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8/13/2019 Applications of Trigonometry to Triangles
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Lpp`afltaoks oj
]rabokodntryto ]ralkb`ns
=.=
Aktromuftaok
Rn orabakl`̀ y aktromufnm trabokodntry usakb rabct-lkb`nm tralkb`ns. Cownvnr, tcn sueinft cls lpp`a-
fltaoks ak mnl`akb watc lky tralkb`ns sufc ls tcosn tclt dabct lrasn ak survnyakb, klvabltaok or tcnstumy oj dnfclkasds.Ak tcas Vnftaok wn scow cow, bavnk fnrtlak akjordltaok leout l tralkb`n, wn flk usn lppropraltn ru`ns,fl``nm tcn Vakn ru`n lkm tcn Fosakn ru`n, to ju``y –so`vn tcn tralkb`n‑ a.n. oetlak tcn `nkbtcs oj l``tcn samns lkm tcn sazn oj l`` tcn lkb`ns oj tclt tralkb`n.
Wrnrnquasatns
Enjorn stlrtakb tcas Vnftaok you scou`m . . .
• clvn l hkow`nmbn oj tcn elsafs oj trabokodntry
• en lwlrn oj tcn stlkmlrm trabokodntrafamnktatans
@nlrkakb Outfodns
Ok fodp`ntaok you scou`m en le`n to . . .
• usn trabokodntry ak nvnrymly satultaoks
• ju``y mntnrdakn l`` tcn samns lkm lkb`ns lkmtcn lrnl oj lky tralkb`n jrod plrtal`akjordltaok
CN@D (577>)9Vnftaok =.=9 Lpp`afltaoks oj ]rabokodntry to ]ralkb`ns
>6
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<. Lpp`afltaoks oj trabokodntry to tralkb`ns
Lrnl oj l tralkb`n
]cn lrnl V oj lky tralkb`n as bavnk ey V 4 <
5 Ò(elsn)
Ò(pnrpnkmafu`lr cnabct) wcnrn –pnrpnkmafu`lr
cnabct‑ dnlks tcn pnrpnkmafu`lr mastlkfn jrod tcn samn fl``nm tcn –elsn‑ to tcn opposatn vnrtnx. ]cus
jor tcn rabct-lkb`nm tralkb`n scowk ak Jaburn 66(l) V 4 <
5 e l. Jor tcn oetusn-lkb`nm tralkb`n
scowk ak Jaburn 66(e) tcn lrnl as V 4 <
5ec.
l
e
L
E
F
fl
e
L
E
F
f
F
c
Mγ γ
(l) (e)
Jaburn 66
Aj wn usn F to mnkotn tcn lkb`n LFE ak Jaburn 66(e) tcnk
sak(<37 ∕ F ) 4 c
l (tralkb`n EF M as rabct-lkb`nm)
. .. c 4 l sak(<37 ∕ F ) 4 l sak F (snn tcn brlpc oj tcn sakn wlvn or nxplkm sak(<37 ∕ f))
. .. V 4 <
5 e l sak F <(l)
Ey otcnr sada`lr fokstruftaoks wn fou`m mndokstrltn tclt
V 4 <
5 l f sak E <(e)
lkm
V 4 <
5 e f sak L <(f)
Kotn tcn plttnrk cnrn9 ak nlfc jordu`l jor tcn lrnl tcn lkb`n akvo`vnm as tcn okn entwnnk tcn samnswcosn `nkbtcs offur ak tclt nxprnssaok.
F`nlr`y aj F as l rabct-lkb`n (so sak F 4 <) tcnk
V 4 <
5 e l ls jor Jaburn 66(l).
Kotn9 jrod kow ok wn wa`` kot bnknrl``y wratn –≬‑ eut usn tcn dorn usul` –4‑.
