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# Analytic Trigonometry MATH 144

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This 58 page Class Notes was uploaded by Breanne Schaden PhD on Saturday October 3, 2015. The Class Notes belongs to MATH 144 at Boise State University taught by Tommy Conklin in Fall. Since its upload, it has received 21 views. For similar materials see /class/217998/math-144-boise-state-university in Mathematics (M) at Boise State University.

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Date Created: 10/03/15

from A exander J Ham Basic Calculus From Archimedes to Newton to Its Role in Science Sp ngenVer ag Newark 1998 Chapter 1 The Greeks Measure the Universe themtu use 11 The Pythagoreans Measure Length The Pymagmean 2 ve atesthe myng a We sees We are Theuvem assens mm m 3 quotgm mang e see Hgmew DJHE equamy a2 51 mm mh mun mm m nThe Pymagmean Theuvemwuvks m reverse a su Neme y Wm sume mang e me eng hs uHhe swdes saus1y the yexanensmp 2 2 2 a 5 we smes WW I 1 n appearstha mpe stve chevs usee msvm m eany censuucnen TakeJuv examp e a rape ma has a engm e112 unns nguve Wi men smce me engths m the sees sausvyme ve auunshwp 4 31 169 5 me Pythaguvean Theuvem assensthat e ang e am 5 3 quotgm ang e The censmcnen e we peypendmu avwaHs can begm 11111111 Assume m1 1 1s sums ummusngm We W111 ssymm shagm ssgmsm 1smeaswabem 11 um1s1sngm m unmsu 11 15 K Wham m and n 1 4 186 m WHD E uv admna unns u Fm mstance 11072 mm 1 15 the mch than the segments uHengths 267 T mches 14 1551 951 6312933 29 7 29 3657 7 3657 pmtuved dwec y mm 15 measmame and um 11 has 12mm F1gu7214 1sms vesuhmg hypmenuse h msssmsmsv We W111 522 suvpnse7 m1 1 157mm ms and 2st2 Nuwss1 x m andy Assume m1 1 1s Thenthe 12mm u h 15 sumsnanmm Mpusmve 171123275 EyPythaguvas smeuvem 2 2 2 2 39 2 aydsmgasnummsmsw n Wu ubserve m1 x2 2y2 Factunng 2 ant m x as manynmss as pussm e gwes x 2 W x add 1n m same Way y 2 y2 W101 y add may 17151371123 x WEYE equ H 2 2 4322 43 56a2 andx43 Eyesubsmuuunwege Zn12 22 y2 Baby 172 h n172 ms Mes m expunenuatmg 22ax12 22b1y12 Notice that 2a 2 2b1 since 2a is even and 2b1 is odd Therefore either 2a gt 2b10r 2a lt 2b1 1 Assume that 2a gt 2b1 So ZZHZMUXIZ y12 It follows that y2 is even Since y1 is odd it has the form y1 2k1 for some integer k So y2 4k2 4k1 4k2 k1 which is the sum of an even integer and 1 But this means that y2 is odd Thus y1 is both even and odd This is impossible 2 Assume next that 2a lt 2b1 This implies that Zam a x12 ylz By the argument in Step 1 it is now x2 that is both even and odd This too is impossible Reflect over what has been done The discussion started with the supposition that h is measurable in u Then the argument moved in a strictly logical way to impossible consequences The inescapable conclusion is that h cannot be measurable in u The preceding proof was not taken from an old Greektext It is however from the point of view ofthe precision ofthe logic and the flow of its argument very much in the spirit of Greek mathematics The point is this While the hypotenuse h is a perfectly valid geometric construction it cannot be measured with the numbers ofthe Pythagoreans In particular their numerical considerations ran into limitations that geometrical ones did not The suggestion presents itself that this was an important reason why Greek geometry and trigonometry flourished in a way that numerical analyses and algebra did not Pythagoras and his followers had founded a Greek colony in today39s southern Italy in the 6th century BC They formed a cult based on the philosophical principle that mathematics is the underlying explanation of all things from the relationship between musical notes to the movements of the planets ofthe solar system Indeed they held that all reality finds its ultimate explanation in numbers and mathematics As we have just seen however the numbers ofthe Pythagoreans were unable to come to grips with the very basic matter of measuring length It is believed that when the Pythagoreans realized this a crisis ensued within their ranks that contributed to their downfall2 In any case it seems somewhat ironic that the Pythagorean theorem an assertion about the lengths of certain segments derives its name from a school or person that did not possess a number system with which length could always be measured Today we can put it this way The Pythagorean number system did not have enough numbers It consisted only ofthe m numbers of the form ie the rational numbers and it did not include other real numbers In fact the preceding n demonstration of the nonmeasurability of h shows that J5 is a number that is not rational Take m n 1 It is in other words irrational On the other hand we know that 514142 14i1L4L 1 10 100 1000 10000 and hence that 2 can be constructed in terms of a decimal expansion This infinite process gives rise to the number line Fix a unit of length and take a straight line that runs infinitely in both directions Fix a point and label it 0 Mark off a point one unit to the right of 0 and label it 1 Continue in this way to get 2 3 Do a similar thing on the left of 0 but use 1 2 to label the points Continue in this way with tenths of units hundredths of units and so on to achieve the following relationship Every point on the line corresponds to a real number in other words a number given by a decimal expansion and every such number corresponds to a point on the line See