of 550 ft, directl bove stright highw. Two motorists re driving crs on the highw on opposite sides of the plne, nd the ngle of depression to one cr is 5 nd to the other is 5. How fr prt re the crs? 57. Determining Distnce If both crs in Eercise 56 re on one side of the plne nd if the ngle of depression to one cr is 8 nd to the other cr is 5,how fr prt re the crs? 58. Height of lloon hot-ir blloon is floting bove stright rod. To estimte their height bove the ground, the blloonists simultneousl mesure the ngle of depression to two consecutive mileposts on the rod on the sme side of the blloon. The ngles of depression re found to be 0 nd. How high is the blloon? 59. Height of Mountin To estimte the height of mountin bove level plin, the ngle of elevtion to the top of the mountin is mesured to be. One thousnd feet closer to the mountin long the plin, it is found tht the ngle of elevtion is 5. Estimte the height of the mountin. 60. Height of loud over To mesure the height of the cloud cover t n irport, worker shines spotlight upwrd t n ngle 75 from the horizontl. n observer 600 m w mesures the ngle of elevtion to the spot of light to be 45. Find the height h of the cloud cover. 8 4 h 45* 75* 600 m 486 HPTER 6 Trigonometric Functions of ngles

22 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 487 SETION 6. Trigonometr of Right Tringles Distnce to the Sun When the moon is ectl hlf full, the erth, moon, nd sun form right ngle (see the figure). t tht time the ngle formed b the sun, erth, nd moon is mesured to be If the distnce from the erth to the moon is 40,000 mi, estimte the distnce from the erth to the sun. sun moon erth 64. Prll To find the distnce to nerb strs, the method of prll is used. The ide is to find tringle with the str t one verte nd with bse s lrge s possible. To do this, the str is observed t two different times ectl 6 months prt, nd its pprent chnge in position is recorded. From these two observtions, E SE cn be clculted. (The times re chosen so tht E SE is s lrge s possible, which gurntees tht E OS is 90.) The ngle E SO is clled the prll of the str. lph enturi, the str nerest the erth, hs prll of Estimte the distnce to this str. (Tke the distnce from the erth to the sun to be mi.) E 6. Distnce to the Moon To find the distnce to the sun s in Eercise 6, we needed to know the distnce to the moon. Here is w to estimte tht distnce: When the moon is seen t its zenith t point on the erth, it is observed to be t the horizon from point (see the figure). Points nd re 655 mi prt, nd the rdius of the erth is 960 mi. () Find the ngle u in degrees. (b) Estimte the distnce from point to the moon. erth moon O E Distnce from Venus to the Sun The elongtion of plnet is the ngle formed b the plnet, erth, nd sun (see the figure). When Venus chieves its mimum elongtion of 46., the erth, Venus, nd the sun form tringle with right ngle t Venus. Find the distnce between Venus nd the sun in stronomicl Units (U). ( definition, the distnce between the erth nd the sun is U.) S 655 mi 6. Rdius of the Erth In Eercise 7 of Section 6. method ws given for finding the rdius of the erth. Here is more modern method: From stellite 600 mi bove the erth, it is observed tht the ngle formed b the verticl nd the line of sight to the horizon is Use this informtion to find the rdius of the erth. Venus sun U å erth Discover Discussion 66. Similr Tringles If two tringles re similr, wht properties do the shre? Eplin how these properties mke it possible to define the trigonometric rtios without regrd to the size of the tringle. HPTER 6 Trigonometric Functions of ngles 487

23 57050_06_ch06_p qd 07/04/008 06:00 PM Pge HPTER 6 Trigonometric Functions of ngles SUGGESTED TIME ND EMPHSIS clss. Essentil mteril. 6. Trigonometric Functions of ngles In the preceding section we defined the trigonometric rtios for cute ngles. Here we etend the trigonometric rtios to ll ngles b defining the trigonometric functions of ngles. With these functions we cn solve prcticl problems tht involve ngles which re not necessril cute. Trigonometric Functions of ngles Let POQ be right tringle with cute ngle u s shown in Figure (). Plce u in stndrd position s shown in Figure (b). P P(, ) hpotenuse opposite r Figure O djcent () Q O (b) Q Then P P, is point on the terminl side of u. In tringle POQ, the opposite side hs length nd the djcent side hs length. Using the Pthgoren Theorem, we see tht the hpotenuse hs length r. So sin u r cos u r tn u The other trigonometric rtios cn be found in the sme w. These observtions llow us to etend the trigonometric rtios to n ngle. We define the trigonometric functions of ngles s follows (see Figure ). P(, ) r 0 Definition of the Trigonometric Functions Let u be n ngle in stndrd position nd let P, be point on the terminl side. If r is the distnce from the origin to the point P,, then Figure sin u r cos u r tn u 0 csc u r 0 sec u r 0 cot u 0 POINTS TO STRESS. Finding the reference ngle for given ngle.. Using the reference ngle to evlute trigonometric functions.. The Pthgoren nd reciprocl identities of the trigonometric functions. 488 HPTER 6 Trigonometric Functions of ngles

24 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 489 SETION 6. Trigonometric Functions of ngles 489 Reltionship to the Trigonometric Functions of Rel Numbers You m hve lred studied the trigonometric functions defined using the unit circle (hpter 5). To see how the relte to the trigonometric functions of n ngle, let s strt with the unit circle in the coordinte pln. P(, ) 0 P(, ) is the terminl point determined b t. Let P, be the terminl point determined b n rc of length t on the unit circle. Then t subtends n ngle u t the center of the circle. If we drop perpendiculr from P onto the point Q on the -is, then tringle OPQ is right tringle with legs of length nd, s shown in the figure. t Now, b the definition of the trigonometric functions of the rel number t, we hve the definition of the trigonometric functions of the ngle u, we hve If u is mesured in rdins, then u t. (See the figure below.) ompring the two ws of defining the trigonometric functions, we see tht the re identicl. In other words, s functions, the ssign identicl vlues to given rel number (the rel number is the rdin mesure of u in one cse or the length t of n rc in the other). sin t cos t sin u opp hp cos u dj hp P(, ) t 0 P(, ) r 0 Tringle OPQ is right tringle. The rdin mesure of ngle is t. Wh then do we stud trigonometr in two different ws? ecuse different pplictions require tht we view the trigonometric functions differentl. (See Focus on Modeling, pges 459, 5, nd 575, nd Sections 6., 6.4, nd 6.5.) IN-LSS MTERILS Students often mke the mistke of ssuming tht the trig functions re liner functions. Since the clss will probbl involve computing few trig functions, one cn kill two birds with one stone b computing sin p p p sin nd sin to demonstrte tht, for emple, sin p Return + p b Z sin p + sin p + p b,,. to this point severl times; the will be grteful when the void tht mistke in lter courses. HPTER 6 Trigonometric Functions of ngles 489

25 57050_06_ch06_p qd 07/04/008 06:00 PM Pge HPTER 6 Trigonometric Functions of ngles EXMPLES sin p = cot p 6 LTERNTE EXMPLE Find cos 60. NSWER LTERNTE EXMPLE b Find tn 40. NSWER cos 7p 6 = - tn p 4 sec 7p csc 7p = - = = = O Figure The following mnemonic device cn be used to remember which trigonometric functions re positive in ech qudrnt: ll of them, Sine, Tngent, or osine. Sine Tngent P(, ) P'(', ') Q ll Q' osine You cn remember this s ll Students Tke lculus. _ (_, ) Figure 4 90* 0 Figure 5 r 5* 0 0* r 45* (, ) (, ) Since division b 0 is n undefined opertion, certin trigonometric functions re not defined for certin ngles. For emple, tn 90 / is undefined becuse 0. The ngles for which the trigonometric functions m be undefined re the ngles for which either the - or -coordinte of point on the terminl side of the ngle is 0. These re qudrntl ngles ngles tht re coterminl with the coordinte es. It is crucil fct tht the vlues of the trigonometric functions do not depend on the choice of the point P,. This is becuse if P, is n other point on the terminl side, s in Figure, then tringles POQ nd P OQ re similr. Evluting Trigonometric Functions t n ngle From the definition we see tht the vlues of the trigonometric functions re ll positive if the ngle u hs its terminl side in qudrnt I. This is becuse nd re positive in this qudrnt. [Of course, r is lws positive, since it is simpl the distnce from the origin to the point P,.] If the terminl side of u is in qudrnt II, however, then is negtive nd is positive. Thus, in qudrnt II the functions sin u nd csc u re positive, nd ll the other trigonometric functions hve negtive vlues. You cn check the other entries in the following tble. Signs of the Trigonometric Functions Qudrnt Positive functions Negtive functions I ll none II sin, csc cos, sec, tn, cot III tn, cot sin, csc, cos, sec IV cos, sec sin, csc, tn, cot We now turn our ttention to finding the vlues of the trigonometric functions for ngles tht re not cute. Emple Finding Trigonometric Functions of ngles Find () cos 5 nd (b) tn 90. Solution () From Figure 4 we see tht cos 5 /r. ut cos 45 /r, nd since cos 45 /, we hve cos 5 (b) The ngles 90 nd 0 re coterminl. From Figure 5 it s cler tht tn 90 tn 0 nd, since tn 0 /, we hve tn 90 IN-LSS MTERILS There is the dnger tht students will use the mnemonic given in the tet (ll Students Tke lculus) s w of voiding ll understnding of the sign of the vrious trigonometric functions in different qudrnts. Mke sure tht the re ble to rticulte wh, for emple, the cosine function is positive in qudrnts I nd IV nd negtive in qudrnts II nd III. 490 HPTER 6 Trigonometric Functions of ngles