>= CN@D (577>)9Rorheooh =9 ]rabokodntry
8/13/2019 Applications of Trigonometry to Triangles
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]cn Vakn ru`n
]cn Vakn ru`n as l jordu`l wcafc, aj wn lrn bavnk fnrtlak akjordltaok leout l tralkb`n, nkle`ns us toju``y –so`vn tcn tralkb`n‑ a.n. oetlak tcn `nkbtcs oj l`` tcrnn samns lkm tcn vl`un oj l`` tcrnn lkb`ns.]o scow tcn ru`n wn kotn tclt jrod tcn jordu`ln (<l) lkm (<e) jor tcn lrnl V oj tcn tralkb`n LEF
ak Jaburn 66 wn clvnel sak F 4 lf sak E or
e
sak E 4
f
sak F
Vada`lr`y usakb (<e) lkm (<f)
lf sak E 4 ef sak L or l
sak L 4
e
sak E
Hny Woakt <3]cn Vakn Uu`n
Jor lky tralkb`n LEF wcnrn l as tcn `nkbtc oj tcn samn opposatn lkb`n L, e tcn samn `nkbtc opposatnlkb`n E lkm f tcn samn `nkbtc opposatn lkb`n F stltns
l
sak L 4
e
sak E 4
f
sak F
^sn oj tcn Vakn ru`n
]o en le`n to ju``y mntnrdakn l`` tcn lkb`ns lkm samns oj l tralkb`n at jo``ows jrod tcn Vakn ru`n tcltwn dust hkow
natcnr two lkb`ns lkm okn samn 9 (hkowakb two lkb`ns oj l tralkb`n rnl``y dnlks tclt l``tcrnn lrn hkowk sakfn tcn sud oj tcn lkb`ns as <37◪)
or two samns lkm lk lkb`n opposatn okn oj tcosn two samns.
Nxldp`n 6Vo`vn tcn tralkb`n LEF bavnk tclt l 4 65 fd, e 4 =: fd lkm lkb`n E 4 :6.5>◪.
Vo`utaok
^sakb tcn rst plar oj nqultaoks ak tcn Vakn ru`n (Hny Woakt <3) wn clvn
65
sak L 4
=:
sak:6.5>◪. .. sak L 4
65
=: sak :6.5>◪ 4 7.:5<5
so L 4 sak∕<
(7.:5<5) 4 63.=◪
(ey fl`fu`ltor)
CN@D (577>)9Vnftaok =.=9 Lpp`afltaoks oj ]rabokodntry to ]ralkb`ns
>>
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Vo`utaok (foktm.)
Qou scou`m, cownvnr, kotn flrnju``y tclt enflusn oj tcn jord oj tcn brlpc oj tcn sakn jukftaok tcnrnlrn two lkb`ns entwnnk 7◪ lkm <37◪ wcafc clvn tcn sldn vl`un jor tcnar sakn a.n. x lkm (<37∕ x).Vnn Jaburn 6=.
x <37◪∕ x
sak γ
γ
Jaburn 6=
Ak our nxldp`n
L 4 sak∕<(7.:5<5) 4 63.=◪
or
L 4 <37◪ ∕ 63.=◪ 4 <=<.:◪.
Cownvnr sakfn wn lrn bavnk tclt lkb`n E as :6.5>◪, tcn vl`un oj <=<.:◪ jor lkb`n L as f`nlr`yadpossae`n.
]o fodp`ntn tcn proe`nd wn sadp`y kotn tclt
F 4 <37◪ ∕ (63.=◪ + :6.5>◪) 4 23.6>◪
]cn rndlakakb samn f as fl`fu`ltnm jrod tcn Vakn ru`n, usakb natcnr l lkm sak L or e lkm sak E.
]lsh sh
Jakm tcn `nkbtc oj samn f ak Nxldp`n 6.
Qour so`utaok
Lkswnr
^sakb, jor nxldp`n, l
sak L 4
f
sak F
wn clvn f 4 lsak F
sak L 4 65 Òsak 23.6>◪
7.:5<5 4 65
Ò7.121=
7.:5<5 4 >7.=> fd.
>: CN@D (577>)9Rorheooh =9 ]rabokodntry
8/13/2019 Applications of Trigonometry to Triangles
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]cn ldeabuous flsnRcnk, ls ak Nxldp`n 6, wn lrn bavnk two samns lkm tcn kok-akf`umnm lkb`n oj l tralkb`n, plrtafu`lrflrn as rnquarnm.Vupposn tclt samns e lkm f lkm tcn lkb`n E lrn bavnk. ]cnk tcn lkb`n F as bavnk ey tcn Vakn ru`n ls
L
E
F sak F 4 f
sak E
e
e
f l
Jaburn 6>Zlraous flsns flk lrasn9
(a) f sak E ; e
]cas adp`ans tclt f sak E
e ; < ak wcafc flsn ko tralkb`n nxasts sakfn sak F flkkot nxfnnm <.
(aa) f sak E 4 e
Ak tcas flsn sak F 4 f sak E
e 4 < so F 4 17◪.
(aaa) f sak E 0 e
Cnkfn sak F 4 f sak E
e 0 <.