the illustration for J5 14142in Figure 15 v37 1 42 4 v H I 2 is 4 A figo q o Hw A F gt oi oi 4 7 4 o x 2 x 5 mm 15 rm m n q uccm mu m 15m camuyy Tm means mum m m have me cuuvdma er a su caHed Caneswanrp ane Thevemve mvuughum 12 The Measure of Angles The ssue s m we Aha amuum See Ham 1 E mm y u V 4 MW 7 See mm 5 swmh degvees n w v m 2mg an m H mm numbev u degvees n s knuwn mm m ang es Manymang e add up m mm mm 8mm 5 s The vaman measuve mum ang e 5 5 mm m be We vanu We wwwme 5 An mmedwate questmn 0m anses r be a meamngM mm H mm m mm 2 x S mm 5 mm anmhevcwde Let R be Msvadmsand S m mm DHHE 3mm 5 mm See nguve 1 a s i 7 7 Tm sthe mmquot mm must be addvessed Cunswdevwha uHuws m be a muugm expenmemr vathenhan sumemmg 0m 5 named um m pvacuce Let n be a pusmve may 1 can be say quEIEI uvAEI EIEIEI and panmunthewedge mm 1 equamweces Cunnemthe quota depend an m 1 0m yuu have pmked The case 1 4 5 Shawn m mum m Obsevve ma each mum smaHev mangu avwedges muse Wm swdes u ength r 5 sman each mum Flow 1 y R 391 L1 mm r mn n R y same Way Ength 1 add up m appmwmate y 5 Sn the numbev m1 5 nEaHy equaHu 5 and heveluve 5 quotEa r uh mi y mi 5 y 2mm m The symbu hm 5 shuvHuv hm aha ye eysm the exesmghh pmcess The symbu w vEvasEMs hhvhuy Mm 1 M 31321 aye We get ma he 11m meansthat h by bemg aken aygeh 5 bemg pushed m hwy Pvuceedmg m exac y the same Way Wuh the Smce dquot D e mahy hwe seethat R s 3 mm u Eng hs n sthevemve a ma numbev r R hm n m m 1 11 ng121 M Cunsmevm ang e Em and ubserve 0m Ms vaman measuve s cucumveyence ng121 12 Usmg m cwde We geHhaHhe Yemen measuve mam 5 2mm m Smce m Yemen 2 Vuvmu a 3 mm cwcumlevence u a cwde uhadms r mm u 111 Tamem Smce in m m 1 mm andcemev 7r C and dwde ms uppav ssmmms mm mm aqua pans as Shawn m F1gu121 13 Each uHhe ang1ss1s equaHu g LaAandB M h m mm 1suscs1ssmang1s ACAB 1ssqm1a1sva1 whyv hsvsmvs AB has 12mm Chuuseme pmm P unms suds such m1 012 segment CP msec sme ang e LACE and dvaw 012 mm A B m 012 211212 s1 P Sweethe b1sectquP and Ir ms 1mm A B ave pevpendmu av by a bash pvupeny ums momma ang1ss s1 A and 8 ave bum squs1 m g 11 uHquthaHhemang es ACA P and ACB P ave 2mm Put A P PB 7 Sweethethvee ang esu ACA B ave equaMHuHqu m1 3643 15 eqm a eva Thwava CA has ength 2pm bythe Pythaguvean heuvemapphed 1n 1 2 ACPA 1HuHuw51ha1 1 22 Thus 312 1 and mm Sn A B has1engm 22 Smce m 73 7339 7r 3 mscussmnrsee s1su Exevmse 15 mm F1gu121 131mm 1113 lt length arcAB ltA B Nuw muhmwmmugh by 1 get Th1s cunespundsm 3 lt7rlt3 47 mvu ved 11 uses many mms1nsng1ss ms ead Mammaw shuwmm Th1s gwes 31408 lt7r lt31429 The tuned expansmn beams yr es abhsh tha 7r 15 an wvatmna numbev 14159 Mums uuandh1s1s mmnu 1 A A V W l g lt N sx mm m 13 Eratosthenes Measures the Earth um um measuvens Mnthe wen m n m vaemcd p mmv usmg a mm M and de evmmedme ang e s m mum 145m b275 Evamsthenesknewa suma syene as b He was mugmy due sum MNexandHa at a dws ance u abum sun m es rng 115W 575 and 500 EZSEIEI The ve waseasy Onthe une hand 15 314 E 71 L 2 180 360 013rad ans Em un he mm mm 55 Thevemva r r r E 3850m1135 s 013 c u mhvnn Mumquot w 395m m es V law A 17 14 Right Triangles Gveek vendmun u nutatmnrm the Gveeks Cunsmeyme ngm mang e m Wm 15 meg gwen ang e 5 de ne m sme uHuwmg vauus u 2quotng 5mg 7 a cosy39 bhb cusmef and angenmu be We and Lang 51 c055 h h j u n FrymulJv39 Ir Lqun n as anew 1 1 sway SeeFME Thevemve 5 TM 5 eas y seen 5 n duesn ma evwhmh Hgmmang e e used m cumpme Them ave many manmwesmahe ate smS 2055 and Lang Fmexampb byPythaguvas smeuvem azbz 2 2 b SD L h h and 01212th sm2 5eos n s eusmmavy m We sm2 5 ms ead BMW 255 ewmenqsm 5 Wewm nuwcumpme smS c055 and Lana meme standavdva ues e1 5 Censmeyme ngmmang e m Hgmew 9 7r Smee m s susc es me acme ang es ave each 45 or 7 vamans Thevemve 7r 1 7r 1 7r sm eos and tan 4 72 4 2 4 Fryuml 13 F ygxml My a 2 h J3 See Ham 1 2n R ecang mum ang es u an equ amvamang e ave each 2qua m Next km 225 h 3 5U39uvvadans weseetha 7r J3 7r 1 Ir J3 sm cos and tan 3 2 3 2 3 7r Smce each mum smaHev ang es aHhe mp s E n uHqumat Figure 1 211 T Mm 221 Observe 0m 5m UcosUand tanU u m make sense mm swmp e veasun manheve s nu quotgm mang e Wm an ang e umm muwa en y yamans Suppusemuwevema 5 svevy smaH A uukauhe Hgmmang ewnh hypmenu521 m Ham 1 21 shuwsthat sxn5a sa suvevysmaH Sn w 5 5 ne u then smS sa su c usem 22ch Take 5 sequenuaHysmaHevand maHev s Tm vma es me hypmenuse duwnwavds and pushes a m 22ch Thevemve smS a s pushed m 22ch We summanze m bywmmg 11m 5m 5 0 M 5m 0 0 Tm me nous un 7r me ang e a m ngm 1 21 Asm hypmenuse vma es xs pushedto E and m m pmcess 7515 pushedto 1 m hm nmauun hmsm 1 lxmcos 0 suwe 52 c0501 and 505 training n sum mm mm 3 u o u 1 2 3 y y 72 a 1 L 39 a g 1 a radJans Smce s gt a MuHquma 5gtsm5 Thevemve m mu HM m and w 5 smaH 5 anew F um s 1 22 Take 3142 Tab 213 smS and a but un y w 5 5 taken m vamans Sn when cumpuung smS make suve 0m yum ca cu amv s m yaman mm and mm degvee made 15 Aristarchus Sizes Up the Universe centered or heliocentric that the Sun is fixed and that the Earth revolves around the Sun and rotates about its own axis in the process What will interest us is Aristarchus39s use of quotcosmicquot trigonometry in his treatise On the Magnitudes and Distances of the Sun and Moon His analysis rests on the following hypotheses and observations7 AD The Moon receives its light from the Sun B0 The Moon revolves in a circle about the Earth with the Earth at the center C0 When an observer on Earth looks out at a precise