26 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 49 SETION 6. Trigonometric Functions of ngles 49 From Emple we see tht the trigonometric functions for ngles tht ren t cute hve the sme vlue, ecept possibl for sign, s the corresponding trigonometric functions of n cute ngle. Tht cute ngle will be clled the reference ngle. Reference ngle Let u be n ngle in stndrd position. The reference ngle u ssocited with u is the cute ngle formed b the terminl side of u nd the -is. Figure 6 shows tht to find reference ngle it s useful to know the qudrnt in which the terminl side of the ngle lies. DRILL QUESTION If u = 7p find its reference 6, ngle u. In wht qudrnt is u? nswer u = p is in the second 6. u qudrnt. Figure 6 The reference ngle u for n ngle u 0 = π 0 Emple Finding Reference ngles Find the reference ngle for () u 5p nd (b) u 870. Solution () The reference ngle is the cute ngle formed b the terminl side of the ngle 5p/ nd the -is (see Figure 7). Since the terminl side of this ngle is in qudrnt IV, the reference ngle is LTERNTE EXMPLE Find the reference ngle for u = p 7. NSWER p 7 Figure 7 u p 5p p 0 870* (b) The ngles 870 nd 50 re coterminl [becuse ]. Thus, the terminl side of this ngle is in qudrnt II (see Figure 8). So the reference ngle is u Evluting Trigonometric Functions for n ngle Figure 8 To find the vlues of the trigonometric functions for n ngle u, we crr out the following steps.. Find the reference ngle u ssocited with the ngle u.. Determine the sign of the trigonometric function of u b noting the qudrnt in which u lies.. The vlue of the trigonometric function of u is the sme, ecept possibl for sign, s the vlue of the trigonometric function of u. SMPLE QUESTION Tet Question Is it possible for two different ngles to hve the sme reference ngle? nswer Yes HPTER 6 Trigonometric Functions of ngles 49

27 57050_06_ch06_p qd 07/04/008 06:00 PM Pge HPTER 6 Trigonometric Functions of ngles LTERNTE EXMPLE Find sin 0. NSWER - 40* 0 Figure 9 S sin 40 is negtive. T Emple Using the Reference ngle to Evlute Trigonometric Functions Find () sin 40 nd (b) cot 495. Solution () This ngle hs its terminl side in qudrnt III, s shown in Figure 9. The reference ngle is therefore , nd the vlue of sin 40 is negtive. Thus sin 40 sin 60 Sign Reference ngle 0 495* (b) The ngle 495 is coterminl with the ngle 5, nd the terminl side of this ngle is in qudrnt II, s shown in Figure 0. So the reference ngle is , nd the vlue of cot 495 is negtive. We hve cot 495 cot 5 cot 45 LTERNTE EXMPLE 4 Find sin 4p 6. NSWER - Figure 0 S tn 495 is negtive, T so cot 495 is negtive. 4π Figure S sin 6p is negtive. T 0 Emple 4 Using the Reference ngle to Evlute Trigonometric Functions 6p Find () sin nd (b) sec p. 4 b Solution () The ngle 6p/ is coterminl with 4p/, nd these ngles re in qudrnt III (see Figure ). Thus, the reference ngle is 4p/ p p/. Since the vlue of sine is negtive in qudrnt III, we hve oterminl ngles Sign Reference ngle sin 6p sin 4p sin p oterminl ngles Sign Reference ngle 0 π _ 4 Figure S p cos 4 is positive, T p so sec is positive. 4 (b) The ngle p/4isinqudrnt IV, nd its reference ngle is p/4 (see Figure ). Since secnt is positive in this qudrnt, we get Trigonometric Identities p sec 4 b sec p 4 Sign Reference ngle The trigonometric functions of ngles re relted to ech other through severl importnt equtions clled trigonometric identities. We ve lred encountered the 49 HPTER 6 Trigonometric Functions of ngles

28 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 49 SETION 6. Trigonometric Functions of ngles 49 reciprocl identities. These identities continue to hold for n ngle u, provided both sides of the eqution re defined. The Pthgoren identities re consequence of the Pthgoren Theorem.* Fundmentl Identities Reciprocl Identities csc u sin u sec u cos u cot u tn u tn u sin u cos u cot u cos u sin u Pthgoren Identities sin u cos u tn u sec u cot u csc u r 0 (, ) Proof Let s prove the first Pthgoren identit. Using r (the Pthgoren Theorem) in Figure, we hve sin u cos u r b r b r r r Thus, sin u cos u. (lthough the figure indictes n cute ngle, ou should check tht the proof holds for ll ngles u.) See Eercises 59 nd 60 for the proofs of the other two Pthgoren identities. Figure Emple 5 Epressing One Trigonometric Function in Terms of nother () Epress sin u in terms of cos u. (b) Epress tn u in terms of sin u, where u is in qudrnt II. Solution () From the first Pthgoren identit we get sin u cos u where the sign depends on the qudrnt. If u is in qudrnt I or II, then sin u is positive, nd hence sin u cos u wheres if u is in qudrnt III or IV, sin u is negtive nd so sin u cos u LTERNTE EXMPLE 5b Epress tn u in terms of sin u, where u is in qudrnt III. NSWER sin(u) - - sin (u) *We follow the usul convention of writing sin u for sin u. In generl, we write sin n u for sin u n for ll integers n ecept n. The eponent n will be ssigned nother mening in Section 7.4. Of course, the sme convention pplies to the other five trigonometric functions. HPTER 6 Trigonometric Functions of ngles 49

29 57050_06_ch06_p qd 07/04/008 06:00 PM Pge HPTER 6 Trigonometric Functions of ngles (b) Since tn u sin u/cos u, we need to write cos u in terms of sin u. prt () cos u sin u nd since cos u is negtive in qudrnt II, the negtive sign pplies here. Thus tn u sin u cos u sin u sin u LTERNTE EXMPLE 6 If tn u = nd u is in qudrnt IV, find cos u. NSWER 5 If ou wish to rtionlize the denomintor, ou cn epress cos u s # Emple 6 Evluting Trigonometric Function If tn u nd u is in qudrnt III, find cos u. Solution We need to write cos u in terms of tn u. From the identit tn u sec u, we get sec u tn u. In qudrnt III, sec u is negtive, so Thus sec u tn u cos u sec u tn u 9 œ Figure 4 Solution This problem cn be solved more esil using the method of Emple of Section 6.. Recll tht, ecept for sign, the vlues of the trigonometric functions of n ngle re the sme s those of n cute ngle (the reference ngle). So, ignoring the sign for the moment, let s sketch right tringle with n cute ngle u stisfing tn u (see Figure 4). the Pthgoren Theorem the hpotenuse of this tringle hs length. From the tringle in Figure 4 we immeditel see tht cos u /. Since u is in qudrnt III, cos u is negtive nd so cos u LTERNTE EXMPLE 7 If sec u = 6 nd u is in qudrnt IV, find the other five trigonometric functions of u. NSWER sin (u) = - 5 6, cos (u) = 6, tn (u) = -5, cot (u) = - 5, csc (u) = œ Figure 5 Emple 7 Evluting Trigonometric Functions If sec u nd u is in qudrnt IV, find the other five trigonometric functions of u. Solution We sketch tringle s in Figure 5 so tht sec u. Tking into ccount the fct tht u is in qudrnt IV, we get sin u csc u res of Tringles cos u sec u tn u cot u We conclude this section with n ppliction of the trigonometric functions tht involves ngles tht re not necessril cute. More etensive pplictions pper in the net two sections. 494 HPTER 6 Trigonometric Functions of ngles

30 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 495 SETION 6. Trigonometric Functions of ngles 495 h Figure 6 b () b h =80* _ (b) The re of tringle is bse height. If we know two sides nd the included ngle of tringle, then we cn find the height using the trigonometric functions, nd from this we cn find the re. If u is n cute ngle, then the height of the tringle in Figure 6() is given b h b sin u. Thus, the re is bse height b sin u If the ngle u is not cute, then from Figure 6(b) we see tht the height of the tringle is h b sin80 u b sin u This is so becuse the reference ngle of u is the ngle 80 u. Thus, in this cse lso, the re of the tringle is bse height b sin u re of Tringle The re of tringle with sides of lengths nd b nd with included ngle u is b sin u 0* cm 0 cm Figure 7 6. Eercises Emple 8 Finding the re of Tringle Find the re of tringle shown in Figure 7. Solution The tringle hs sides of length 0 cm nd cm, with included ngle 0. Therefore b sin u 0 sin 0 5 sin 60 5 cm Reference ngle LTERNTE EXMPLE 8 Find the re of tringle shown in the figure below. NSWER 7 # 8 0* 8 Find the reference ngle for the given ngle.. () 50 (b) 0 (c) 0. () 0 (b) 0 (c) 780. () 5 (b) 80 (c) () 99 (b) 99 (c) 59 p p 5. () (b) (c) p 4 6 4p p 6. () (b) (c) 4 p 6 5p 7. () 7 (b).4p (c).4 8. ().p (b). (c) 0p HPTER 6 Trigonometric Functions of ngles 495