Ls dnktaoknm nlr`anr tcnrn lrn two possae`n vl`uns oj lkb`n F ak tcn rlkbn 7 to <37◪, okn lfutn lkb`n(0 17◪) lkm okn oetusn (entwnnk 17◪ lkm <37◪.) ]cnsn lkb`ns lrn F < 4 x lkm F 5 4 <37 ∕ x. VnnJaburn 6:.Aj tcn bavnk lkb`n E as brnltnr tclk 17◪ tcnk tcn oetusn lkb`n F 5 as kot l possae`n so`utaok enflusn,
oj foursn, l tralkb`n flkkot possnss two oetusn lkb`ns.
e
L
E
f
e
F <F 5E F <F 5
Jaburn 6:Jor E `nss tclk 17◪ tcnrn lrn sta`` two possaea`atans.Aj tcn bavnk samn e as brnltnr tclk tcn bavnk samn f, tcn oetusn lkb`n so`utaok F 5 as kot possae`n enflusntcnk tcn `lrbnr lkb`n wou`m en opposatn tcn sdl``nr samn. (]cas wls tcn satultaok ak Nxldp`n 6.)]cn kl` flsn
e 0 f, E 0 17◪
mons bavn rasn to two possae`n vl`uns F <, F 5 oj tcn lkb`n F lkm as rnjnrrnm to ls tcn ldeabuousflsn. Ak tcas flsn tcnrn wa`̀ en two possae`n vl`uns l< lkm l5 jor tcn tcarm samn oj tcn tralkb`nforrnspokmakb to tcn two lkb`n vl`uns
L< 4 <37◪ ∕ (E + F <)L5 4 <37◪ ∕ (E + F 5)
CN@D (577>)9Vnftaok =.=9 Lpp`afltaoks oj ]rabokodntry to ]ralkb`ns
>2
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]lsh sh
Vcow tclt two tralkb`ns t tcn jo``owakb mltl jor l tralkb`n LE F 9
l 4 =.> fd e 4 2 fd L 4 6>◪
Oetlak tcn samns lkm lkb`n oj eotc possae`n tralkb`ns.
Qour so`utaok
Lkswnr
Rn clvn, ey tcn Vakn ru`n, sak E 4 e sak L
l 4
2sak6>◪
=.> 4 7.3155
Vo E 4 sak∕< 7.3155 ∕ :6.<>◪ (ey fl`fu`ltor) or <37 ∕ :6.<>◪ 4 <<:.3>◪.
Ak tcas flsn, eotc vl`uns oj E lrn akmnnm possae`n sakfn eotc vl`uns lrn `lrbnr tclk lkb`n L (samn eas `okbnr tclk samn l). ]cas as tcn ldeabuous flsn watc two possae`n tralkb`ns.
E 4 E< 4 :6.<>◪ E 4 E5 4 <<:.3>◪
F 4 F < 4 3<.3>◪ F 4 F 5 4 53.<>◪
f 4 f< wcnrn f<
sak 3<.3>◪4
=.>
sak6>◪f 4 f5 wcnrn
f5
sak 53.<> 4
=.>
sak6>◪
f< 4 =.>Ò 7.1311
7.>26: f5 4
=.> Ò 7.=2<3
7.>26:
4 2.2:: fd 4 6.27< fd
Qou flk f`nlr`y snn tclt wn clvn okn lfutn lkb`nm tralkb`n LE<F < lkm okn oetusn lkb`nm LE5F 5forrnspokmakb to tcn bavnk mltl.
>3 CN@D (577>)9Rorheooh =9 ]rabokodntry
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]cn Fosakn ru`n
]cn Fosakn ru`n as lk l`tnrkltavn jordu`l jor –so`vakb l tralkb`n‑ LEF . At as plrtafu`lr`y usnju` jortcn flsn wcnrn tcn Vakn ru`n flkkot en usnm, a.n. wcnk two samns oj tcn tralkb`n lrn hkowk tobntcnrwatc tcn lkb`n entwnnk tcnsn two samns.
Foksamnr tcn two tralkb`ns LEF scowk ak Jaburn 62.
l
E
f
L
f
E
LF
l
LL M
F M
e e
(l) (e)
Jaburn 62
Ak Jaburn 62(l) usakb tcn rabct-lkb`nm tralkb`n LEM, EM 4 f sak L.
Ak Jaburn 62(e) usakb tcn rabct-lkb`nm tralkb`n LEM, EM 4 f sak(χ∕
L) 4 f sak L.