half moon the angle LEAS in Figure 123 is 90 At that moment the angle AMES can be measured to equal 87 D0 At the instant of a total eclipse of the Sun the Moon and the Sun as viewed from the Earth subtend the same angle and this angle can be measured to be 2 Refer to Figure 124 E0 During a lunar eclipse the shadow indicated in Figure 125 has width 4 times the radius rM of the Moon This was based on how long the Moon was observed to be in the Earth39s shadow What did Aristarchus deduce from these observations Let rE the radius of the Earth rM the radius of the Moon rs the radius of the Sun DM the distance from the Earth to the Moon DS the distance from the Earth to the Sun M39an m mumw mm us rim 2 125 r E v H 1 me czii 7r D 7r Ham125namesmyecnynumubswanun C Obsevve hat 3 7 Thevemve i sm3 smi Smce 0 DS 60 3 If ram 60 60 60 esumate we ake 3 14 such esumates mum 7r 1 D If 1 60 60 20 DS 60 20 60 m thwsway Anstavchus amved aHhe 25mm D M 20 5 P212 nexHu ubsewatmn D and F gu12124 Thws s he snuatmn uHhe su av echpse F gu121 27 s an abuvatmn u ngu121 A m n ea E aw u Ham 1 24 395 D5 Muun Eyswm avmang es i 7 and hevemve 3914 DM r 7239 r 7239 7239 Observe also that S smlquot Since 1 radians i sm m Taking 7239 314 it follows that M 180 DM 180 180 7r 1 rM 72 1 0017 Note that is very close to 0017 So to make things simple we Will take Sln 6 DM 180 60 So Aristarchus obtained the approximation D 60 FM From observation E and Figure 125 he obtained Figure 128 This figure shows a light ray that is tangent to both the Sun and the Earth The radii ofthe Sun and the Earth drawn into the figure are both perpendicular to this light ray The extension of the radius ofthe Moon indicated in the figure is perpendicular to this light ray as well Because the two triangles with the quotdottedquot bases are similar it follows that rE ZVM DMI rSrE Ds Recalling that DM rM DS rS we get rE 27M rM rs rE rs I After crossmultiplying rSrE 2737M erS erE So 7er erE 3rSrM Dividing this last equation through by r r r rer gives us E E 3 Since S 20 we have rs 20rM By substitution we get r rM 327 5 rE 20rE r5221r5 rM 202quotM 202quotM 207M Therefore r E 2 and 7M 21 7 7M Ergh r r r 7 Since S S i 20 7 it follows that rE rM rE 20 rs 77E With Eratosthenes39s value of VE 3850 miles Aristarchus got the approximations rM 1350 miles and rs 27000 miles D D Smee M 60 he 2511ma2d 1ha1 DM 609 80000 m11es she1hsemhg1h1sva1ue1hm 5 20 he gm 0 DM D 1600000 mxles rm 1 1 9 1 s11gh11ye111eyeh1ahsweys Fuvexamp121nstead e1 he EIb amEd ahe1hs1eaeu1 395 20 20 60 he had lt39A Huw2v211h2 essehee u1h1saha1ys1shasheeh 1e1a1hee 60 DM 45 Tah1e1 1a1h21 1111 H 1 1he vadms unhe Sun Eanh ave u11hy1ae1ms u31Ear1d 5Dvesp2m1ve1y ml 14 1 ventuusarrhm 21mm vs 1 93 x w uu1es1 1 1 i 7 7 7 1 111 W11 Wmquot we 1 1m smasher w 11 maalivenh 11m 111111 1 gum mm A1u1m mm as 1391Luhmllmcnmndr whey 1m mm serge mmhmm mumm 11111111 n 1h 11 and M The mm mm mncrmw 11 xlrm11mimsmessgtnc Hmu nnmguwmnud hguhehm 1 1 1m oxen 39ahW 18139 MM htmme 1h h m 1h m h 1 1h11h1yh111 hhhn quot1111 y 1 1 1h hhh 7 enhe 2 cunecwame 1a1h used 1h1mma11uh ahuunhe ubserved pa1h e11 Venus aemss1he Sun 16 The Sandreckoner Nexandna she 1he1e seems 11111e duubt 1ha1 he smmee w11h1he sueeessms quuehe A11e1h1s s10d125 he 1emhee 1e Late in the 3rd century BC Syracuse became embroiled in the struggle between Rome and Carthage for control ofthe western Mediterranean In his Parallel Lives the historian Plutarch about 46126 AD recounts Archimedes39s efforts in the defense of the city against the Romans When therefore the Romans assaulted the walls in two places at once fear and consternation stupefied the Syracusans believing that nothing was able to resist that violence and those forces But when Archimedes began to ply his engines he at once shot against the land forces all sorts of missile weapons and immense masses of stone that came down with incredible noise and violence against which no man could stand for they knocked down those upon whom they fell in heaps breaking all their ranks and les In the meantime huge poles thrust out from the walls over the ships sunk some by the great weights which they let down from on high upon them others they lifted up into the air by an iron hand and soon such terror had seized upon the Romans that if they did but see a little rope or a piece of wood from the wall instantly crying out that there it was again Archimedes was about to let fly some engine at them they turned their backs and fled The Roman attack on Syracuse was repelled A lengthy siege was later successful and Syracuse was conquered and destroyed in 212 BC Archimedes perished during the destruction Plutarch relates several versions of his death The one most widely cited finds Archimedes oblivious to the city39s capture absorbed in the study of a particular diagram that he had sketched in the sand When a Roman soldier confronted him Archimedes requested time to complete his deliberations The impatient soldier however ran him through with his sword The work of Archimedes is impressive as we shall soon see Plutarch speaks of Archimedes39s purer speculations and studies the superiority ofwhich to all others is unquestioned and in which the only doubt can be whether the beauty and grandeur of the subjects examined or the precision and cogency ofthe methods and means of proof most deserve our admiration It is not possible to find in all geometry more difficult and intricate