31 57050_06_ch06_p qd 07/04/008 06:00 PM Pge HPTER 6 Trigonometric Functions of ngles 9 Find the ect vlue of the trigonometric function. 9. sin sin 5. cos 5. cos 60. tn sec csc cot 0 7. cos sec 0 9. tn cos 660. sin p. sin 5p. sin p 4. cos 7p 7p 5. cos 6. tn 5p b 6 7. sec 7p 8. csc 5p cos 7p. tn 5p. 4 p cot 4 b sin p 6 6 Find the qudrnt in which u lies from the informtion given.. sin u 0 nd cos u 0 4. tn u 0 nd sin u 0 5. sec u 0 nd tn u 0 6. csc u 0 nd cos u Write the first trigonometric function in terms of the second for u in the given qudrnt. 7. tn u, cos u; u in qudrnt III 8. cot u, sin u; u in qudrnt II 9. cos u, sin u; u in qudrnt IV 40. sec u, sin u; u in qudrnt I 4. sec u, tn u; u in qudrnt II 4. csc u, cot u; u in qudrnt III 4 50 Find the vlues of the trigonometric functions of u from the informtion given. 4. sin u 5, u in qudrnt II cos u, u in qudrnt III 45. tn u 4, cos u sec u 5, sin u csc u, u in qudrnt I 48. cot u 4, sin u cos u 7, tn u tn u 4, sin u 0 5. If u p/, find the vlue of ech epression. () sin u, sin u (b) sin u, sin u (c) sin u, sinu 5. Find the re of tringle with sides of length 7 nd 9 nd included ngle Find the re of tringle with sides of length 0 nd nd included ngle Find the re of n equilterl tringle with side of length tringle hs n re of 6 in, nd two of the sides of the tringle hve lengths 5 in. nd 7 in. Find the ngle included b these two sides. 56. n isosceles tringle hs n re of 4 cm, nd the ngle between the two equl sides is 5p/6. Wht is the length of the two equl sides? Find the re of the shded region in the figure Use the first Pthgoren identit to prove the second. [Hint: Divide b cos u.] 60. Use the first Pthgoren identit to prove the third. pplictions 0* 6. Height of Rocket rocket fired stright up is trcked b n observer on the ground mile w. () Show tht when the ngle of elevtion is u, the height of the rocket in feet is h 580 tn u. (b) omplete the tble to find the height of the rocket t the given ngles of elevtion. u h mi h π IN-LSS MTERILS Problems such s Eercises re crucil when students tke clculus. It is importnt tht the go beond memorizing the nswers nd trul understnd where the nswers come from. 496 HPTER 6 Trigonometric Functions of ngles

32 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 497 SETION 6. Trigonometric Functions of ngles Rin Gutter rin gutter is to be constructed from metl sheet of width 0 cm b bending up one-third of the sheet on ech side through n ngle u. () Show tht the cross-sectionl re of the gutter is modeled b the function u 00 sin u 00 sin u cos u (b) Grph the function for 0 u p/. (c) For wht ngle u is the lrgest cross-sectionl re chieved? Find the rnge nd height of shot put thrown under the given conditions. () On the erth with 0 ft/s nd u p/6 (b) On the moon with 0 ft/s nd u p/6 H R 0 cm 0 cm 6. Wooden em rectngulr bem is to be cut from clindricl log of dimeter 0 cm. The figures show different ws this cn be done. () Epress the cross-sectionl re of the bem s function of the ngle u in the figures. (b) Grph the function ou found in prt (). (c) Find the dimensions of the bem with lrgest crosssectionl re. 0 cm 66. Sledding The time in seconds tht it tkes for sled to slide down hillside inclined t n ngle u is d t 6 sin u where d is the length of the slope in feet. Find the time it tkes to slide down 000-ft slope inclined t 0. d width depth 0 cm 0 cm 67. eehives In beehive ech cell is regulr hegonl prism, s shown in the figure. The mount of w W in the cell depends on the pe ngle u nd is given b W cot u 0.65 csc u 64. Strength of em The strength of bem is proportionl to the width nd the squre of the depth. bem is cut from log s in Eercise 6. Epress the strength of the bem s function of the ngle u in the figures. 65. Throwing Shot Put The rnge R nd height H of shot put thrown with n initil velocit of 0 ft/s t n ngle u re given b R 0 sinu g H 0 sin u g On the erth g ft/s nd on the moon g 5. ft/s. ees instinctivel choose u so s to use the lest mount of w possible. () Use grphing device to grph W s function of u for 0 u p. (b) For wht vlue of u does W hve its minimum vlue? [Note: iologists hve discovered tht bees rrel devite from this vlue b more thn degree or two.] HPTER 6 Trigonometric Functions of ngles 497

33 57050_06_ch06_p qd 07/04/008 06:00 PM Pge HPTER 6 Trigonometric Functions of ngles 68. Turning orner steel pipe is being crried down hllw 9 ft wide. t the end of the hll there is rightngled turn into nrrower hllw 6 ft wide. () Show tht the length of the pipe in the figure is modeled b the function Lu 9 csc u 6 sec u (b) Grph the function L for 0 u p/. (c) Find the minimum vlue of the function L. (d) Eplin wh the vlue of L ou found in prt (c) is the length of the longest pipe tht cn be crried round the corner. 9 ft 6 ft 69. Rinbows Rinbows re creted when sunlight of different wvelengths (colors) is refrcted nd reflected in rindrops. The ngle of elevtion u of rinbow is lws the sme. It cn be shown tht u 4b where sin k sin b nd 59.4 nd k. is the inde of refrction of wter. Use the given informtion to find the ngle of elevtion u of rinbow. (For mthemticl eplntion of rinbows see lculus, 5th Edition, b Jmes Stewrt, pges ) Discover Discussion 70. Using lcultor To solve certin problem, ou need to find the sine of 4 rd. Your stud prtner uses his clcultor nd tells ou tht sin On our clcultor ou get sin Wht is wrong? Wht mistke did our prtner mke? 7. Viète s Trigonometric Digrm In the 6th centur, the French mthemticin Frnçois Viète (see pge 49) published the following remrkble digrm. Ech of the si trigonometric functions of u is equl to the length of line segment in the figure. For instnce, sin u 0 PR 0, since from OPR we see tht sin u opp hp For ech of the five other trigonometric functions, find line segment in the figure whose length equls the vlue of the function t u. (Note: The rdius of the circle is, the center is O, segment QS is tngent to the circle t R, nd SOQ is right ngle.) S 0 PR 0 0 OR 0 0 PR 0 0 PR 0 R O P Q 498 HPTER 6 Trigonometric Functions of ngles

34 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 499 SETION 6. Trigonometric Functions of ngles 499 DISOVERY PROJET c c b Thles used similr tringles to find the height of tll column. (See pge 48.) b Similrit In geometr ou lerned tht two tringles re similr if the hve the sme ngles. In this cse, the rtios of corresponding sides re equl. Tringles nd in the mrgin re similr, so b c b c Similrit is the crucil ide underling trigonometr. We cn define sin u s the rtio of the opposite side to the hpotenuse in n right tringle with n ngle u, becuse ll such right tringles re similr. So the rtio represented b sin u does not depend on the size of the right tringle but onl on the ngle u.this is powerful ide becuse ngles re often esier to mesure thn distnces. For emple, the ngle formed b the sun, erth, nd moon cn be mesured from the erth. The secret to finding the distnce to the sun is tht the trigonometric rtios re the sme for the huge tringle formed b the sun, erth, nd moon s for n other similr tringle (see Eercise 6 in Section 6.). In generl, two objects re similr if the hve the sme shpe even though the m not be the sme size.* For emple, we recognize the following s representtions of the letter becuse the re ll similr. d d If two figures re similr, then the distnces between corresponding points in the figures re proportionl. The blue nd red s bove re similr the rtio of distnces between corresponding points is. We s tht the similrit rtio is s. To obtin the distnce d between n two points in the blue, we multipl the corresponding distnce d in the red b. So d sd or d d Likewise, the similrit rtio between the first nd lst letters is s 5, so 5.. Write short prgrph eplining how the concept of similrit is used to define the trigonometric rtios.. How is similrit used in mp mking? How re distnces on cit rod mp relted to ctul distnces?. How is our erbook photogrph similr to ou? ompre distnces between different points on our fce (such s distnce between ers, length of * If the hve the sme shpe nd size, the re congruent, which is specil cse of similrit. HPTER 6 Trigonometric Functions of ngles 499

