Convenions nd Noion: GeoTrig Noes 04-14 Hello ll, his revision inludes some numeri exmples s well s more rigonomery heory. This se of noes is inended o ompny oher uorils in his series: Inroduion o EDA, Fivenumer BoxPlos, Business Trend Anlysis. As onus for hose who wn lile exr, I hve pu in n revied ppliion of Euler s formul o derive few rig ideniies. This is definiely supplemenl meril u, I ouldn resis poining ou how ll rig ideniies n e derived from his formul! Convenions nd Noion: I will use he following symols nd noions: 1. - denoes n ngle (in degrees) 2. s - denoes n ngle (in degrees) 3.,,, d, e, f, g, h, k, m - denoe lenghs 4. A, B, X, Y- denoe veors, h is, direed line segmens generlly, u we will speifilly use hese o represen smple or populion oservion veors or sisil prmeers. See he veor disussion in hese noes for more deil. 5. A, B, X, Y - denoes he lengh of hese veors. 6. {1,2}, {1,2}, {x1,x2}, {y1,y2} - denoe oordines, nd will e used o show posiions in 2-dimensionl spe. These re he x, y oordines of he end poin of veor in 2-D spe, or generl poin in 2-D spe. 7. {1,2,3}, {1,2, 3}, {x1,x2,x3}, {y1,y2, y3}- will e used o show posiions in 3-dimensionl spe. These re he x, y, z oordines of he end poin of veor in 3-D spe, or generl poin in 3-D spe. 8. To indie vrile rised o power, I will use he symol ^. So, if I wn o show he squre of he vrile, I will wrie ^2. If is o e ued, I wrie ^3. 9. To indie squre roo I will use he noion Sqr[x]. For exmple, Sqr[16] 4. I ould lso wrie his s 16^(1/2) or 16^0.5 10. To indie (slr) mulipliion I will use *. So 4*3 12. 11. For veor mulipliion I will use he Do produ, indied y do,. For exmple, he do produ of he wo veors {2,4} nd {-3, 12} is wrien s {2,4} {-3, 12} nd resuls in he numer (2* -3) + ( 4 * 12) 42. As you n see, eh omponen of he firs veor is muliplied y he orresponding omponen of he seond veor nd hese produs re dded up. Firs Things Firs - Pyhogors nd His Tringle I wn you o e le o se nd prove he Pyhgoren heorem whih sys h for righ ringle, he sum of he squres of he sides djen o he righ ngle, equls he squre of he opposie side (hypoenuse). Consider he digrm elow (Figure 1 on pge 2): I would like o show one wy h demonsres he onlusion of he heorem. Firs drw squre where eh side is mde up of n lengh plus lengh, rrnged s shown in he digrm. This mens h eh side is + unis long. Then look he digrm nd onsider he seps: C:\ooks\QPk2008\geoTrig.fm 4/14/08 r.r 1
Firs Things Firs - Pyhogors nd His Tringle 1. ol re of he squre, from n ouside perspeive, is (+)^2 (+) * (+) ^2 + ^2 + 2 * * 2. now drw he four inerior ringles s shown, nd he inerior squre h hs on eh side. (noie h I don know wh is, his poin, u I know eh of he long sides is he sme, so ll hem. 3. eh inerior ringle hs re of ( * )/2 nd here re four of hese h, if you dd hem up, gives n re of 2 * *. (rememer, rengle hs re of lengh imes widh, or if he lengh were nd he widh, hen he re would e *. Now, he ringle shown is hlf he re of h rengle.) 