U!JJl!Hlfflir. .. ; ... .,. -·- ~ ~ _

TPJ1f0HOMETPJ1JA ITPABOYfJIOf TPOYfJIA

'

Tpu:roHOMeTpnjcKe cl>y1uai;nje omTpHX yrnosa (346)

A .LJ;eqnrnm.:i;wje TpwroHoMeTpwjcKMX cpy1na.:i;wja ormpor yrna (346)

E HeKe je,n;ttocTaBHe ocoforne TpwroHoMeTpwjcKMX cp)'HKIJ;Hja (349)

Bpeµ;HOCTH TpHrOHOMeTpnjcKHX ci>ymai;Hja omTpHX ymosa (350)

A Bpe,n;HOCTH TpwroHoMeTpwjcKmc cp)'HKIJ;Hja yrnoBa o,n; 45°, 30° w 60°; yrroTpe5a KaJIK)'llaTopa IIpHJIHKOM H3paq)'HaBaJhe Bpe,n;HOCTH TpwroHoMeTpwjcKMX cp)'HKIJ;Hja ormpHX yrnoBa (350)

OcHOBHH TpnroHoMeTpnjcKH nµ;eHTHTeTH (352)

E Be3e Mel)y TpwroHoMeTpwjcKHM cp)'HKIJ;HjaMa; ornoBHH TpHrOHOMeTpHjCKH H,D;eHn1TeTH (352)

IlpHMeHe TpHroHoMeTpnjcKHX cl>YHKD;Hja (354)

A PeIIIaBalhe rrpaBoyrnor Tpoyrna (354)

E IlpwMeHe TpwroHOMeTpwje y reoMeTpwjw (356)

, ~ . ,.a·--(_4Q

, ir ....... ·~

.... !fe-1&-r.1

A

Pe'>: wpuloHOMeiUpuja e Ip'IKOr nopeIGia n

cacraB;neHa je o.:i: pe'Ilt wpufatt = wpoyuzo It

Meiiip<m = Me{Jefbe..

Iloc.rre;:i;u:na ;:q>yror

CTaBa Gill'IHOCill

~.\.Ko ;rBa rrpasoyma

rpoyma HMajy jeµa.H uCTif orrrrap yrao,

oH;Jajepa.3.Mepa.JBe crpa.Hn~e y je4HOM

rpoymy jeµ;Ra.Ka pa.n!epu o;rroBapajytrnx

crpamma y ilPYroM rpoyrny.

reopeMa

346

TpJIToHoMeTp:ujcKe <l>y111ru;:uje omTp:ux yrrrosa

IlpHJt:ep 1.

TanecoBa JVJ,eja ~ MepH BHCJIBe ocSjeKaTa Mepeh.H A}')KJ'IHe H>HXOBHX ceHKH 3acH0BaHa

je, Kao IIITO CMO Ben BH):\eillf, Ha ,n;pyroM CTaBY CJIWIHOCTH. 3arrpaBO, y cSmrn KOM

rpeeyncyTOKOM ,n;aea rrpaBoyrJIH TpoyrnoBH Koje cpopMHpajy ocSjeKTH ca CBOjHM

ceHKaMa, Mel)yco8Ho cy CJIWIID1.

IloornrpajMo nojeAHOCTaBJbeH}' CillfKY.

/1 s~

CRH yoqeHH rrpaiwyrJIH TpoyrnoBH HMajy HCTe yrnoBe. Crre~ttjanHo, yrao a rro,n; KojHM

cywieBH 3p~ nap;ajy Ha mo 3aje,n;mr<nrn: je CBHMa H>HMa. TaKol)e, y CBaKOM o,n; OBHX

rpoyrnoBa Hera je H pa3Mepa KaTeTe Kaja je HacrrpaM yrna a H KaTeTe Ha Kojy OBaj yrao

Hane)l(e, rj.

v v v _!_ =_1_ =-1. = t . • s1 Sz SJ

33.KJb)"IaI< ,n;o Kora CMO )l;OIIIJIH y rrpeTXO,ll;HOM rrpHMepy HCTHqeMO rne,n;ehoM TeopeMOM

Koja µ;J-IpeKTHO CJie,n;H H3 ,n;pyror CTaBa CJIHqHQCTH.