Ak Jaburn 62(l) ML 4 f fos L . .. FM 4 e ∕ f fos L
Ak Jaburn 62(e) ML 4 f fos(<37 ∕L) 4 ∕f fos L . .. FM 4 e + LM 4 e ∕ f fos L
Ak eotc flsns, ak tcn rabct-lkb`nm tralkb`n EMF
(EF )5 4 (FM)5 + (EM)5
Vo, usakb tcn leovn rnsu`ts,
l5 4 (e∕ f fos L)5 + f5(sak L)5 4 e5 ∕ 5ef fos L + f5(fos5 L + sak5 L)
bavakb
l5 4 e5 + f5 ∕ 5ef fos L (6)
Nqultaok (6) as okn jord oj tcn Fosakn ru`n. F`nlr`y at flk en usnm, ls wn stltnm leovn, to fl`fu`ltntcn samn l aj tcn samns e lkm f lkm tcn akf`umnm lkb`n L lrn hkowk.
Kotn tclt aj L 4 17◪, fos L 4 7 lkm (6) rnmufns to Wytclborls‑ tcnornd.
]wo sada`lr jordu`ln to (6) jor tcn Fosakn ru`n flk en sada`lr`y mnravnm - snn jo``owakb Hny Woakt9
CN@D (577>)9Vnftaok =.=9 Lpp`afltaoks oj ]rabokodntry to ]ralkb`ns
>1
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Hny Woakt <1
Fosakn Uu`n
Jor lky tralkb`n watc samns l, e, f lkm forrnspokmakb lkb`ns L, E, F
l5 4 e5 + f5 ∕ 5ef fos L fos L 4 e5 + f5 ∕ l5
5ef
e5 4 f5 + l5 ∕ 5fl fos E fos E 4 f5 + l5 ∕ e5
5fl
f5 4 l5 + e5 ∕ 5ef fos F fos F 4 l5 + e5 ∕ f5
5le
Nxldp`n =Vo`vn tcn tralkb`n wcnrn e 4 2.77 fd, f 4 6.>1 fd, L 4 =2◪.
Vo`utaok
Vakfn two samns lkm tcn lkb`n L entwnnk tcnsn samns as bavnk wn dust rst usn tcn Fosakn ru`n aktcn jord (6l)9
l5 4 (2.77)5 + (6.>1)5 ∕ 5(2.77)(6.>1)fos=2◪ 4 =1 + <5.333 ∕ 6=.522 4 52.:<7
so l 4∙
52.:<7 4 >.5>> fd.
Rn flk kow dost nlsa`y usn tcn Vakn ru`n to so`vn okn oj tcn rndlakakb lkb`ns9
2.77
sak E 4
>.5>>
sak =2◪so sak E 4
2.77sak=2◪
>.5>> 4 7.12=5
jrod wcafc E 4 E< 4 2:.1:◪ or E 4 E5 4 <76.7=◪.
Lt tcas stlbn at as kot oevaous wcafc vl`un as forrnft or wcntcnr tcas as tcn ldeabuous flsn lkm eotcvl`uns oj E lrn possae`n.]cn two possae`n vl`uns jor tcn rndlakakb lkb`n F lrn
F < 4 <37◪ ∕ (=2◪ + 2:.1:◪) 4 >:.7=◪
F 5 4 <37◪ ∕ (=2 + <76.7=) 4 51.1:◪
Vakfn jor tcn samns oj tcas tralkb`n e ; l ; f tcnk sada`lr`y jor tcn lkb`ns wn dust clvnE ; L ; F so tcn vl`un F 5 4 51.1:◪ as tcn forrnft okn jor tcn tcarm samn.
]cn Fosakn ru`n flk l`so en lpp`anm to sodn tralkb`ns wcnrn tcn `nkbtcs l, e lkm f oj tcn tcrnn samnslrn hkowk lkm tcn ok`y fl`fu`ltaoks knnmnm lrn kmakb tcn lkb`ns.
:7 CN@D (577>)9Rorheooh =9 ]rabokodntry
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]lsh sh
L tralkb`n LEF cls samns
l 4 2fd e 4 << fd f 4 <5 fd.
Oetlak tcn vl`uns oj l`` tcn lkb`ns oj tcn tralkb`n. (^sn Hny Woakt <1.)
Qour so`utaok
LkswnrVupposn wn km lkb`n L rst usakb tcn jo``owakb jordu`l jrod Hny Woakt <1
fos L 4 e5 + f5 ∕ l5
5ef
Cnrn fos L 4 <<5 + <55 ∕ 25
5Ò << Ò <5 4 7.3<3 so L 4 fos∕<(7.3<3) 4 6>.<◪
(]cnrn as ko otcnr possaea`aty entwnnk 7◪ lkm <37◪ jor L. Ko –ldeabuous flsn‑ lrasns usakb tcnFosakn ru`n!)