4. he inerior squre, wih unis on side, hs re of *, or ^2 5. ol re of he squre from n inside perspeive is ^2 + 2** 6. Sine i s he sme squre, he ouside nd inside perspeives mus mh, so: 7. ^2 + 2** ^2 + ^2 + 2** 8. The 2** erms nel nd he onlusion follows: 9. ^2 ^2 + ^2 Ouside perspeive, re of squre is: ( + )^2 ^2 + ^2 + 2 * Inside perspeive, re of squre is: 4 (* )/2 + ^2 2** + ^2 Equing he wo perspeives: 2** + ^2 + ^2 + ^2 + 2** Therefore: ^2 ^2 + ^2 FIGURE 1. Pyhgors Theorem: Demonsrion Exmple of Pyhgors Theorem Jus o ge his o sink in, here is numeri exmple of he heorem. le 3, 4, hen wh is? The ouer squre would e (3+4) * (3+ 4) 9 + 2* 3*4 + 16 49 Eh - ringle on he inside hs re (3 * 4)/2 6 C:\ooks\QPk2008\geoTrig.fm 4/14/08 r.r 2
There re four of hese for ol re of 4* 6 24 The inerior squre of side hs re * ^2 (whih we don know ye) So, (3+4)(3+4) 4 * (3*4)/2 + ^2 2* 3 * 4 + ^2 9 + 2* 3 * 4 + 16 2*3*4 + ^2 9+ 16 ^2 25 ^2 So, ^2 ^2 +^2 3^2 + 4^2 5 (his is well know exmple of ringle, lled 3-4-5 ringle, go figure! You n see how hese ringles work y drwing few. The mos ommon is he 3-4-5 ringle s we jus looked. Oher ringles h work ou o whole numers re: 5-12-13 nd 7-24-25. The nex seion inrodues he si rig funions h you will need for he reminder of your life. You will e le o use his digrm (Figure 2 on pge 4), o predi nd iner-rele he ehvior of he vrious rig funions, geomerilly. This digrm will llow you o see he rig funions in erms of physil lenghs of sides of righ ringles. I dvise you o refully sudy he digrm elow nd ommi he firs hree funions, Sin, Cos, nd Tn, o memory. Afer sudying he digrm, see if you n re ou he onsequenes of hnges o he ngle. For exmple, when pprohes zero, wh hppens o eh of he funions suh s Sin[] or Tn[]? You n ell wh hppens y imgining how he lenghs of he vrious lines hnge. Wh hppens when pprohes 90 degrees? For exmple, when pprohes zero, Sin[] pprohes zero (h is, lengh goes o zero), while Cos[] pprohes 1 (h is, line goes o 1 ). However, he funion 1/Sin[] will low up o lrger nd lrger vlues s goes o zero. This funion, 1/Sin[], is lled Cosen of, revied s Cs[] nd will e desried in he nex digrm. Similrly, he Tn[] will go o zero s does, u Co[] will low up. Chek ou he oher funions for heir ehvior nd see if you n menlly re ou he onsequenes of he funions s inreses or dereses y looking how he vrious lenghs,,, d, e, g ehve. Noe h ll of hese lenghs re he sides of righ ringles. Some Common Angles, Their Trig Funions nd Lengh Equivlenes (Ignore he ls hree olumns for now, I will desrie hose in he nex seions). Look row one of he le nd you will see h Cos[0 degrees] 1. This is lso represened y he lengh of side, whih exends o he edge of he irle {1,0}, when he ngle is zero. This mkes he lengh of 1. Similrly, Sin[0] is 0 nd is represened y he lengh of, whih you n see will go o zero s pprohes zero. Sin[45] nd Cos[45] re speil vlues sine he re equl. In his se, nd oh hve lengh of Sqr[2]/2 0.7071. To hek ou hese vlues, jus drw hose ringles on grph pper wih he ngles given y proror. Mke sure h you pik unis so h he hypoenuse is uni lengh. Th is, if you drw your ringle 30 degrees nd he hypoenuse is 4 inhes, hen h is he uni of lengh. If he opposie side is 2 inhes, you hve jus verified h Sin[30] 2/4 1/2. As noher hek on hese ls wo vlues for Sin nd Cos of 45 degrees, onsider h euse of symmery, lengh is equl o lengh. This hppens sine 45 degrees is hlfwy eween 0 nd 90. Sine 1, hen: C:\ooks\QPk2008\geoTrig.fm 4/14/08 r.r 3