35 57050_06_ch06_p qd 07/04/008 06:00 PM Pge HPTER 6 Trigonometric Functions of ngles nose, distnce between ees, nd so on) to the corresponding distnces in photogrph. Wht is the similrit rtio? 4. The figure illustrtes method for drwing n pple twice the size of given pple. Use the method to drw tie times the size (similrit rtio ) of the blue tie. 5. Give conditions under which two rectngles re similr to ech other. Do the sme for two isosceles tringles. 6. Suppose tht two similr tringles hve similrit rtio s. () How re the perimeters of the tringles relted? (b) How re the res of the tringles relted? c b h s sc sb sh If the side of squre is doubled, its re is multiplied b. 7. () If two squres hve similrit rtio s, show tht their res nd hve the propert tht s. (b) If the side of squre is tripled, its re is multiplied b wht fctor? (c) plne figure cn be pproimted b squres (s shown). Eplin how we cn conclude tht for n two plne figures with similrit rtio s, their res stisf s. (Use prt ().) 500 HPTER 6 Trigonometric Functions of ngles

36 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 50 SETION 6.4 The Lw of Sines 50 If the side of cube is doubled, its volume is multiplied b. 8. () If two cubes hve similrit rtio s, show tht their volumes V nd V hve the propert tht V s V. (b) If the side of cube is multiplied b 0, b wht fctor is the volume multiplied? (c) How cn we use the fct tht solid object cn be filled b little cubes to show tht for n two solids with similrit rtio s, the volumes stisf V s V? 9. King Kong is 0 times s tll s Joe, norml-sized 00-lb gorill. ssuming tht King Kong nd Joe re similr, use the results from Problems 7 nd 8 to nswer the following questions. () How much does King Kong weigh? (b) If Joe s hnd is in. long, how long is King Kong s hnd? (c) If it tkes squre rds of mteril to mke shirt for Joe, how much mteril would shirt for King Kong require? Figure b c 6.4 The Lw of Sines In Section 6. we used the trigonometric rtios to solve right tringles. The trigonometric functions cn lso be used to solve oblique tringles, tht is, tringles with no right ngles. To do this, we first stud the Lw of Sines here nd then the Lw of osines in the net section. To stte these lws (or formuls) more esil, we follow the convention of lbeling the ngles of tringle s,,, nd the lengths of the corresponding opposite sides s, b, c, s in Figure. To solve tringle, we need to know certin informtion bout its sides nd ngles. To decide whether we hve enough informtion, it s often helpful to mke sketch. For instnce, if we re given two ngles nd the included side, then it s cler tht one nd onl one tringle cn be formed (see Figure ()). Similrl, if two sides nd the included ngle re known, then unique tringle is determined (Figure (c)). ut if we know ll three ngles nd no sides, we cnnot uniquel determine the tringle becuse mn tringles cn hve the sme three ngles. (ll these tringles would be similr, of course.) So we won t consider this lst cse. SUGGESTED TIME ND EMPHSIS clss. Essentil mteril. () S or S (b) SS (c) SS (d) SSS Figure In generl, tringle is determined b three of its si prts (ngles nd sides) s long s t lest one of these three prts is side. So, the possibilities, illustrted in Figure, re s follows. POINTS TO STRESS. Using the Lw of Sines to solve tringles.. Understnding which cses re mbiguous nd which cses re unmbiguous. HPTER 6 Trigonometric Functions of ngles 50

37 57050_06_ch06_p qd 07/04/008 06:00 PM Pge HPTER 6 Trigonometric Functions of ngles IN-LSS MTERILS There is n interesting bit of trivi bout the Mount Everest epedition described in the tet: Sir George Everest mesured the pek of the mountin to be 9,000 ft ectl. He didn t wnt people to think it ws n estimte (to onl two deciml plces of ccurc) so he lied nd sid he mesured it to be 9,00 ft. See lso Section. of this book. DRILL QUESTION Find in the following digrm: 6* 60* nswer L 4.9 LTERNTE EXMPLE Two people who live 0 miles prt look t n overhed irplne simultneousl. The ngle of elevtion reltive to the first person is 80 degrees, nd the ngle of elevtion reltive to the second person is 5 degrees. Wht is the distnce between the first person nd the irplne? NSWER 5. miles 5 57* c Figure b Los ngeles Figure 4 b h=b ß 75* 60* c=40 mi Phoeni se One side nd two ngles (S or S) se Two sides nd the ngle opposite one of those sides (SS) se Two sides nd the included ngle (SS) se 4 Three sides (SSS) ses nd re solved using the Lw of Sines; ses nd 4 require the Lw of osines. The Lw of Sines The Lw of Sines ss tht in n tringle the lengths of the sides re proportionl to the sines of the corresponding opposite ngles. The Lw of Sines In tringle we hve sin sin b sin c Proof To see wh the Lw of Sines is true, refer to Figure. the formul in Section 6. the re of tringle is b sin. the sme formul the re of this tringle is lso c sin nd bc sin. Thus Multipling b /bc gives the Lw of Sines. Emple Trcking Stellite (S) stellite orbiting the erth psses directl overhed t observtion sttions in Phoeni nd Los ngeles, 40 mi prt. t n instnt when the stellite is between these two sttions, its ngle of elevtion is simultneousl observed to be 60 t Phoeni nd 75 t Los ngeles. How fr is the stellite from Los ngeles? In other words, find the distnce in Figure 4. Solution Whenever two ngles in tringle re known, the third ngle cn be determined immeditel becuse the sum of the ngles of tringle is 80. In this cse, (see Figure 4), so we hve sin b sin 60 b bc sin c sin b sin sin c b sin sin 60 sin Lw of Sines Substitute Solve for b The distnce of the stellite from Los ngeles is pproimtel 46 mi. EXMPLES S. = 0, = 40, c = 00. = 0.75 rd, = 0.8 rd, c = 00 NSWERS. = 0, = 40, = 0, L 5., b L 68.4, c = 00. = 0.75 rd, = 0.8 rd, L p -.55 L.59 rd, L 68.8, b L 7.75, c = HPTER 6 Trigonometric Functions of ngles

38 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 50 SETION 6.4 The Lw of Sines 50 c=80.4 0* Figure 5 b 5* Emple Solving Tringle (S) Solve the tringle in Figure 5. Solution First, Since side c is known, to find side we use the reltion sin Similrl, to find b we use sin b sin c c sin sin sin c b c sin sin The mbiguous se 80.4 sin 0 sin sin 5 sin Lw of Sines Solve for Lw of Sines Solve for b LTERNTE EXMPLE Solve the tringle. Find the solution for the vlue of j, nd the solutions for the vlues of the sides nd b. 0* c=8. 5* NSWER 5,., 6.6 b In Emples nd unique tringle ws determined b the informtion given. This is lws true of se (S or S). ut in se (SS) there m be two tringles, one tringle, or no tringle with the given properties. For this reson, se is sometimes clled the mbiguous cse. To see wh this is so, we show in Figure 6 the possibilities when ngle nd sides nd b re given. In prt () no solution is possible, since side is too short to complete the tringle. In prt (b) the solution is right tringle. In prt (c) two solutions re possible, nd in prt (d) there is unique tringle with the given properties. We illustrte the possibilities of se in the following emples. b b b b SMPLE QUESTION Tet Question Wh does the book s tht SS is the mbiguous cse? nswer It is possible to form two different tringles given two sides nd n opposite ngle. Figure 6 The mbiguous cse () (b) (c) (d) 7 7 œ 45* Figure 7 Emple SS, the One-Solution se Solve tringle, where 45, 7, nd b 7. Solution We first sketch the tringle with the informtion we hve (see Figure 7). Our sketch is necessril tenttive, since we don t et know the other ngles. Nevertheless, we cn now see the possibilities. We first find. sin sin b sin b sin 7 7 sin 45 b b Lw of Sines Solve for sin LTERNTE EXMPLE Solve tringle, where j = 45, = 9, nd b = 9.0. Find the length of the side c, nd the vlues of ngles nd. NSWER =0, = 05, c = 7.9 IN-LSS MTERILS One cn demonstrte the mbiguous cse phsicll, using two rulers tped together to mke fied ngle, nd piece of string. See Figure 6 in the tet. HPTER 6 Trigonometric Functions of ngles 50

