applications of trigonometry to triangles

8/13/2019 Applications of Trigonometry to Triangles

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Lppàfltaoks oj

]rabokodntryto ]ralkb`ns

=.=

Aktromuftaok

Rn orabakl`̀ y aktromufnm trabokodntry usakb rabct-lkb`nm tralkb`ns. Cownvnr, tcn sueinft cls lppà-

fltaoks ak mnlàkb 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 plrtalàkjordltaok

CN@D (577>)9Vnftaok =.=9 Lppàfltaoks 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




]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 vlùn 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`òws 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>◪.

Voùtaok

^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)


>>




Voùtaok (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 vlùn 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 vlùn 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 soùtaok

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




]cn ldeabuous flsnRcnk, ls ak Nxldp`n 6, wn lrn bavnk two samns lkm tcn kok-akfùmnm 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 adpàns 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 nlrànr tcnrn lrn two possae`n vlùns 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 soùtaok 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 possaeaàtans.Aj tcn bavnk samn e as brnltnr tclk tcn bavnk samn f, tcn oetusn lkb`n soùtaok 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 vlùns F <, F 5 oj tcn lkb`n F lkm as rnjnrrnm to ls tcn ldeabuousflsn. Ak tcas flsn tcnrn wa`̀ en two possae`n vlùns l< lkm l5 jor tcn tcarm samn oj tcn tralkb`nforrnspokmakb to tcn two lkb`n vlùns

L< 4 <37◪ ∕ (E + F <)L5 4 <37◪ ∕ (E + F 5)


>2




]lsh sh

Vcow tclt two tralkb`ns t tcn jo`òwakb 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 soùtaok

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 vlùns oj E lrn akmnnm possae`n sakfn eotc vlùns lrn `lrbnr tclk lkb`n L (samn eas òkbnr 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




]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


>1




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◪.

Voùtaok

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 vlùn as forrnft or wcntcnr tcas as tcn ldeabuous flsn lkm eotcvlùns oj E lrn possae`n.]cn two possae`n vlùns 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 vlùn F 5 4 51.1:◪ as tcn forrnft okn jor tcn tcarm samn.

]cn Fosakn ru`n flk l`so en lppànm 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




]lsh sh

L tralkb`n LEF cls samns

l 4 2fd e 4 << fd f 4 <5 fd.

Oetlak tcn vlùns oj l`` tcn lkb`ns oj tcn tralkb`n. (^sn Hny Woakt <1.)

Qour soùtaok

LkswnrVupposn wn km lkb`n L rst usakb tcn jo`òwakb 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 possaeaàty 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`òwakb 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◪


:<




Nxnrfasns

<. Mntnrdakn tcn rndlakakb lkb`ns lkm samns jor tcn jo`òwakb 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`àkb lt <5 hkots (klutafl` da`ns pnr cour) tcnotcnr lt <7 hkots. ]cn jlstnr scap dlaktlaks l enlrakb oj K =2◪R, tcn sòwnr okn l enlrakbV 57◪R. Fl`fu`ltn tcn snplrltaok oj tcn scaps lt dammly. (Cakt9 Mrlw lk lppropraltn malbrld.)

=. ]cn frlkh dnfclkasd scowk enòw cls lk lrd OL oj `nkbtc 67 dd rotltakb lktafòfhwasnleout 7 lkm l fokknftakb rom LE oj `nkbtc :7 dd. E dovns lòkb tcn corazoktl` àkn

OE. Rclt as tcn `nkbtc OE wcnk OL cls rotltnm ey <

3 oj l fodp`ntn rnvoùtaok jrod tcn

corazoktl`?

L

EO

:5 CN@D (577>)9Rorheooh =9 ]rabokodntry




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 vlùn 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 akfùmnm 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 possaeaàty 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 vlùns 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


:6




Lkswnrs foktakunm

5. Rn usn tcn Fosakn ru`n rst̀ y to km tcn lkb`n opposatn tcn òkbnst 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`àkb) 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 rnvoùtaok)

]cn oetusn vlùn 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


applications of trigonometry to triangles

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