questions or more simple and lucid explanation No amount of investigation of yours would succeed in attaining the proof and yet once seen ou immediately believe you would have discovered it by so smooth and so rapid a path he leads you to the conclusion required Archimedes was also the quintessential eccentric scientist His deep absorption in thought made him forget his food and neglect his person to that degree that when he was occasionally carried by absolute violence to bathe or have his body anointed he used to trace geometrical figures in the ashes of the fire and diagrams in the oil on his body A famous episode recounts how after a particularly satisfying discovery Archimedes ran through the streets of Syracuse in naked celebration shouting quotEureka Eurekaquot Eureka or c39vpC39Ka is Greek for quotl have found itquot It is of course difficult to separate fact from fiction and reality from legend in Plutarch39s account of Archimedes39s remarkable talents as inventor of machines of war The ingenuity of Archimedes39s speculations about geometry and physics however can be corroborated Much of his work has survived in transmitted form We turn now to Archimedes39s scheme for writing large numbers You will encounter the basic Greek number system in the exercises It allows for numbers as large as 99 999 999 999910 000 9999 6146M g but not larger Archimedes enlarged the basic Greek number system into an incredible scheme He introduced the number MM 10 00010 000 100 000 000 108 and referred to numbers up to MM as numbers of the rst order He then built the following tower of quotordersquot and quotperiodsquot of numbers In modern notation First order The numbers N with IS N lt108 108391 Second order The numbers N with 108391 108 S N lt 1016 108392 Third order The numbers Nwith 108392 1016 S N lt1024 108393 Fourth order The numbers N with 108393 1024 S N lt1032 108394 71 8 108 th order The numbers N with 10810 S N lt1032 10839108 These are the numbers of the first period This is only the beginning The numbers of the second period are also partitioned into orders 81081 First order The numbers N with 10839108 S N lt1032 10 Second order The numbers N with 108391 8 S N lt1032 108I1082 Third order The numbers Nwith 108391 8 S N lt1032 108I1083 Fourth order The numbers N with 108391 83 S N lt1032 108I1084 108 th order The numbers N witthBI oaH ail S N lt1032 108I108108 The second period ends with one less than the number 108008 10839108 108108108 1081082 The third fourth fifth etc periods follow Finally with the 108th period and the number 108108 108 10839wm the array stops To put all of this into perspective note that the visible universe is thought to have on the order of 103 atoms about 999 of them hydrogen and helium This is Eddington39s number after the British astrophysicist Arthur Stanley Eddington Since 1080 10 this is the first number of the 11th order ofthe first period The point is not so much the usefulness of Archimedes39s scheme but rather the grandiose nature of his speculations Archimedes in other words thought big Archimedes looked for a context in which to illustrate the utility of his cosmic array of numbers Having found it he was evidently very pleased to address his manuscript The Sandreckoner to his benefactor the king of Syracuse He began by giving the king an astronomy lesson Aristarchus brought out a book consisting of some of the hypotheses in which the premises lead to the result that the universe is many times greater than that now so called His hypotheses are that the fixed stars and the Sun remain unmoved that the Earth revolves about the Sun in the circumference of a circle the Sun lying in the middle of the orbit and that the sphere of the fixed stars situated about the same center as the Sun is so great Then he stated his purpose I say then that even if a sphere were made up of sand as great as Aristarchus supposes the sphere of the fixed stars to be I shall still prove that of the numbers named by me some exceed in multitude the number of grains of sand in a mass which is equal in magnitude to the sphere referred to provided that the following assumptions are made In other words Archimedes imagined the entire cosmos to be packed with sand and proposed to count the number of grains of sand in question Our description of Archimedes39s discussion will use earlier notation rE for the radius of the Earth rM for the radius ofthe Moon rs for the radius of the Sun and DS for the distance between the Sun and the Earth What assumptions did Archimedes make The first concerned rE rE S 47 500 miles We have seen that Eratosthenes39s rather accurate estimate was 3850 miles So here Archimedes thought too big Remember however that it was his purpose to display the vastness of his number scheme Following Aristarchus Archimedes supposed that r Mltr E Recalling that Aristarchus had shown that rs is about 20 times greaterthan rM he next assumed that r 30r S M This is in fact too small A look at Table 14 shows after a quick calculation that rs 400FM Because rM 30VM lt 30rE m 3047500 Archimedes got rs 1 425 000 miles While these inequalities are based on a combination of earlier estimates and pure speculation Archimedes next turned to careful observation and delicate geometrical arguments He used a long rod with a small disc fastened at its end He pointed it in the direction of the Sun just after sunrise and carefully measured the