AKo je a HeKH oIIITap yrao, Ta,n;a 3a cBaKH rrpaBoyrnH Tpoyrao 'IBjH je je,n;aH onITap

yrao a, pa3Mepa KaTeTe Kaja je HacrrpaM yrna a H KaTeTe Ha Kojy OBaj yrao HaJie:>Ke

jecTe KOHCTaHTHa (cTaJIHa).

HapaBHO, aeanorua TeopeMa Ba)l(H H y CJI)"Iajy Ka,n;a ce YMecTo KaTeTa rrocMaTpa HeKH

,n;pyrn nap crp~a rrpaBoyrnor Tpoyrna, Tj. XHIIOTeHy3a H je,n;Ha o,n; KaTeTa.

OAI OA2 OAJ --=--=--=··· A1B1 A1B2 AJBJ OA 1 OA2 OAJ --=--=--=··· OBI OB2 OBJ

AIBl A1B2 AJBJ --=--=--=··· OBI OB2 OBJ

,

HeKa je a HeK11 ounap yrao (0° <a< 90°) 11 ABC HeKM rrpaBoyrIDf rpoyrao ca rrpaBHM yrnoM y TeMetty C 11 yrnoM a y TeMetty A. Kao IIITO je yo8J1'Iajetto, KaTeTy BC Koja je ttacrrpaMHa yrny a o8eJie)lmneMo ca a, KaTeTy AC Ha Kojy HaJie)l(e yrao a 03Hatru.beMo ca b, a XMrroTetty3y AB ca c.

XJ1IlOTeHY3a c a ttacrrpaMHa KaTeTa

,/

a

b Hanerna KaTeTa

Pa3Mepa ttacrrpaMHe KaTeTe a 11 ttanerne KaTeTe b Ha3MBa ce Ta.Hreuc yrna a H o8eJie)l(aBa ce ca tg a:

a tga=-.

b

Pa3Mepa Hanerne KaTeTe b 11 HacrrpaMHe KaTeTe a Ha3MBa ce KOTa.Hreec yrna a n o8eJie)l(aBa ce ca ctg a:

b ctga=-.

a

Pa3Mepa HacrrpaMHe KaTeTe a 11 XMITOTetty3e c Ha311Ba ce cmryc ynrn a H o8eJie)l(aBa ceca sin a:

. a Sill a=-.

c

Pa3Mepa Hanerne KaTeTe b 11 XMrroTetty3e c Ha3HBa ce KoCHeyc yrna a 11 o8eJie)l(aBa ceca cos a:

L-

IlpHMep2.

b cos a=-.

c

HeKa cy a 11 p OIIITPH yrnoB11 npaBoyrnor Tpoyrna 'IMje cy CTpamfl\e 3, 4 H 5, rrpu qeMY je a< p.

3 4 tga=-. tg P=-.

4 3 4 3

ctga=-. 3 ctg ~=-. 3 4 3 . ~ 4 sin a=-.

4 Sill =-. 5 5 4 3

cos a=-. cos p =-:-· • 5 ::>

2 " n

tg a=~·

KC\T(ffrr

c/a ~

b .. ~

ctg a= a ·

C.illifC

b

in a=-.

~a ~

h -iJ

cos a=-.

347

-D

Ilo:rceha.Mo Ha :i:oroBop :::ra yrao 11 H>eroBy .Mepy y cren:eHKM.a 03Haq<rn~m

UCTIIM CJl.MbOilO~ I.

reopeMa

IlpHMep 3.

Om>e.llJIMO crpamt:Qe rrpa.Boyrnor rpoyrna ABC aKO je c = 1 Ocm H sin a= 0,6.