39 57050_06_ch06_p qd 07/04/008 06:00 PM Pge HPTER 6 Trigonometric Functions of ngles IN-LSS MTERILS It is tempting to do ll emples in degrees. Students should lso be eposed to the reltivel unfmilir rdin, if possible. We consider onl ngles smller thn 80, since no tringle cn contin n ngle of 80 or lrger. The supplement of n ngle u (where 0 u 80 ) is the ngle 80 u. Which ngles hve sin? From the preceding section we know tht there re two such ngles smller thn 80 (the re 0 nd 50 ). Which of these ngles is comptible with wht we know bout tringle? Since 45, we cnnot hve 50, becuse So 0, nd the remining ngle is Now we cn find side c. sin b sin c c b sin sin 7 sin 05 sin 0 7 sin 05.5 Lw of Sines Solve for c In Emple there were two possibilities for ngle, nd one of these ws not comptible with the rest of the informtion. In generl, if sin, we must check the ngle nd its supplement s possibilities, becuse n ngle smller thn 80 cn be in the tringle. To decide whether either possibilit works, we check to see whether the resulting sum of the ngles eceeds 80. It cn hppen, s in Figure 6(c), tht both possibilities re comptible with the given informtion. In tht cse, two different tringles re solutions to the problem. LTERNTE EXMPLE 4 Solve tringle if j = 45., = 67., nd b = 85.. NSWER (5, 8.7,.), (8, 6.7, 7.4) ln Oddie/PhotoEdit Surveing is method of lnd mesurement used for mpmking. Surveors use process clled tringultion in which network of thousnds of interlocking tringles is creted on the re to be mpped. The process is strted b mesuring the length of bseline between two surveing sttions. Then, using n instrument clled theodolite, the ngles between these two sttions nd third sttion re mesured. The Lw of Sines is then used to clculte the two other sides of the tringle formed b the three sttions. The clculted sides re used s bselines, nd the process is repeted over nd over to crete network of tringles. In this method, the onl distnce mesured is the initil bseline; ll (continued) Emple 4 SS, the Two-Solution se Solve tringle if 4., 86., nd b Solution From the given informtion we sketch the tringle shown in Figure 8. Note tht side m be drwn in two possible positions to complete the tringle. From the Lw of Sines Figure 8 sin b sin b= sin =86. =86. 4.* There re two possible ngles between 0 nd 80 such tht sin Using the SIN ke on clcultor (or INV SIN or RSIN ), we find tht one of these ngles is pproimtel The other is pproimtel We denote these two ngles b nd so tht 65.8 nd EXMPLES SS: For ech of the following, hve students drw tringles to tr to guess how mn solutions there re.. = 80, b = 00, = 0. = 80, b = 0, = 00. = 80, b =, = 0 4. = 0. rd, b = 50, = 40 NSWERS. No solution. One solution: = 80, 5.65, 94.5, = 00, b = 0, c 0.5. Two solutions: = 80, 8., 6.8, = 0, b =, c 5.; = 80, 96.8,., = 0, b =, c Two solutions: = 0. rd, 0.5 rd,.69 rd, = 40, b = 50, c 87.8; = 0. rd,.89 rd, 0.05 rd, = 40, b = 50, c HPTER 6 Trigonometric Functions of ngles

40 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 505 SETION 6.4 The Lw of Sines 505 other distnces re clculted from the Lw of Sines. This method is prcticl becuse it is much esier to mesure ngles thn distnces. heck bse Thus, two tringles stisf the given conditions: tringle nd tringle. Solve tringle : Thus c sin 86. sin sin sin 4. Find Lw of Sines Solve tringle : seline One of the most mbitious mpmking efforts of ll time ws the Gret Trigonometric Surve of Indi (see Problem 8, pge 55) which required severl epeditions nd took over centur to complete. The fmous epedition of 8, led b Sir George Everest, lsted 0 ers. Rnging over trecherous terrin nd encountering the dreded mlri-crring mosquitoes, this epedition reched the foothills of the Himls. lter epedition, using tringultion, clculted the height of the highest pek of the Himls to be 9,00 ft. The pek ws nmed in honor of Sir George Everest. Tod, using stellites, the height of Mt. Everest is estimted to be 9,08 ft. The ver close greement of these two estimtes shows the gret ccurc of the trigonometric method. Thus Tringles nd re shown in Figure 9. Figure c sin 86. sin sin sin 4. b= * =86. 4.* 65.8* c Å57.8 b=48.6 Find Lw of Sines.7* =86. 4.* 4.* c Å05. The net emple presents sitution for which no tringle is comptible with the given dt. 4* Figure 0 70 Emple 5 SS, the No-Solution se Solve tringle, where 4, 70, nd b. Solution To orgnize the given informtion, we sketch the digrm in Figure 0. Let s tr to find. We hve sin sin b sin b sin sin Lw of Sines Solve for sin Since the sine of n ngle is never greter thn, we conclude tht no tringle stisfies the conditions given in this problem. LTERNTE EXMPLE 5 Solve tringle, where j = 4, = 40, nd b = 89. Find the length of the side c nd the vlues of ngles nd. NSWER No solution HPTER 6 Trigonometric Functions of ngles 505

41 57050_06_ch06_p qd 07/04/008 06:00 PM Pge HPTER 6 Trigonometric Functions of ngles 6.4 Eercises 6 Use the Lw of Sines to find the indicted side or ngle u * 98.4* 4.6* 7 8.* * Solve the tringle using the Lw of Sines * 70* * *.4 80* 65 0* 6 Sketch ech tringle nd then solve the tringle using the Lw of Sines.. 50, 68, c 0., 0, c 50. 0, 65, b 0 4., 95, , 5, b , 00, c * 67* * 6.5 0* 8* 7 6 Use the Lw of Sines to solve for ll possible tringles tht stisf the given conditions. 7. 8, b 5, , c 40, , c 45, 5 0. b 45, c 4, 8. b 5, c 0, 5. 75, b 00, 0. 50, b 00, , b 80, , c 5, 9 6. b 7, c 8, For the tringle shown, find () D nd (b) D. 8. For the tringle shown, find the length D. 9. In tringle, 40, 5, nd b 0. () Show tht there re two tringles, nd,tht stisf these conditions. (b) Show tht the res of the tringles in prt () re proportionl to the sines of the ngles nd,tht is, 0. Show tht, given the three ngles,, of tringle nd one side, s, the re of the tringle is pplictions 0 5* re of ^ sin re of ^ sin re sin sin sin. Trcking Stellite The pth of stellite orbiting the erth cuses it to pss directl over two trcking sttions nd, which re 50 mi prt. When the stellite is on one 0 D 8 D 0* 5* 506 HPTER 6 Trigonometric Functions of ngles

42 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 507 SETION 6.4 The Lw of Sines 507 side of the two sttions, the ngles of elevtion t nd re mesured to be 87.0 nd 84., respectivel. () How fr is the stellite from sttion? (b) How high is the stellite bove the ground? 6. Rdio ntenn short-wve rdio ntenn is supported b two gu wires, 65 ft nd 80 ft long. Ech wire is ttched to the top of the ntenn nd nchored to the ground, t two nchor points on opposite sides of the ntenn. The shorter wire mkes n ngle of 67 with the ground. How fr prt re the nchor points? 7. Height of Tree tree on hillside csts shdow 5 ft down the hill. If the ngle of inclintion of the hillside is to the horizontl nd the ngle of elevtion of the sun is 5, find the height of the tree. 87.0* 84.*. Flight of Plne pilot is fling over stright highw. He determines the ngles of depression to two mileposts, 5 mi prt, to be nd 48, s shown in the figure. () Find the distnce of the plne from point. (b) Find the elevtion of the plne. 5 5 ft. Distnce cross River To find the distnce cross river, surveor chooses points nd, which re 00 ft prt on one side of the river (see the figure). She then chooses reference point on the opposite side of the river nd finds tht 8 nd 5. pproimte the distnce from to. * 5 mi 00 ft 8* 5* 48* 8. Length of Gu Wire communictions tower is locted t the top of steep hill, s shown. The ngle of inclintion of the hill is 58. gu wire is to be ttched to the top of the tower nd to the ground, 00 m downhill from the bse of the tower. The ngle in the figure is determined to be. Find the length of cble required for the gu wire lculting Distnce Observers t P nd Q re locted on the side of hill tht is inclined to the horizontl, s shown. The observer t P determines the ngle of elevtion to hot-ir blloon to be 6. t the sme instnt, the observer t Q mesures the ngle of elevtion to the blloon to be 7. If P is 60 m down the hill from Q, find the distnce from Q to the blloon. å 4. Distnce cross Lke Points nd re seprted b lke. To find the distnce between them, surveor loctes point on lnd such tht He lso mesures s ft nd s 57 ft. Find the distnce between nd. 5. The Lening Tower of Pis The bell tower of the cthedrl in Pis, Itl, lens 5.6 from the verticl. tourist stnds 05 m from its bse, with the tower lening directl towrd her. She mesures the ngle of elevtion to the top of the tower to be 9.. Find the length of the tower to the nerest meter. P * Q 60 m HPTER 6 Trigonometric Functions of ngles 507

43 57050_06_ch06_p qd 07/04/008 06:00 PM Pge HPTER 6 Trigonometric Functions of ngles 40. lculting n ngle wter tower 0 m tll is locted t the top of hill. From distnce of 0 m down the hill, it is observed tht the ngle formed between the top nd bse of the tower is 8. Find the ngle of inclintion of the hill. 8* 0 m hve mesure 60. () Show tht the rdius r of the common fce is given b r b b [Hint: Use the Lw of Sines together with the fct tht n ngle u nd its supplement 80 u hve the sme sine.] (b) Find the rdius of the common fce if the rdii of the bubbles re 4 cm nd cm. (c) Wht shpe does the common fce tke if the two bubbles hve equl rdii? 0 m b r D 4. Distnces to Venus The elongtion of plnet is the ngle formed b the plnet, erth, nd sun (see the figure). It is known tht the distnce from the sun to Venus is 0.7 U (see Eercise 65 in Section 6.). t certin time the elongtion of Venus is found to be 9.4. Find the possible distnces from the erth to Venus t tht time in stronomicl Units (U). Venus sun Discover Discussion 4. Number of Solutions in the mbiguous se We hve seen tht when using the Lw of Sines to solve tringle in the SS cse, there m be two, one, or no solution(s). Sketch tringles like those in Figure 6 to verif the criteri in the tble for the number of solutions if ou re given nd sides nd b. Venus å U erth 4. Sop ubbles When two bubbles cling together in midir, their common surfce is prt of sphere whose center D lies on the line pssing throught the centers of the bubbles (see the figure). lso, ngles nd D ech riterion Number of Solutions b b b sin b sin b sin 0 If 0 nd b 00, use these criteri to find the rnge of vlues of for which the tringle hs two solutions, one solution, or no solution. SUGGESTED TIME ND EMPHSIS clss. Essentil mteril. 6.5 The Lw of osines The Lw of Sines cnnot be used directl to solve tringles if we know two sides nd the ngle between them or if we know ll three sides (these re ses nd 4 of the preceding section). In these two cses, the Lw of osines pplies. POINTS TO STRESS. Using the Lw of osines to solve tringles.. Using Heron s Formul to find the re of tringle. 508 HPTER 6 Trigonometric Functions of ngles