angle a that the Sun subtends in the 90 90 sky He determined that lt a lt or in radian measure 200 164 1 The implied estimate a m 3 for the socalled angular diameter of the Sun is accurate Aristarchus took it to be 2 Refer to Figure 124 Archimedes imagined an observer on the Earth looking out at the Sun just after sunrise He constructed the diagram of Figure 129 and positioned the observer at the point E Two tangent lines are drawn from E to the Sun Note that a is the angle that these tangent lines determine at E From C the center ofthe Earth he placed two more tangent lines to the Sun and let be the angle that they determine Taking the center of the Sun to be 0 he obtained a circular arc by rotating CO and he let A and B be the two points of intersection ofthis arc with the tangents from C arcAB Since arcAB 2A0 2rS DS m mm 127 Smceme Sun SVavawayand E and c ve awe y c use ubsevvemat m Avcmmedesshuwed much muve Eyavevy m mm smE s tanE 100 forU ltylt5lt heven edthat ltaltE smy y any Sn Avcmmedes ub amed me appmmauuns nsemngme mequahues ISS3047500 m esand i gm heluundma D5 ltWlt 160x106 m es Hemevemve ubtamed 7r D5 lt160x10 muss Dlt16x10 2 muss Jam 5 I Arc Ina125 Aumal p p mmthun mm mm pg pm n uuLhLu aw Lb x m mi cs 24 x mm mm W Lpim m up eniumu m mm W the m pp u p mm Mn N mm n p pmm p p we wwwm v m Amy mm mm W pm my 11m an N1th m WWW rim m m up quotMu aquot mum m 5mm 7 mm W 2 Aha spheve u vadms D See Ham 1 am He began mp a puppy seed and assumed mm a spheve mg m u a m mm m p m ngarbvaadth NEW 4 4 d 3 r is 3703 or 373 m evms u Ms mamem d n uHqumat We mamem Ma spheve s mcveased by a emu 339 3339 339 szv mam has a vu ume c1403 mes gveamman pm we puppy seed Theveluve m can be mad mp 10001 3 WWW y gvamsulsand Cummumgnthwsway Avcmmedescundudedmat1063 gyamsuv sand W on a spheve uv vadms 1 6x10 m es Smce Dlt16x10 2 m es m 5 wave mp enuugh m on m spheve m 2 H x v m M MW A 3n m m X m amuum Mcusmm manenhat 57 mm Eddmgtun sth cemmy 251mm 17 Postscript As1he uh andthemstamunes say me Mh Tthh 1mmn M h dusev 11 1s The smauenhe change 1h pusmun 1he anhey away 11 1s MW 1 111 m N 1 H and E h 115 mm abum 1he Sun See F1gu12131Whenth2 Eanh 1s a1Emahe hme enhe pes1hehce1A agamsnhe med me 1 h V have shmed 1e say C2 wheh1he Eanh1s a1 E Wheh1he Eanh1s back a1 E measme1he ahg1e 54 2 he1weeh1he pusmuns C and c2 SmEE 1he ems 1h 1he eehs1euaheh ave vevylamhe hhes 5 62 and EC2 ave payaue1 Sn AEAE 5 Theahg1e p A Le1 DA A Th1 S m A mhheh mhes Seerah1e15 Ashunumevs ye1ey1e1h1sms1ahee as1he astronomicaMMMAU Sn 1 AU 93x10 mhes can he appmwmated hy1he Ewcu astc mshuWn 1h F1guye132 LE mg RM he1he steHav pavaHax e1 A h hamah measmehueuews 1ha1 pm DA Wham DA 15 gNEn 1h AU 81 1sscunu1siuv1 60 mmme whmh 15 i 811 degvee Sn 1 3600 seconds Thevemve 60 1 mam mseconds 2x105 seconds 7r 7r Sc 1112 pavaHax p 15 cunvened 1mm vamans 1n secunds by pm 2x10 pm Maw 32 nquot 1mm v 1131815th 171 mm 1 6x10 m es m 8712 yea Sn We 1m 1 LV 75 x 18 2 m es S1ncs1 AU 793 x 18 m es 11 uHqu m1 2 12 6 6AU Thevemve lAU 6X10 6 1 93x10 93x10 asmHqu LY15X10 5 LY The equamy pm can nuw be cunvened ILY pg 2x105pM 2x105LAU2x10515x105LY in DA DA 121218124112 mstance DA mm anm m 182 Sun 171 11gmysa1s1s Wham pm 15012 pavaHax 81 A measuved m secund The pavaHax 81m 1am s1a1 Pmma Csmaun 1s abum 8 7E 81711220115 1sms neavs amtspavaHax15012 1aygss1 mssmng pm 076 1mm pvecedmg 3 equauun shuwsthanhe mstance m Pmma Csmaun 1s appmwma e y m secunds m 5000 ALY Smce 1LY6X10 muss 8mm Centaun 1s abuut 24x10 m es d1s1amsee T312121 5 SteHav pavaHaxesweve measuved W101 guud accuvacy m 1112 cemmy Fm mm 1 1m R 1 E1 Cygn11n1838 and came wnhm 188182 madam mus 818 27 secunds Puppmg p5 0 27 1mm equauumust devwed we 1nd 18am Cygm 1s 18 9 LV away S1110 m 811gmss1 513111135 3 pavaHax uvabuum 38 secunds Th1s we ds a mstance abumE LV 3 j mm L my mm m Fzmu c 1 33 Exercises IA The Greek Number Sstem ens Hundreds 710 iota p100 rho ppa Sigma lambda mu upsxlon nu xx ommon 0 p 47800 omega 59 theta 90 koppa 7 90 sampx Thanh mm km Fm examme KS 25 M 37 um 488 Tu deswgnate huusandthe um symbms WEYE used Wow a shake Fm examme 7 000 A 7 4 e evsth2Gveekssumeumesputab 7 umeva s Edmowy 10000anM fawnma myriad was us d cumbmed WM 0012 symbu s as VuHqu 8 20000 1507177054 14100008567 148567 0M 40010000 4000000 mysM mE 84500 000 2384 8452 384 geumeuy 1vvmemnumbevsas 842 34547 2875739 usmgmereeksystem lB Greek Algebra Problems 2 5 are taken from The Greek Anthology8 2 The Muses stole and divided among themselves in different proportions the apples l was bringing from Helicon Clio got the fifth part and Euterpe the twelfth but divine Thalia the eighth Melpomene carried offthe twentieth part and Terpsichore the fourth and Erato the seventh Polyhymnia robbed me of thirty apples and Urania of a hundred and twenty and Calliope went off with a load ofthree hundred apples So I come to thee with lighter hands bringing these fifty apples that the goddesses left me How many apples did I bring from Helicon 3 Make me a crown weighing sixty minae mixing gold and brass and with them tin and much wrought iron Let the gold and brass together form twothirds the gold and tin together threefourths and the gold and iron threefifths Tell me how much gold you must put in how much brass how much tin and how much iron so as to make the whole crown weigh sixty minae A number of references describe the mina as a unit of weight roughly equal to one pound So it seems that this crown was