^2 1 ^2 + ^2 2 * ^2 (sine ) So ^2 1/2 nd 1/Sqr[2] Sqr[2]/2 ANGLE (degrees) 0 0 1 0 VERY BIG 1 VERY BIG 30 1/2 Sqr[3]/2 1/Sqr[3] 2 2/Sqr[3] Sqr[3]/1 45 Sqr[2]/2 Sqr[2]/2 1 2/Sqr[2] 2/Sqr[2] 1 60 Sqr[3]/2 1/2 Sqr[3] 2/Sqr[3] 2 1/Sqr[3] 90 1 0 VERY BIG 1 VERY BIG VERY BIG Sin[] Cos[] Tn[] d Cs[] g Se[] e Co[] f The Big Three Trig Funions: Sin, Cos, Tn {0,1} UNIT irle, every poin on he irle is uni disne from origin (Noe: lengh of 1) Sin[] / /1 Cos[] / /1 Tn[] / lso: Tn[] d/1 d 1 d {1,0} Noe h lenghs nd d oh pproh zero s ngle pprohes zero. Also, noe h lengh d lows up o lrger nd lrger vlues s pprohes 90 degrees. FIGURE 2. The Big Three Trig Funions, Sine, Cosine, nd Tngen Complemens of Angles nd Their Reled Trig Funions The Sin of n ngle nd he Cosine of n ngle re inimely reled s you n see from he digrm ove. They re lso reled in simple wy when you roe righ ringle s we will see from he digrm elow. A roion of referene ringle y 90 degrees les you lk ou ngles nd heir omplemens. Look he ringle on he righ in he digrm elow nd you will see Cos[s] 3/5. Now onsider roing h ringle y 90 degrees, ounerlokwise. Th will give you he ringle on he lef. See how he legs of he ringle swih? Now onsider Cos[s + 90] nd see how h ends up s sking for he Cos[] -4/5. Noie h he osine is negive in his 2nd qudrn. Looking he roed ringle you n see h Cos[s+90] -Sin[s] -4/5 C:\ooks\QPk2008\geoTrig.fm 4/14/08 r.r 4
2nd qudrn 1s qudrn These kinds of digrms showing ngles nd heir omplemens le you disover ll of he rig omplemens. For exmple, (180 -s) is he ngle. Go hed nd see wh oher onneions you n eslish given h knowledge! The Seond Three Trig Funions: Cs, Se, Co This is he sme drwing s ove, u wih he oher hree rig funions inluded, ogeher wih heir represenive lenghs. Agin, he pln is o express rig funions in erms of simple lenghs. Before showing hese hree ddiionl funions, le me sep k nd give some kground for hese funions. A sen line of urve is line h inerses wo or more poins on he urve. The word sen omes from he Lin sere, for o u. You n use he sen line o pproxime he ngen o urve, some poin P y running he sen line hrough suessively loser poins on he urve o P. The limi of hese lines is he direion of ngen line o he urve he poin P. A hord is segmen of sen line where oh ends lie on he urve. These nex hree funions re he reiprols of he ig hree. Cosen[] Cs[] 1/Sin[] 3 Sen[] 1/Cos[] s (-1,0} 4 {0,0} Congen[] Co[] 1/Tn[] s 5 This le presens few ngles,, nd heir rig funions: Noe h Sqr[x] mens he squre roo of x. Some Common Angles, Their Trig Funions nd Lengh Equivlenes 5 3 Now you n see he omplee equivlenes eween he lenghs of he sides nd rig funionl vlues. For hese lulions I hve se side 1. For exmple, he Sin[0 degrees] 0 nd is equivlen o he lengh of side. The Cs[0] is he reiprol of he Sin[0] nd lows up. You n lso see his from he f h side g ges rirrily long. 4 {1,0} FIGURE 3. Complemenry Sin nd Cos: Cos[s + 90] Cos[] -4/5 -Sin[s] ANGLE (degrees) 0 0 1 0 VERY BIG 1 VERY BIG 30 1/2 Sqr[3]/2 1/Sqr[3] 2 2/Sqr[3] Sqr[3]/1 Sin[] Cos[] Tn[] d Cs[] g Se[] e Co[] f C:\ooks\QPk2008\geoTrig.fm 4/14/08 r.r 5