44 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 509 SETION 6.5 The Lw of osines 509 Figure b c The Lw of osines In n tringle (see Figure ), we hve b c bc cos b c c cos c b b cos DRILL QUESTION The sides of tringle re = 5, b = 8, nd c =. Find the ngle opposite side. nswer 0.07 (b ç, b ß ) b (0, 0) c (c, 0) Figure Proof To prove the Lw of osines, plce tringle so tht is t the origin, s shown in Figure. The coordintes of the vertices nd re c, 0 nd (b cos, b sin ), respectivel. (You should check tht the coordintes of these points will be the sme if we drw ngle s n cute ngle.) Using the Distnce Formul, we get b cos c b sin 0 b cos bc cos c b sin b cos sin bc cos c b c bc cos ecuse sin cos This proves the first formul. The other two formuls re obtined in the sme w b plcing ech of the other vertices of the tringle t the origin nd repeting the preceding rgument. In words, the Lw of osines ss tht the squre of n side of tringle is equl to the sum of the squres of the other two sides, minus twice the product of those two sides times the cosine of the included ngle. If one of the ngles of tringle, s, is right ngle, then cos 0 nd the Lw of osines reduces to the Pthgoren Theorem, c b. Thus, the Pthgoren Theorem is specil cse of the Lw of osines. Figure 88 ft 8.4* ft Emple Length of Tunnel tunnel is to be built through mountin. To estimte the length of the tunnel, surveor mkes the mesurements shown in Figure. Use the surveor s dt to pproimte the length of the tunnel. Solution To pproimte the length c of the tunnel, we use the Lw of osines: c b b cos cos c Lw of osines Substitute Use clcultor Tke squre roots LTERNTE EXMPLE tunnel is to be built through mountin. To estimte the length of the tunnel, surveor mkes the mesurements shown in the figure below. Use the surveor s dt to pproimte the length of the tunnel. Thus, the tunnel will be pproimtel 47 ft long. IN-LSS MTERILS Hve the clss ppl the Lw of osines to the specil cse where is right ngle. Note tht the Lw of osines is ctull generliztion of the Pthgoren Theorem. 55 ft NSWER * 7 ft HPTER 6 Trigonometric Functions of ngles 509

45 57050_06_ch06_p qd 4/0/08 0: PM Pge HPTER 6 Trigonometric Functions of ngles LTERNTE EXMPLE The sides of tringle re = 76.56, b = 9.70, nd c = 6.4. Find the ngles of the tringle. b=9.70 =76.56 b=8 c= Figure 4 OS =5 Emple SSS, the Lw of osines The sides of tringle re 5, b 8, nd c (see Figure 4). Find the ngles of the tringle. Solution We first find. From the Lw of osines, we hve b c bc cos. Solving for cos, we get cos b c bc Using clcultor, we find tht 8. In the sme w the equtions c=6.4 NSWER = 9.7, =.87, = OR INV OS OR R OS cos c b c cos b c b give 9 nd. Of course, once two ngles re clculted, the third cn more esil be found from the fct tht the sum of the ngles of tringle is 80. However, it s good ide to clculte ll three ngles using the Lw of osines nd dd the three ngles s check on our computtions. LTERNTE EXMPLE Solve tringle, where j = 46., b = 8.6, nd c = 4.4 (see the figure below). Emple SS, the Lw of osines Solve tringle, where 46.5, b 0.5, nd c 8.0. Solution We cn find using the Lw of osines. b c bc cos cos b= * NSWER 0.5, 6., 97.4 c=4.4 b= * Å. 46.5* 5.* c=8.0 Figure 5 Thus, We lso use the Lw of osines to find nd, s in Emple. cos c b c cos b c b Using clcultor, we find tht 5. nd 98.. To summrize: 5., 98., nd.. (See Figure 5.) EXMPLE SS: = 5, b = 0, c = 0 NSWER.75,.,.9 We could hve used the Lw of Sines to find nd in Emple, since we knew ll three sides nd n ngle in the tringle. ut knowing the sine of n ngle does not uniquel specif the ngle, since n ngle u nd its supplement 80 u both hve the sme sine. Thus we would need to decide which of the two ngles is the correct choice. This mbiguit does not rise when we use the Lw of osines, becuse ever ngle between 0 nd 80 hs unique cosine. So using onl the Lw of osines is preferble in problems like Emple. EXMPLE SSS: = 4, b = 8, c = NSWER 6.,.9, HPTER 6 Trigonometric Functions of ngles

46 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 5 SETION 6.5 The Lw of osines 5 Nvigtion: Heding nd ering In nvigtion direction is often given s bering, tht is, s n cute ngle mesured from due north or due south. The bering N 0 E, for emple, indictes direction tht points 0 to the est of due north (see Figure 6). N N N N W 0 E W 60 E W 70 E W 50 E S N 0 E S N 60 W S S 70 W S S 50 E Figure 6 00 mi 0* Figure 7 40* 00 mi nother ngle with sine is ut this is clerl too lrge to be in. Emple 4 Nvigtion pilot sets out from n irport nd heds in the direction N 0 E, fling t 00 mi/h. fter one hour, he mkes course correction nd heds in the direction N 40 E. Hlf n hour fter tht, engine trouble forces him to mke n emergenc lnding. () Find the distnce between the irport nd his finl lnding point. (b) Find the bering from the irport to his finl lnding point. Solution () In one hour the plne trvels 00 mi, nd in hlf n hour it trvels 00 mi, so we cn plot the pilot s course s in Figure 7. When he mkes his course correction, he turns 0 to the right, so the ngle between the two legs of his trip is So b the Lw of osines we hve b # 00 # 00 cos 60 87, Thus, b The pilot lnds bout 96 mi from his strting point. (b) We first use the Lw of Sines to find. sin 00 sin sin 00 # sin Using the ke on clcultor, we find tht From Figure 7 we see tht the line from the irport to the finl lnding site points in the direction est of due north. Thus, the bering is bout N 6.6 E. SIN LTERNTE EXMPLE 4 To find the distnce cross smll lke, surveor hs tken the mesurements shown in the figure below. Find the distnce cross the lke (in miles) using this informtion..84 mi NSWER.44 miles 4.8*.66 mi HPTER 6 Trigonometric Functions of ngles 5

47 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 5 5 HPTER 6 Trigonometric Functions of ngles SMPLE QUESTION Tet Question Is it possible to find the re of tringle if the side lengths re known, but the ngles re not? nswer Yes Figure 8 c b The re of Tringle n interesting ppliction of the Lw of osines involves formul for finding the re of tringle from the lengths of its three sides (see Figure 8). Heron s Formul The re of tringle is given b ss s bs c where s b c is the semiperimeter of the tringle; tht is, s is hlf the perimeter. Proof We strt with the formul b sin from Section 6.. Thus 4 b sin 4 b cos Pthgoren identit Fctor Net, we write the epressions cos nd cos in terms of, b nd c. the Lw of osines we hve Similrl 4 b cos cos cos b c b cos b c b b b c b b c b b c b c b Lw of osines dd ommon denomintor Fctor c bc b cos b Difference of squres Substituting these epressions in the formul we obtined for gives To see tht the fctors in the lst two products re equl, note for emple tht b c b c s c c b c b c 4 b b b c b c c b c b ss cs bs c bc b b Heron s Formul now follows b tking the squre root of ech side. IN-LSS MTERILS Notice tht the bilit to esil find res of tringles cn be used to help find res of polgons, becuse the cn be decomposed into tringles. 5 HPTER 6 Trigonometric Functions of ngles

48 57050_06_ch06_p qd 07/04/008 06:00 PM Pge 5 SETION 6.5 The Lw of osines 5 5 ft 5 ft 80 ft Emple 5 re of Lot businessmn wishes to bu tringulr lot in bus downtown loction (see Figure 9). The lot frontges on the three djcent streets re 5, 80, nd 5 ft. Find the re of the lot. Solution The semiperimeter of the lot is s LTERNTE EXMPLE 5 businessmn wishes to bu tringulr lot in bus downtown loction (see the figure below). The lot frontges on the three djcent streets re 05, 80, nd 45 ft. Find the re of the lot. Heron s Formul the re is Figure ,45.6 Thus, the re is pproimtel 7,45 ft. 6.5 Eercises 45 ft 80 ft 8 Use the Lw of osines to determine the indicted side or ngle u... 9* * * 8 88* * Solve tringle *..0, b 4.0, 5. b 60, c 0, 70. 0, b 5, c 4. 0, b, c 6 5. b 5, c 6, , c 50, , b 65, , 6, Find the indicted side or ngle u. (Use either the Lw of Sines or the Lw of osines, s pproprite.) * 85*.. 0* 50 00* * NSWER,70 ft 05 ft HPTER 6 Trigonometric Functions of ngles 5