intended for no ordinary mortal 4 Throw me in silversmith besides the bowl itself the third of its weight and the fourth and the twelfth and casting them into the furnace stir them and mixing them all up take out please the mass and let it weigh one mina The first thing is to decide what the question is 5 Brickmaker I am in a great hurry to erect this house Today is cloudless and I do not require many more bricks but I have all I want but three hundred Thou alone in one day couldst make as many but thy son left off working when he had finished two hundred and thy soninlaw when he had made two hundred and fifty Working all together in how many hours can you make these Hint If there is not enough information supply it IC The Quadratic Formula 5 6 Consider the quadratic polynomial x2 5x 4 Take the x coefficient 5 divide it by 2 to get squaring gives 5 2 5 2 5 2 5 2 Now rewrite x2 5x4 as follows x2 5x4x2 5x3 4 2 2 2 2 2 Checkthat x2 5x x gj Therefore x2 5x4x 4 2 x 2 E 2 2 44 529 x 2 4 We have done what is called quotcompleting the squarequot for the polynomial x2 5x4 Now answer the following i For which values of x is x2 5x 4 0 2 ii What is the least value that x2 5x4 can have 7 Solve the equation 3x2 21x12 0 by completing the square for the polynomial x2 7x 4 8 Let a b and c be constants and consider the equation ax2 bxc 0 What is x equal to if a 0 If a 0 use the strategy of Problems 6 and 7 to show that the solutions are given by the quadratic formula b ixlbz 4610 2a x ID Rational and Real Numbers 9 The matter of measurability is perhaps most concretely illustrated by the consideration ofthe numbers that underlie our 2 monetary system All dollar amounts are expressed in the form x1 0 So only rational numbers indeed only certain rational numbers are allowed In a supermarket one will occasionally find say 3 items for a dollar A single item is not measurable within the system Why not 10 Consider the numbers 1333333 2676767 and 4728728728 Show that they are rational numbers Hint Let r be the number In the first case consider r lOr 5 468 11 What are the decimal expansions ofthe rational numbers I and E 2 Hint Carry out the divisions Note It turns out that a real number r is rational precisely if its decimal expansion has a repetitive pattern after some point For example 23459999999 2346 234 and 52363636 52 198 are rational numbers So is 3534672638 12 Recall that a prime number is a positive integer p gt1 that has no divisors except 1 and p Euclid pursued the study of prime numbers in Book 9 ofthe Elements The following fact about prime numbers is known as the Fundamental Theorem of Arithmetic Every positive integer n is a product 71 pf pf pf of powers of distinct prime numbers p1p2 p and there is only one way of doing this aside from rearranging the order of the factors For example 54 2333 2133 233 is the unique factorization of 54 into prime powers i Determine the factorizations of 28 192 and 143 into powers of distinct primes ii Use the Fundamental Theorem of Arithmetic to show that 3 is irrational iii Show that a positive integer n is a square precisely when all the exponents of its factorization into primes are even integers iv Let n be a positive integer Use the Fundamental Theorem of Arithmetic to show that J is rational only if n is square IE Angles and Circular Arcs 13 Fill in the following blanks i 1radian degrees ii 1 degree radians iii 7850 radians iV 1238 radians degrees 1a m We cwcmav seem u Heme 1 33 45 57 3 Wha sthe engm uHhe arcAB7 58 anwnh Hgmew 13 uHhe ex Takemetangemmme wde a A and 2 P be Ms pumtuhmevsectmnwnh A P ayF gmewamnspxaus mema aye APgAPUrP P Shuwma AP ltA P Cuncmdetha he mm 14 meeuahnes an AP A P tandAB A B 2 ave p auswb e m w n m 1n secund th Wm speed n122pevsecund duesthe vuck y em 17 The cwde m Rama 1 35 hascemev 0 and a vadms UVZVEM The anuwwsvma mg c uckwwse a a yam Mane Th mms A and D gwen manhe anuw veewes7 5 huuvstu mme mm A m D IF Basic Trigonometry 13 Use the appvupnate mang es m m m We va ues 7r x eos n cos 6 7r m eos 1v 3 7r v anii m 4 19 Use Figuve 1 21 uHhe ieme deievmme We luiiuwmg hmns 57 x hmeos M Z iiiusvaiewnh a diagvamihai w 5 gt5gt 0 Men 5055 lteos5 and cane an 1 055 Veniyihe Mammy 5325 mien 21 The seeam u 5 is de ned by 535 22 Cumpave use a caicuiaiuv the vaiues eve sm s ian aim 15101radians sma tan n 51001radians sma m 510001radians sma IG Distances and Sizes in the Universe 23 Cumpuie W 395 DM and 135 using Ansiavchus savgumemand 3 Keep r5 aasn miies b in hypuihesis make 89w ms1eae e1 87 The angie measuve is caiied mmme is equai m u c In hypothesis D take instead of 2 So the angle in Figure 127 is instead of 1 d In hypothesis E take SVM instead of 4rM Round off to get an accuracy up to 4 decimal places Compare your conclusions with the modern values from Table 14 24 Both Aristarchus and Archimedes assumed that the Earth is a sphere Is this reasonable in view ofthe mountain ranges on its surface You are given that the radius rE of the Earth is 3950 miles that the height of Mount Everest is 29028 feet that 1 mile 5280 feet9 and that the radius of a basketball is 47 inches lfthe Earth were shrunk to the size of a basketball how high would Mount Everest be Is this higher than one ofthe little mounds officially called a pebble on a basketball These have a height of about 002 inches 25 Since 3950 miles 20856000 feet and 47 inches 039 feet the shrinking factor in Problem 24 is r 039 20856000 0 about