Lkotcnr lkb`n E or F fou`m kow en oetlaknm usakb tcn Vakn ru`n or tcn Fosakn ru`n.
^sakb tcn jo``owakb jordu`l jrod Hny Woakt <19
fos E 4 f5 + l5 ∕ e5
5fl 4
<55 + 25 ∕ <<5
5Ò <5 Ò 2 4 7.=51 so E 4 fos∕<(7.=51) 4 :=.:◪
Vakfn L + E + F 4 <37◪ wn flk mnmufn F 4 37.6◪
CN@D (577>)9Vnftaok =.=9 Lpp`afltaoks oj ]rabokodntry to ]ralkb`ns
:<
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Nxnrfasns
<. Mntnrdakn tcn rndlakakb lkb`ns lkm samns jor tcn jo``owakb tralkb`ns9
l
f :
L
E F
<67◪
57◪
(l)
L
E F l
6 =
37
◪
F
(e)
<7 e
<5
L
E F 5:◪
(f)
(m) ]cn tralkb`ns LEF watc E 4 >7◪, e 4 >, f 4 :. (]lhn spnfal` flrn cnrn!)
5. Mntnrdakn l`` tcn lkb`ns oj tcn tralkb`ns LEF wcnrn tcn samns clvn `nkbtcs l 4 2, e 4 ::lkm f 4 1
6. ]wo scaps `nlvn l port lt 3.77 ld, okn trlvn``akb lt <5 hkots (klutafl` da`ns pnr cour) tcnotcnr lt <7 hkots. ]cn jlstnr scap dlaktlaks l enlrakb oj K =2◪R, tcn s`ownr okn l enlrakbV 57◪R. Fl`fu`ltn tcn snplrltaok oj tcn scaps lt dammly. (Cakt9 Mrlw lk lppropraltn malbrld.)
=. ]cn frlkh dnfclkasd scowk en`ow cls lk lrd OL oj `nkbtc 67 dd rotltakb lktaf`ofhwasnleout 7 lkm l fokknftakb rom LE oj `nkbtc :7 dd. E dovns l`okb tcn corazoktl` `akn
OE. Rclt as tcn `nkbtc OE wcnk OL cls rotltnm ey <
3 oj l fodp`ntn rnvo`utaok jrod tcn
corazoktl`?
L
EO
:5 CN@D (577>)9Rorheooh =9 ]rabokodntry
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Lkswnrs
<.
(l) ^sakb tcn Vakn ru`n l
sak <67◪
4 :
sak 57◪
4 f
sak F
. Jrod tcn two `njt-clkm nqultaoks
l 4 :sak <67◪
sak 57◪ <6.==.
]cnk, sakfn F 4 67◪, tcn rabct clkm plar oj nqultaoks bavn f 4 :sak67◪
sak57◪ 3.22.
(e) Lblak usakb tcn Vakn ru`n l
sak L 4
=
sak 37◪4
6
sak F so sak F 4
6
= sak 37◪ 4 7.263:
tcnrn lrn two possae`n lkb`ns sltasjyakb sak F 4 7.263: or F 4 sak∕<(7.263:).
]cnsn lrn =2.:<◪ lkm <37◪ ∕ =2.:<=◪ 4 <65.61◪. Cownvnr tcn oetusn lkb`n vl`un asadpossae`n cnrn enflusn tcn lkb`n E as 37◪ lkm tcn sud oj tcn lkb`ns wou`m tcnk nxfnnm
<37◪
Cnkfn f 4 =2.7<◪
so L 4 <37◪ ∕ (37
◪
+ =2.:<◪
) 4 >5.61◪
.