49 57050_06_ch06_p qd 07/04/008 06:0 PM Pge HPTER 6 Trigonometric Functions of ngles * Find the re of the tringle whose sides hve the given lengths. 7. 9, b, c 5 8., b, c 9. 7, b 8, c 9 0., b 00, c 0 4 Find the re of the shded figure, correct to two decimls * * * Three circles of rdii 4, 5, nd 6 cm re mutull tngent. Find the shded re enclosed between the circles * 0 5* 60* Prove tht in tringle These re clled the Projection Lws. [Hint: To get the first eqution, dd the second nd third equtions in the Lw of osines nd solve for.] pplictions 7. Surveing To find the distnce cross smll lke, surveor hs tken the mesurements shown. Find the distnce cross the lke using this informtion..8 mi b cos c cos b c cos cos c cos b cos 40.* 8. Geometr prllelogrm hs sides of lengths nd 5, nd one ngle is 50. Find the lengths of the digonls. 9. lculting Distnce Two stright rods diverge t n ngle of 65. Two crs leve the intersection t :00 P.M., one trveling t 50 mi/h nd the other t 0 mi/h. How fr prt re the crs t :0 P.M.? 40. lculting Distnce cr trvels long stright rod, heding est for h, then trveling for 0 min on nother rod tht leds northest. If the cr hs mintined constnt speed of 40 mi/h, how fr is it from its strting position? 4. Ded Reckoning pilot flies in stright pth for h 0 min. She then mkes course correction, heding 0 to the right of her originl course, nd flies h in the new direction. If she mintins constnt speed of 65 mi/h, how fr is she from her strting position? 4. Nvigtion Two bots leve the sme port t the sme time. One trvels t speed of 0 mi/h in the direction N 50 E nd the other trvels t speed of 6 mi/h in direction S 70 E (see the figure). How fr prt re the two bots fter one hour? N.56 mi N 50 E W 50 S 70 E S 70 E 54 HPTER 6 Trigonometric Functions of ngles

50 57050_06_ch06_p qd 07/04/008 06:0 PM Pge 55 SETION 6.5 The Lw of osines Nvigtion fishermn leves his home port nd heds in the direction N 70 W. He trvels 0 mi nd reches Egg Islnd. The net d he sils N 0 E for 50 mi, reching Forrest Islnd. () Find the distnce between the fishermn s home port nd Forrest Islnd. (b) Find the bering from Forrest Islnd bck to his home port. Forrest Islnd 47. Fling Kites bo is fling two kites t the sme time. He hs 80 ft of line out to one kite nd 40 ft to the other. He estimtes the ngle between the two lines to be 0. pproimte the distnce between the kites. 80 ft 0 40 ft 0 50 mi Egg Islnd 0 mi 70 Home port 48. Securing Tower 5-ft tower is locted on the side of mountin tht is inclined to the horizontl. gu wire is to be ttched to the top of the tower nd nchored t point 55 ft downhill from the bse of the tower. Find the shortest length of wire needed. 44. Nvigtion irport is 00 mi from irport t bering N 50 E (see the figure). pilot wishing to fl from to mistkenl flies due est t 00 mi/h for 0 minutes, when he notices his error. () How fr is the pilot from his destintion t the time he notices the error? (b) Wht bering should he hed his plne in order to rrive t irport? irport 55 ft 5 ft mi 49. ble r steep mountin is inclined 74 to the horizontl nd rises 400 ft bove the surrounding plin. cble cr is to be instlled from point 800 ft from the bse to the top of the mountin, s shown. Find the shortest length of cble needed. irport 45. Tringulr Field tringulr field hs sides of lengths, 6, nd 44 d. Find the lrgest ngle. 46. Towing rge Two tugbots tht re 0 ft prt pull brge, s shown. If the length of one cble is ft nd the length of the other is 0 ft, find the ngle formed b the two cbles. 800 ft 74* 400 ft ft 0 ft 0 ft 50. N Tower The N Tower in Toronto, nd, is the tllest free-stnding structure in the world. womn on the observtion deck, 50 ft bove the ground, wnts to determine the distnce between two lndmrks on the ground below. She observes tht the ngle formed b the lines of sight to these two lndmrks is 4. She lso observes tht the ngle between the verticl nd the line of sight to one of the HPTER 6 Trigonometric Functions of ngles 55

51 57050_06_ch06_p qd 07/04/008 06:0 PM Pge HPTER 6 Trigonometric Functions of ngles lndmrks is 6 nd to the other lndmrk is 54. Find the distnce between the two lndmrks. 5. Lnd Vlue Lnd in downtown olumbi is vlued t $0 squre foot. Wht is the vlue of tringulr lot with sides of lengths, 48, nd 90 ft? Discover Discussion 5. Solving for the ngles in Tringle The prgrph tht follows the solution of Emple on pge 50 eplins n lterntive method for finding nd, using the Lw of Sines. Use this method to solve the tringle in the emple, finding first nd then. Eplin how ou chose the pproprite vlue for the mesure of. Which method do ou prefer for solving n SS tringle problem, the one eplined in Emple or the one ou used in this eercise? 6 Review oncept heck. () Eplin the difference between positive ngle nd negtive ngle. (b) How is n ngle of mesure degree formed? (c) How is n ngle of mesure rdin formed? (d) How is the rdin mesure of n ngle u defined? (e) How do ou convert from degrees to rdins? (f) How do ou convert from rdins to degrees?. () When is n ngle in stndrd position? (b) When re two ngles coterminl?. () Wht is the length s of n rc of circle with rdius r tht subtends centrl ngle of u rdins? (b) Wht is the re of sector of circle with rdius r nd centrl ngle u rdins? 4. If u is n cute ngle in right tringle, define the si trigonometric rtios in terms of the djcent nd opposite sides nd the hpotenuse. 5. Wht does it men to solve tringle? 6. If u is n ngle in stndrd position, P, is point on the terminl side, nd r is the distnce from the origin to P, write epressions for the si trigonometric functions of u. 7. Which trigonometric functions re positive in qudrnts I, II, III, nd IV? 8. If u is n ngle in stndrd position, wht is its reference ngle u? 9. () Stte the reciprocl identities. (b) Stte the Pthgoren identities. 0. () Wht is the re of tringle with sides of length nd b nd with included ngle u? (b) Wht is the re of tringle with sides of length, b, nd c?. () Stte the Lw of Sines. (b) Stte the Lw of osines.. Eplin the mbiguous cse in the Lw of Sines. Eercises Find the rdin mesure tht corresponds to the given degree mesure.. () 60 (b) 0 (c) 5 (d) 90. () 4 (b) 0 (c) 750 (d) 5 4 Find the degree mesure tht corresponds to the given rdin mesure. 5p p 9p. () (b) (c) (d) HPTER 6 Trigonometric Functions of ngles

52 57050_06_ch06_p qd 07/04/008 06:0 PM Pge 57 HPTER 6 Review 57 5 p 4. () 8 (b) (c) (d) 6 5. Find the length of n rc of circle of rdius 8 m if the rc subtends centrl ngle of rd. 6. Find the mesure of centrl ngle u in circle of rdius 5 ft if the ngle is subtended b n rc of length 7 ft. 7. circulr rc of length 00 ft subtends centrl ngle of 70. Find the rdius of the circle. 8. How mn revolutions will cr wheel of dimeter 8 in. mke over period of hlf n hour if the cr is trveling t 60 mi/h? 9. New York nd Los ngeles re 450 mi prt. Find the ngle tht the rc between these two cities subtends t the center of the erth. (The rdius of the erth is 960 mi.) 0. Find the re of sector with centrl ngle rd in circle of rdius 5 m.. Find the re of sector with centrl ngle 5 in circle of rdius 00 ft.. sector in circle of rdius 5 ft hs n re of 5 ft. Find the centrl ngle of the sector.. potter s wheel with rdius 8 in. spins t 50 rpm. Find the ngulr nd liner speeds of point on the rim of the wheel. p 5 (b) Find the ngulr speed of the wheels. (c) How fst (in mi/h) is the cr trveling? 5 6 Find the vlues of the si trigonometric rtios of u Find the sides lbeled nd, correct to two deciml plces * 7 5 Ger 5 Rtio st 4. nd.0 rd.6 4th 0.9 5th 0.7 5* in. 0* 0* 0* 4 4. In n utomobile trnsmission ger rtio g is the rtio ngulr speed of engine g ngulr speed of wheels Solve the tringle... The ngulr speed of the engine is shown on the tchometer (in rpm). certin sports cr hs wheels with rdius in. Its ger rtios re shown in the following tble. Suppose the cr is in fourth ger nd the tchometer reds 500 rpm. () Find the ngulr speed of the engine. 0* 60* 0 HPTER 6 Trigonometric Functions of ngles 57