Show that ifthe radius rM 50000000 distance DS to the Sun and the distance D to the nearest star were shrunk by this factor we would have approximately ofthe Moon the distance DM to the Moon the radius rs of the Sun the VE 47 inches the radius of a basketball as we have already seen rM 137 inches about the radius of a baseball which is 143 inches rs 45 feet DM 25 feet DS 186 miles 1 D million miles 2 26 Shrink the Sun to the size of a basketball What is the shrinking factor Shrink the rest ofthe solar system by this factorand compute rE rM DM rs DS and D inthis case 27 Reconstruct Archimedes39s argument that 1063 grains of sand more than fill the sphere of the universe Assume that 2 finger breadth is 3 of an inch 28 The near stars Barnard 65115 and Ross 780 have stellar parallaxes of 055 027 and 021 seconds respectively Determine their distances in light years Notes 1 This theorem was probably already known to the Babylonians in 1700 80 more than 1000 years before the time of Pythagoras about 570 to 500 BC 2 The Pythagorean philosophy was not without later influence however The book by GL Hershey Pythagorean Palaces Cornell University Press Ithaca NY and London 1976 was written to establish the fact that in quotthe Italian Renaissance domestic architecture was largely ruled by Pythagorean principlesquot 3 This is a question that occupied the mathematicians of antiquity and is still relevant today Our discussion will give only a very elementary perspective 4 When a segment such as A P or PB 39 appears in a mathematical expression the reference here and elsewhere in this text will always be to its length 5 A gnomon is simply a straight stick or rod The word comes from gnomon quotto knowquot in ancient Greek Our words prognosis and physiognomy are derived from it 6 The Greeks used stadia not miles Ten stadia are the equivalent of about one mile 7 The detail on the Earth in the figures that follow should not mislead the reader into thinking that the Greeks ofthe third century BC understood the scope and shape of Africa Northern Europe and the Atlantic Ocean The Greeks were very familiar with the territory near the Mediterranean Sea and the armies of Alexander the Great advanced all the way to today39s India and Afghanistan However a true concept ofthe extent ofthe continental land masses and oceans began to develop only with the voyages of discovery in the 15th century 8 The Greek Anthology is a collection of Greek poems gongs and riddles Some of these were compiled as early as the 7th century BC and others as late as the 10th century AD Harvard University Press produced anew edition of The Greek Anthology in 1993 9 Mount Everest is about 55 miles high As an aside note the Earth is actually not quite a sphere The Earth39s rotation has caused an quotequatorial bulgequot so that the Earth39s diameter through the equator is about 26 miles more than that through the poles 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Hal II 1Irrl 1rlml lul nil mii 31 l u u39Ei39E L mama11 35 u gum 13 u IIL22 n Ij ju 139 1 5 l mmmmmmumpmmume mwmmllmmmi mmtm m mum hmlmli EMFM 1339 Elle 2m 211311quot un t illIv 2 Him 3LIENL huff m w 1um m mam mmw mug HE HF 1 3 in Illl 3 i 1 gt 35 w all 53 Sh I r h j Eu h App utinm 3 Commlam Jhmpmjunlmnm maimmmllmmu mmgkdww Eu Emma 51m lm v ddy alum1mm it 21139 Slim Immarmm mm lhw z isy 4L A m nmhumhiimamwmm manning ll may k dummng mm d Emilinhjmmthm mmu ung b trimmlmk 41 My A y g ml ihh m 39 g m i ml l wm 2251mm MgmMWMM M Jmmw mwlm tym xmvuammz afarHourleil A ummmmm m m mmwi lm mm ii miniiii M Esm i kdn iy lh rd mml mum IE9 mummwyammmm It TmMalia Agnsia mmhnm Mhhlmiuw aqx mmmmm wm39 nmm mlmymrswt mlm WIFE I ll Emmi i y minn hw mlkmym ugmwmmgwm inn Fud gmm miydth l mm M mumwmmdmg Tumll mmjmanmmmimmmmmnmmn h m im armMann hwnmmwmpm 39hlEImtEEHl dhpki imzmmmhm mm awnmanna Amnm rmuggnam l M WmmmmmR u i mimnmmm rmmgmmmm m m zmhnm mmmwmmvm Ill Esm mwMydibmJmmm 111 Ewmwh wnnmmmmIMIrm mulmdjatmljm 1 a Ham Hu ll lg hhw h l lh mm A H 2 h III A I a v 39 o munn Humrm my am uamummmmvdmnmmm Wmmmpjmh mmmmmm ll midi 7TB mm mmm mmwmnmmm mn mwm ymh mmtukm mmmq mhmhmsm um Ewwwh ly lhwmmwh WE Iwmm m WW3 WWW H In 5 W m hl l kim m i mg m I I ixquot VIL39 ain t a J r vm r 39m HM hwmm ad mmlh uMHH mmgvlw 39TMmHm migmmg awurnmmmmwmimmwm Miamim ail I Md mlmFIJ39FhHF Eg ngazmlgmmrmayfdmhhmumwm dh 1mm mmmnmmm raA wlam til nunmum mmunm ib mmmzm W39I 39myl m n has 5 m 2 33 I vii35 5 W 7quot WW NEW quotinquot ring warlike I m mm It n I3 mm in 51 Fa quot 1 Fr 3 F1 tantra d111 53 IF an m alen r 31 Wu 1 quot I FElam Fri43 F p39m lmy mm 9 l mheplw ludm39lilim hpMmI MIMi Mgm im gmmmmmdmhm m hij m l bm quotru m l V a E I g quota lm VIII 1 139 I u l emaci ymfulh h 39H dFn 39 quot A f 39 The Bat Product at Hinton ih i by 12 am ang Ihmtsp mm Mmmm Angles and Radians of a Unit Circle pm in m mm mmmsumm MM mmmm 0 MM m 27 7730 mam 7r3 37r4 Whig2 12J 2mm5 45252 M lt z n yrs ms ms 1 quotE239 67212 m 27 117m 3242 fl2432 pm m cm Mgmw m m m m Danny whammy 0mm 5mm Um mp WW Mnhmnlfdp mm m 77r4 ndZ2 Saw2 Math fnaticsg Help Central Measuring the Salmr 5175th Michael Fowler UVa Physics Department n m5 lecturer We shall SHOW hOW the Greeks made the WEI real measurements Of astronomlca r Where thew r W0 How big is the Earth atosthehes a Greek who Wed rh Alexandra Egypt h thethrro century B c He knew that farto the south h thetoWh of svehe n a t mldday oh June 21 far down h