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Transcription

1 6. Spheril rignmetri Frmule Just s in plne gemetry, there re useful trignmetri frmule whih relte the sides nd vertex ngles f spheril tringles: Csine Frmul [6.1] s ss + sin sin s Sine Frmul [6.] sin sin sin sin sin sin C Fur Prts Frmul [6.3] s s sin tn sin tn nlgue Frmul [6.4] sin s ssin sin s s 1 Psitinl strnmy Pge 13 Leture 3

2 Emplying the spheril trignmetri frmule: he spheril sine frmul nd the spheril sine frmul re the mst frequently used frmule. he fur prts nd nlgue frmule re f sinl utility. he first three frmule relte 4 prts (sides nd verties) f spheril tringle, the nlgue frmul 5 prts. emply these frmule, identify t lest 3 prts f the tringle tht yu knw (4 fr the nlgue frmul) nd the unknwn prt tht yu wnt, nd hek whih frmul the nfigurtin f knwn nd unknwn prts reltes t. Exmples: Use sine Use sine Use fur prts C Use sine Use nlgue Use sine C he mjrity f prlems yu will enunter n e slved using just the sine nd sine frmule. Smetimes this will e multi-stge press where different frmule re used in suessin t slve fr different prts f the spheril tringle. 1 Psitinl strnmy Pge 14 Leture 3

3 Exmple: Use sine Use sine Sine nd sine frmule in plne gemetry: Suppse we nsider n extremely smll spheril tringle, ne where the sides,, nd re ll á 1. If suffiiently smll, this spheril tringle will e lmst flt, i.e. the sme s n rdinry tringle in plne gemetry. We n use the smll ngle pprximtins, equtins (3.1) nd (3.), n the sides f this tringle. Sine sin, et., the sine frmul will then eme sin sin sin C [6.5] i.e. just the fmilir sine frmul frm plne gemetry. Similrly, using s 1 /, the sine frmul emes s ss + sin sin s s + s 1 Psitinl strnmy Pge 15 Leture 3

4 Sine nd re lredy very smll, /4 will e extremely tiny, nd n e sfely negleted. S + s + s [6.6] i.e. just the fmilir sine frmul frm plne gemetry. Nte: Equtins (6.5) nd (6.6) nly pply t plne (flt) tringles, NO spheril tringles. 7. errestril C-rdintes ll spheril -rdinte systems re essentilly similr nsider the system f terrestril ltitude nd lngitude. his ssumes tht the Erth is perfet sphere, rtting rund its nrth-suth xis, i.e. rund the line etween the nrth (N) nd suth (S) gegrphi ples. Greenwih eridin N λ meridins f lngitude φ prllels f ltitude λ φ λ equtr xis f rttin he gret irle with N nd S s its ples is lled the equtr. Smll irles running prllel t the equtr re lled prllels f ltitude. Gret irles pssing thrugh N nd S re lled meridins f lngitude. Prllels f ltitude nd meridins f lngitude interset t right-ngles t eh ther. S 1 Psitinl strnmy Pge 16 Leture 3

5 he psitin f pint n the surfe f the Erth my e speified y its ltitude, φ, nd lngitude, λ. he equtr is nturl zer-pint fr ltitude: φ is the ngulr distne f pint frm the equtr, mesured lng meridin f lngitude running thrugh tht pint. here is n nturlly-urring zer-pint fr lngitude: y nventin we use the meridin pssing thrugh Greenwih. λ is the spheril ngle etween the Greenwih meridin nd the meridin f lngitude running thrugh the pint. he usul nventin is tht φ is psitive nrth f the equtr nd negtive suth, nd tht λ is psitive est f Greenwih nd negtive west. Exmple: Sine they n (nrmlly) ignre surfe fetures, lng-distne mmeril ir rutes re usully gret irle rs (t keep flight time nd fuel use t minimum). Suppse n irrft tkes ff frm sw (55 58 N, 37 5 E) nd lnds t ky (35 46 N, E). () Hw fr des it fly etween these ities () On wht ering will it e flying s it deprts sw nd s it pprhes ky Drw suitle terrestril sphere, shwing the ltin f Greenwih (G), nd mrking n knwn nd desired prts. Identify 3 knwn pints t frm useful spheril tringle. G λ φ N λ n φ S 1 Psitinl strnmy Pge 17 Leture 3

6 N 90 -φ λ -λ 90 -φ n C We knw sides nd their internl ngle, wnt the 3 rd side n use sine frmul. s ss + sin sin s s n smst + sin msin t s N s n s ( 90 φ ) s( 90 φ ) sin( 90 φ ) sin( 90 φ ) ( λ λ ) + s φ 55 58, λ 37 5, φ 35 46, λ Putting in the numers, ( ) s( 34 ) sin ( ) sin( 34 ) s( ) s n s + Slving fr n, we find tht n Definitin: he nutil mile (nm) is the distne whih rrespnds t 1 minute f gret irle r n the Erth s surfe. (1 nm 1.85 km) herefre, nm. nswer: () he irrft flies distne f 4051 nm etween sw nd ky. 1 Psitinl strnmy Pge 18 Leture 3