53 57050_06_ch06_p qd 07/04/008 06:0 PM Pge HPTER 6 Trigonometric Functions of ngles. Epress the lengths nd b in the figure in terms of the trigonometric rtios of u. b 0 8. pilot mesures the ngles of depression to two ships to be 40 nd 5 (see the figure). If the pilot is fling t n elevtion of 5,000 ft, find the distnce between the two ships. 40* 5* 4. The highest free-stnding tower in the world is the N Tower in Toronto, nd. From distnce of km from its bse, the ngle of elevtion to the top of the tower is 8.8. Find the height of the tower. 5. Find the perimeter of regulr hegon tht is inscribed in circle of rdius 8 m. 6. The pistons in cr engine move up nd down repetedl to turn the crnkshft, s shown. Find the height of the point P bove the center O of the crnkshft in terms of the ngle u. P O 7. s viewed from the erth, the ngle subtended b the full moon is Use this informtion nd the fct tht the distnce from the erth to the moon is 6,900 mi to find the rdius of the moon. Q 8 in Find the ect vlue. 9. sin 5 0. csc 9p 4. tn 5. cos 5p 6 p. cot 4. sin 405 b 5. cos csc 8p cot sec p sec p 6 tn p 4 4. Find the vlues of the si trigonometric rtios of the ngle u in stndrd position if the point 5, is on the terminl side of u. 4. Find sin u if u is in stndrd position nd its terminl side intersects the circle of rdius centered t the origin t the point /,. 4. Find the cute ngle tht is formed b the line 0 nd the -is. 44. Find the si trigonometric rtios of the ngle u in stndrd position if its terminl side is in qudrnt III nd is prllel to the line Write the first epression in terms of the second, for u in the given qudrnt. 45. tn u, cos u; u in qudrnt II 46. sec u, sin u; u in qudrnt III 47. tn u, sin u; u in n qudrnt 48. csc u cos u, sin u; u in n qudrnt 58 HPTER 6 Trigonometric Functions of ngles

54 57050_06_ch06_p qd 07/04/008 06:0 PM Pge 59 HPTER 6 Review Find the vlues of the si trigonometric functions of u from the informtion given tn u 7/, sec u sec u 4 40, csc u 9 5. sin u 5, cos u 0 5. sec u 5, tn u 0 5. If tn u for u in qudrnt II, find sin u cos u. 54. If sin u for u in qudrnt I, find tn u sec u. 55. If tn u, find sin u cos u. 56. If cos u /nd p/ u p, find sin u Find the side lbeled * 0* 45* 05* 64. From point on the ground, the ngle of elevtion to the top of tll building is 4.. From point, which is 600 ft closer to the building, the ngle of elevtion is mesured to be 0.. Find the height of the building. 65. Find the distnce between points nd on opposite sides of lke from the informtion shown. 4.* 600 ft. mi 4* 0.* * * * Two ships leve port t the sme time. One trvels t 0 mi/h in direction N E, nd the other trvels t 8 mi/h in direction S 4 E (see the figure). How fr prt re the two ships fter h? 6 8 0* 5.6 mi 66. bot is cruising the ocen off stright shoreline. Points nd re 0 mi prt on the shore, s shown. It is found tht 4. nd Find the shortest distnce from the bot to the shore. Shoreline 4.* 0 mi N * N E 68.9* W E S 4* S 4 E 67. Find the re of tringle with sides of length 8 nd 4 nd included ngle Find the re of tringle with sides of length 5, 6, nd 8. HPTER 6 Trigonometric Functions of ngles 59

55 57050_06_ch06_p qd 07/04/008 06:0 PM Pge HPTER 6 Trigonometric Functions of ngles 6 Test. Find the rdin mesures tht correspond to the degree mesures 0 nd 5. 4p. Find the degree mesures tht correspond to the rdin mesures nd... The rotor bldes of helicopter re 6 ft long nd re rotting t 0 rpm. () Find the ngulr speed of the rotor. (b) Find the liner speed of point on the tip of blde. 4. Find the ect vlue of ech of the following. () sin 405 (b) tn 50 (c) sec 5p (d) csc 5p 5. Find tn u sin u for the ngle u shown. 6. Epress the lengths nd b shown in the figure in terms of u. 4 b 7. If cos u nd u is in qudrnt III, find tn u cot u csc u If sin u 5 nd tn u, find sec u. 9. Epress tn u in terms of sec u for u in qudrnt II. 0. The bse of the ldder in the figure is 6 ft from the building, nd the ngle formed b the ldder nd the ground is 7. How high up the building does the ldder touch? 7* 6 ft 50 HPTER 6 Trigonometric Functions of ngles

56 57050_06_ch06_p qd 07/04/008 06:0 PM Pge 5 HPTER 6 Test 5 4 Find the side lbeled Refer to the figure below. () Find the re of the shded region. (b) Find the perimeter of the shded region m 6. Refer to the figure below. () Find the ngle opposite the longest side. (b) Find the re of the tringle Two wires tether blloon to the ground, s shown. How high is the blloon bove the ground? h 75* 85* 00 ft HPTER 6 Trigonometric Functions of ngles 5

57 57050_06_ch06_p qd 07/04/008 06:0 PM Pge 5 Focus on Modeling Surveing How cn we mesure the height of mountin, or the distnce cross lke? Obviousl it m be difficult, inconvenient, or impossible to mesure these distnces directl (tht is, using tpe mesure or rd stick). On the other hnd, it is es to mesure ngles to distnt objects. Tht s where trigonometr comes in the trigonometric rtios relte ngles to distnces, so the cn be used to clculte distnces from the mesured ngles. In this Focus we emine how trigonometr is used to mp town. Modern mp mking methods use stellites nd the Globl Positioning Sstem, but mthemtics remins t the core of the process. Mpping Town student wnts to drw mp of his hometown. To construct n ccurte mp (or scle model), he needs to find distnces between vrious lndmrks in the town. The student mkes the mesurements shown in Figure. Note tht onl one distnce is mesured, between it Hll nd the first bridge. ll other mesurements re ngles. Figure The distnces between other lndmrks cn now be found using the Lw of Sines. Foremple, the distnce from the bnk to the first bridge is clculted b ppling the Lw of Sines to the tringle with vertices t it Hll, the bnk, nd the first bridge: 0.86 sin 50 sin sin 50 sin 0. mi Lw of Sines Solve for lcultor 5 5 HPTER 6 Trigonometric Functions of ngles

58 57050_06_ch06_p qd 07/04/008 06:0 PM Pge 5 Surveing 5 So the distnce between the bnk nd the first bridge is. mi. The distnce we just found cn now be used to find other distnces. For instnce, we find the distnce between the bnk nd the cliff s follows:. sin 64 sin 50. sin 64 sin mi Lw of Sines Solve for lcultor ontinuing in this fshion, we cn clculte ll the distnces between the lndmrks shown in the rough sketch in Figure. We cn use this informtion to drw the mp shown in Figure. it Hll nk hurch N Fire Hll School 0 /4 / /4 mile Figure To mke topogrphic mp, we need to mesure elevtion. This concept is eplored in Problems 4 6. Problems. ompleting the Mp Find the distnce between the church nd it Hll.. ompleting the Mp Find the distnce between the fire hll nd the school. (You will need to find other distnces first.) HPTER 6 Trigonometric Functions of ngles 5

59 57050_06_ch06_p qd 07/04/008 06:0 PM Pge Focus on Modeling. Determining Distnce surveor on one side of river wishes to find the distnce between points nd on the opposite side of the river. On her side, she chooses points nd D, which re 0 m prt, nd mesures the ngles shown in the figure below. Find the distnce between nd. 50* 40* 0* 0 m 45* D.* 69.4* 00 m 5.6* 4. Height of liff To mesure the height of n inccessible cliff on the opposite side of river, surveor mkes the mesurements shown in the figure t the left. Find the height of the cliff. 5. Height of Mountin To clculte the height h of mountin, ngle, b,nd distnce d re mesured, s shown in the figure below. () Show tht h d cot cot b (b) Show tht sin sin b h d sinb (c) Use the formuls from prts () nd (b) to find the height of mountin if 5, b 9, nd d 800 ft. Do ou get the sme nswer from ech formul? h å 40 ft d 68* 4* 9* 6. Determining Distnce surveor hs determined tht mountin is 40 ft high. From the top of the mountin he mesures the ngles of depression to two lndmrks t the bse of the mountin, nd finds them to be 4 nd 9. (Observe tht these re the sme s the ngles of elevtion from the lndmrks s shown in the figure t the left.) The ngle between the lines of sight to the lndmrks is 68. lculte the distnce between the two lndmrks. 54 HPTER 6 Trigonometric Functions of ngles

60 57050_06_ch06_p qd 07/04/008 06:0 PM Pge 55 Surveing Surveing uilding Lots surveor surves two djcent lots nd mkes the following rough sketch showing his mesurements. lculte ll the distnces shown in the figure nd use our result to drw n ccurte mp of the two lots. 8. Gret Surve of Indi The Gret Trigonometric Surve of Indi ws one of the most mssive mpping projects ever undertken (see the mrgin note on pge 504). Do some reserch t our librr or on the Internet to lern more bout the Surve, nd write report on our findings. ritish Librr HPTER 6 Trigonometric Functions of ngles 55