thrs H oh ho other day of the year The oorht was that the sun was exactly vemcaHy overhead at that we a d at no othertlme m the W h Alexandra the closest rt got w degre s o as oh June 2l wheh ltwas off by ah ahgle hetound to be about 7 2 es by rheasurhg the ha 0W of a vertrcal suck The rh rah rr rh almost exactly due south From thrs and the dlfference h the angle of sunhght at mldday oh June 21 Sunhght Syen The angle between the aunhght and at 3 mu equal the angle between Alexandna ands nf m thee t otthe Eatt Of course Eratosthenes fully recognized that the Earth is spherical in shape and that quotvertically downwardsquot anywhere on the surface just means the direction towards the center from that point Thus two vertical sticks one at Alexandria and one at Syene were not really parallel On the other hand the rays of sunlight falling at the two places were parallel Therefore if the sun39s rays were parallel to a vertical stick at Syene so it had no shadow the angle they made with the stick at Alexandria was the same as how far around the Earth in degrees Alexandria was from Syene According to the Greek historian Cleomedes Eratosthenes measured the angle between the sunlight and the stick at midday in midsummer in Alexandria to be 72 degrees or one ftieth of a complete circle It is evident on drawing a picture of this that this is the same angle as that between Alexandria and Syene as seen from the center of the earth so the distance between them the 5000 stades must be one ftieth ofthe distance around the earth which is therefore equal to 250000 stades about 23300 miles The correct answer is about 25000 miles and in fact Eratosthenes may have been closer than we have stated herewe39re not quite sure how far a stade was and some scholars claim it was about 520 feet which would put him even closer How high is the Moon measure the ahgte to the mooh trom two otttes farapart at the same Wet and oohstruot a stmttar trtang e We Thates measurthg the dtstahoe otthe shto at sea Unfortunatety the ahgte dtfference at the Wet so that method woutdh t Work Neyerthetess h h D t method of t ofa unarechpse whtoh haooehswheh the earth shtetds the mooh trom the sun s ttght Fora Ftash moyte of a unarechpse chck heret T0 oetterytsuattze a tuhar echpse tust tmagthe hotdthg up a quarter dtameter ohe thoh aoorowmatety at the dtstahoe where tttust btocks out the Sun s rays from one eye or course you h w I try thtS ur eyet feet away or 108 thohes tr the ouarterts turther away than that tt ts hot otg ehough to btock out an the suhttght tttt ts otoserthah108thohesttwttttotatty btock the suhttght from some smatt otroutar area whtoh thh the ootht the quarter or course thts ts surrouhded oya fuzzter area oattedthe oehumora x where the suhttght ts pamaHybtocked The tutty shaded area ts oatted the umora Thts ts tatth torshadow UmbreHa ttattah tt t appropnatety you can see these dtfferent shadow areas Questtoh tr you used a dtme thstead of a quartet how farfrom your eye woutd you haye to hotd tt to Now tmagthe you re out th soaoe some dtstahoe from the earth tooktng at the earth s shadow or course t tttu to oomoat to the ouatt t t must be ahgutar stze of 804000 mttes from earth gt 4 To Run NOWt the far ootht atmosphere t feH oh W the Greeks was about Woandrarhatmmea the moon s owh dtamete Note shadow oh oetteh by aotuat oosetvattoh ofa tuhateottose t They wt and a hatt ttgute 0t tt tat a a ttt moon was Wth tt ototett them the moon was ho mnherawaythan108x8 000 864000 mHeS othetwtse th atatt Dot tt ttt tart ttoouto be a 000 thttesa a ttau tt at ttt ooth Howeveh such a my thooh couto Never cause a sotat eottose th tact as the Greeks wett kneW the down the moon s dtstance from earth th thts gure the fact that the the ahgte EAF Nottce how that the tehgth FE ts the otathetetotthe earth s shaoow at the otstahoe of th ON and the tehgth ED ts the otathetetotthe moon The f y setvattoh otthe tuhat eottose that the tatto of FE to ED was 2 5 to t so tooktng at the stthttat tsosoetes thahgtes FAE ahoDcE A A tht Ets from h h Dotttt wt tttatAc must be t 8000 thttes the tutthest ootht otthe oohtoat shadow At ts 864 argument thts ts 3 5 ttthe mH to b 000 mHeS from earth From the above 5 further aWaythan the moon 5 50 the dtstanceto the moon t5 864 0003 5 t5 thhtn a How far away is the Sun and they oroh t do 50 We hm W nr 1 s L they oro rearh trorh thrs approach that the Sun was much turther away than the moon and consequem y srhce rthas the same apparent STZe rt must be much brggertharr ertherthe moon or the eart rrr rr roeator They kneW ofcourse mat Therefore they r a h d exac y haxtmh o w rror mm moon to the observer see thetr we to cohyrhce yourseTt otthrs so rt ah observer oh earthy oh obseryrhg a hat rhooh rh dayhgm measures caremhy the ahgTe between the orrectroh or the moon ahothe orrectroh or the sum the ahgTe a rh the gure he Shomd be abTeto construct a Tohgthrh mang e T omen and so trhothe ratro or the sun s orstahce to the moon s orstahce to Sun EARTH or a degree t h H r w HM atr ad suggestthe sun to be much Targerthah the earth H was orobabTy thrs reahzatroh that Ted Arrstarchus to suggest that the sum ratherthah the earthywas at the center or the unwerse The best Tater Greek The oresehtatroh here rs srrhrTar to that rh Errc Rogers Physrcs tor the Thourrrhg Mmd Prtnceton 1960 h n39 dciubteinohvltim viminia milllav 39 1htm

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