45 Sqr[2]/2 Sqr[2]/2 1 2/Sqr[2] 2/Sqr[2] 1 60 Sqr[3]/2 1/2 Sqr[3] 2/Sqr[3] 2 1/Sqr[3] 90 1 0 VERY BIG 1 VERY BIG VERY BIG {0,1} g e f d {1,0} * Noe: lengh 1 Sin[] / Cos[] / Tn[] / Tn[] d/1d Sin[] 1/g Cos[] 1/e Tn[] 1/f herefore: Cs[]1/Sin[] g Se[]1/Cos[] e Co[] 1/Tn[] f FIGURE 4. The Seond Three Trig Funions: Cosen, Sen, nd Congen Trig Ideniies We will need ouple of ideniies involving he funions shown ove. Sin[]^2 + Cos[]^2 1, h is sine squred plus osine squred, equls 1. Proof: From he drwing, nd definiion, Sin[] / so Sin[]^2 ^2/^2 nd, you n see h Cos[]^2 ^2/^2 So dding hese I ge: ^2/^2 + ^2/^2 (^2 + ^2)/^2 Bu: We know h ^2 + ^2 ^2, so he rio eomes 1 Sin[]^2 + Cos[]^2 1 QED. All Trig Ideniies Follow From Euler s Formul, A Shor Deour This is definiely opionl meril, unless you wn o know how o derive rig ideniies! If you do wn o know more ou rig ideniies I would reommend you find ou ou Euler s formul, lso lled DeMoivre s lw. Looking his formul requires shor exursion ino omplex vri- C:\ooks\QPk2008\geoTrig.fm 4/14/08 r.r 6
les u is no oo ough if you ke i in piees. Jus o whe your ppeie, le me se Euler s formul nd hen use i o find he doule ngle formul for Cos[2 ] nd Sin[2 ], in erms of Cos[] nd Sin[]. Euler s Formul The mos imporn nd euiful formul in ll of mhemis is Euler s, sed elow. I reles he rig funions o omplex numers in mnner h will llow you o derive ll of he rig funions jus y pplying some simple rules. Below is h formul where i is he squre roo of -1 nd e is speil numer h omes up hroughou mhemis nd is equl o 2.71828. I m using he leer x o snd for ny numer. X is usully inerpreed s n ngle in rdin mesure, u i n e ny numer whever. This supremely imporn formul reles omplex numer lying on he uni irle wih ssoied rig funions. x Cos@xD + Sin@xD * noe, x n e ny numer/symol ll, inluding s+, or s y iself Hs+L Cos@D + Sin@D s * Cos@sD + Sin@sD * FIGURE 5. Euler s Formul - where x n e ny numer (or symol) ll Deriving he Trig Ideniy Cos[2 ] Cos[]^2 - Sin[]^2 When you use omplex vriles, you ge wo resuls from eh single derivion. For exmple, in he derivion elow, lhough we only were looking o see how o express Cos[2] in erms of Cos[] nd Sin[], we lso go he doule ngle formul for Sin[2 ] wih no exr effor. From Eulers formul, le x or 2*, depending on wh you need, nd susiue in. Cos @D + Sin @D le x in Euler s formul * ( Cos @D + Sin @D ) * ( Cos @D + Sin @D ) Cos @D 2 + 2 Cos @D Sin @D Sin @D 2 * 2 use lw of exponens e.g. 3^2 * 3^2 3^4 2 Cos @2 D + Sin @2 D le x2 in Euler s formul Cos @D 2 + 2 Cos @D Sin @D Sin @D 2 muliplying omplex numers Now eque rel prs o rel prs nd imginry prs o imginry prs Cos @2 D Sin @2 D Cos @D 2 Sin @D 2 2Cos@D Sin @D his is he doule ngle formul for Cos[2] his is he doule ngle formul for Sin[2] FIGURE 6. Applying Euler s Formul o Derive he Trig Doule Angle Formul C:\ooks\QPk2008\geoTrig.fm 4/14/08 r.r 7