]cnk, l
sak >5.61◪4
=
sak37◪so l 4 =
sak >5.61◪
sak37◪ 6.55
(f) Ak tcas flsn sakfn two samns lkm tcn akf`umnm lkb`n lrn bavnk wn dust usn tcn Fosakn ru`n.]cn lppropraltn jord as
e5 4 f5 + l5 ∕ 5fl fos E 4 <75 + <55 ∕ (5)(<7)(<5) fos 5:◪ 4 53.531=
so e 4∙
53.531= 4 >.65
Foktakuakb wn usn tcn Fosakn ru`n lblak to mntnrdakn sly lkb`n F wcnrn
f5 4 l5 + e5 ∕ 5le fos F tclt as <75 4 <55 + (>.65)5 ∕ 5(<.5)(>.65)fos F
jrod wcafc fos F 4 7.>::6 lkm F 4 >>.><◪ (]cnrn as ko otcnr possaea`aty jor F entwnnk7◪ lkm <37◪. Unfl`̀ tclt tcn fosakn oj lk lkb`n entwnnk 17◪ lkm <37◪ as knbltavn.)Jakl``y, L 4 <37 ∕ (5:◪ + >>.><◪) 4 13.=1◪.
(m) Ey tcn Vakn ru`n
l
sak L 4
>
sak >7◪4
:
sak F . .. sak F 4 :
sak>7◪
> 4 7.1<16
]cnk F 4 sak∕<(7.1<16) 4 ::.35◪ (fl`fu`ltor) or <37◪∕ ::.35◪ 4 <<6.<3◪. Ak tcas flsneotc vl`uns oj F sly F < 4 ::.35◪ lkm F 5 4 <<6.<3◪ lrn possae`n lkm tcnrn lrn twopossae`n tralkb`ns sltasjyakb tcn bavnk mltl. Foktakunm usn oj tcn Vakn ru`n promufns
(a) watc F < 4 ::.35 (lfutn lkb`n tralkb`n) L 4 L< 4 <37∕ (::.35◪ + >7◪) 4 :6.<3◪
l 4 l< 4 >.36
(aa) watc F 5 4 <<6.<3◪ L 4 L5 4 <:.35◪ l 4 l5 4 <.31
CN@D (577>)9Vnftaok =.=9 Lpp`afltaoks oj ]rabokodntry to ]ralkb`ns
:6
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Lkswnrs foktakunm
5. Rn usn tcn Fosakn ru`n rst̀ y to km tcn lkb`n opposatn tcn `okbnst samn. ]cas wa`` tn`` uswcntcnr tcn tralkb`n foktlaks lk oetusn lkb`n. Cnkfn wn so`vn jor f usakb
f5 4 l5 + e5 ∕ 5le fos F 3< 4 =1 + 6:∕ 3=fos F
jrod wcafc 3= fos F 4 = fos F 4 =/3= bavakb F 4 32.52◪.
Vo tcnrn as ko oetusn lkb`n ak tcas tralkb`n lkm wn flk usn tcn Vakn ru`n hkowakb tclt tcnrnas ok`y okn possae`n tralkb`n ttakb tcn mltl. (Rn fou`m foktakun to usn tcn Fosakn ru`n aj wnwascnm oj foursn.) Fcoosakb to km tcn lkb`n E wn clvn
:
sak E 4
1
sak32.52◪
jrod wcafc sak E 4 7.::>1 bavakb E 4 =<.2>◪
. (]cn oetusn flsn jor E as kot possae`n, lsnxp`laknm leovn.) Jakl``y L 4 <37◪ ∕ (=<.2>◪ + 32.52◪) 4 >7.13◪.
6.
L
E
O
K
V
f=3
=757◪
=2◪
Lt dammly (= cours trlvn``akb) scaps L lkm E lrn rnspnftavn`y =3 lkm =7 klutafl` da`ns jrodtcn port O. Ak tralkb`n LOE wn clvn
LOE 4 <37◪ ∕ (=2◪ + 57◪) 4 <<6◪.
Rn dust usn tcn Fosakn ru`n to oetlak tcn rnquarnm mastlkfn lplrt oj tcn scaps. Mnkotakb tcnmastlkfn LE ey f, ls usul`,
f5 4 =35 + =75 ∕ 5(=3)(=7) fos <<6◪ jrod wcafc f5 4 >=7=.=< lkm f 4 26.> klutafl` da`ns.
=. Ey tcn Vakn ru`n 67
sak E 4 :7
sak=> . .. sak E 4 67
:7 sak =>◪
4 7.6>6 so E 4 57.27=◪
.
L
EO =>◪
67dd :7dd (Wosataok ljtnr <
3 rnvo`utaok)
]cn oetusn vl`un oj sak∕<(7.6>6) as adpossae`n. Cnkfn,
L 4 <37◪ ∕ (=>◪ + 57.27=◪) 4 <<=.51:◪.
^sakb tcn sakn ru`n lblak 677.6>6
4 OEsak <<=.51:
jrod wcafc OE 4 22.> dd.
:= CN@D (577>)9Rorheooh =9 ]rabokodntry