Deriving Cos[s+] nd Sin[s+] Le x s + in Eulers formul nd plug in! Hs+L Cos@s + D + Sin@s + D s Cos@s + D HCos@sD + Sin@sDL HCos@D + Sin@DL Cos@sD Cos@D + Cos@D Sin@sD + Cos@sD Sin@D Sin@sD Sin@D now eque rel nd imginry prs Cos@sD Cos@D Sin@sD Sin@D Sin@s + D Cos@D Sin@sD + Cos@sD Sin@D FIGURE 7. Cos[s+] Cos[s] Cos[] - Sin[s] Sin[] Inroduing he Lw of Cosines This nex rule les you find he lengh of side of ringle opposie wo given sides, even when i is no righ ringle. Look he digrm elow (Figure 8 on pge 8: B C B C A Given righ ringle ( 90 degrees) hen: C ^2 A ^2 + B ^2 In words: he squred lengh of C is he squred lengh of B plus he squred lengh of A. A If is no 90 degrees, hen he Pyhgoren heorem doesn (quie) hold. Anoher ideniy is used in his se, whih hs n exr erm ked on: (Lw of Cosines) C ^2 A ^2 + B ^2-2* A * B Cos[] FIGURE 8. Compring Side Lenghs of Righ Tringle wih Generl One Demonsrion of he Lw of Cosines The pln here is o desrie he lengh of side C, of generl ringle, in erms of he lenghs of A, B nd he ngle eween A nd B. I urns ou h i is esier o firs desrie how he squre of C reles o he squres of A nd B. Th is, how C ^2 reles o A ^2 nd B ^2. The reson for C:\ooks\QPk2008\geoTrig.fm 4/14/08 r.r 8
his is h we will e le o use he Pyhgoren heorem s omponen of he demonsrion. See if you gree wih he seps I will show nex: 1. drop perpendiulr from he ip of B o A. Th is he line whose lengh is denoed s d nd mke righ ngle wih A. Noie h his reks up he originl ringle ino wo righ ringles.th is rel progress sine we hve res he originl quesion y onsruing wo more quesions, u whih re simpler. Th is, we hve roken up generl ringle ino wo speil ones (righ ringles) h we know how o work wih. 2. he line segmen mrked e is he lengh from he ip of A o he se of d. 3. C ^2 d^2 + e^2 y Pyhogors (nd now you know how o prove h!). 4. d B Sin[] from he digrm, referring o your rig funion definiion of Sin[] 5. e A - B Cos[] 6. C ^2 d^2 + e^2 B ^2 Sin[]^2 + A ^2 + B ^2 Cos[]^2-2* A * B * Cos[] 7. rememer h sine squred plus osine squred is 1, So, for ou he B ^2 erm from he sine nd osine squres nd you will ge: 8. C ^2 d^2 + e^2 A ^2 + B ^2 {Sin[]^2 + Cos[]^2} -2* A * B * Cos[] 9. C ^2 A ^2 + B ^2-2* A * B QED.(quod er demonsrndom - whih ws wh hd o e proved) A generl ringle used o demonsre he lw of osines B d C A e C ^2 d^2 + e^2 ( B Sin[] )^2 + ( A - B Cos[] )^2 B ^2 {Sin[]^2 + Cos[]^2} + A ^2-2* A * B * Cos [] A ^2 + B ^2-2* A * B Cos[] FIGURE 9. Demonsrion of he Lw of Cosines Summry A his poin you hve he sis of rig nd n move on o he disovery of ddiionl rig ideniies suh s he doule ngle formuls or hlf ngle formuls. We won need hese in our sisis work in his ourse, u hey re hndy for generl d nlysis. From he lw of osines we will oninue on wih seion reled o veors nd heir relions o rig. Th will e in forh- C:\ooks\QPk2008\geoTrig.fm 4/14/08 r.r 9
oming uoril lled Veor Arihmei nd Veor Operions C:\ooks\QPk2008\geoTrig.fm 4/14/08 r.r 10