K · · a s:: • · a 06 · 6 aim Je sm a=-, ,QOuHJaMO p;a Je - = , , TJ. a= cm. c 10

IlpID!eHoM ThrraropJme TeopeMe ,Qofa1jaMo ,Qa je

b =.../c2-a2 =8cm. •

Tae:reHc, KOTaHTeHc, cmryc H Koarnyc npe,QCTaBJbajy TaK03BaHe Tp11:roH0MeTpHjcKe

cpymrn;nje onnpHX yrnosa. ·

L{a 8ncMo OA-pep,Jun1 spe,QHOCT HeKe OA TPHrOHOMeTpHjcKHX

¢}'1IKQJ1ja ;:i;aTor orrrrpor yrrra a, p;OBOJbHO je H3a8paTH OHJIO

KojJ1 rrpasoyrIDI rpoyrao 'IltjH je jep;att yrao jep;HaK a H ttahH

pa3Mepy owoBapajyhux crpaHJm;a. a

je,QHHH~a

IlpHMep4.

HeKa j e 11:aT oIIITa.p yrao a.

Axo ma8epeMo rrpasoyrIDI Tpoyrao qJ1jH je je.n;aH yrao a H

Hanen ra KaTeTa no;zy;n;apHa jeAJfHH~H Mepe, ott.n;a je Jzy)KHHa

Hacnpa.MHe KaTeTe j e)J;HaKa tg a.

----Mepe

tg a

1

IIocrynajyhH aHarrorno, MO)l(eMo (KoHcTp)'KTHBHO) oA-pe,D;HTH H Bpe,QHOCT 8Hno Koje

µ;pyre rpuroHOMeTpHjcKe <f>ytt~:11je .n;aTor yrna a.

J. sin a

ctg a cos a

3a csaKH onITap yrao a,

tg a :11 ctg a Mory OHTH OHrro Koj:11 II03HTHBHH peaJIHH 8pojesH:

0 < tg a < + oo H 0 < ctg a < + oo;

•

sin a l1 cos a Mory bHTH CaMO Il03HTl1BHl1 pearrHH 8pojeBH MalbH o.n; 1: 0 < sin a < 1 H 0 < cos a < 1.

:.·.=====::;;;;;;MW!"""'""'""""--~~ffflllt--'llllllllllllllllllllllf{(l/lllllllllllli~~ii4"1"iiiii7UUWTWffP

I

:.

Jep;HOCTaBHO ce o,n;pel)yje (KOHCTpym.ne) yrao aKO je rro3HaTa Bpe,wiocT j e]lRe

Tp11roHOMeTp11jCKe cpymai;11je Tor yrna.

1 fipHMep5.

Ha rn11u;11 11crro,n; p;aT11 cy yr110B11 qi, 8, 'I' 11 p TaKBH 1.1;a je

tg q> = 0,8, ctg 8 = 0,8, sin \jl = 0,8, cos p = 0,8.

~- -

"'

08jaCHJ1 KOHCTpYKIJ;ttjy CBaKor op; Tp<DKema: yrnoBa . •

AKO je 0 < a1 < a2 < 90°, KaKaB je o,n;Hoc M3Mel)y Bpep;HOCTM TpJIToH0MeTP11jcKHX

cpy1IKu;11ja OBMX yrnoBa?

H11je TeIIIKo yoq11T11 ,n;a ce TaHreHc11 yrnoBa rroBenaBajy aKo ce rroBenaBajy H yr110B11

(BI1,ll;J1 CJIMKY ,n;ecHO ).

a1 < a2 => tg a 1 < tg a2

3a KOTaHreHC B<DKJ1 o8pHyTo: IIOBenaBalbeM yrnoBa CMaJ:byjy ce KOTaHI"eHCJf.

a1 < a2 => ctg a1 > ctg a2

IloBenaBalheM yrnoBa lhMXOBH CHffYCH ce TaKol)e rroBenaBajy.

a1 < a2 => sin a1 < sin a2

Haj3ap;, rroBenaBalheM yrnoBa lhMXOBM KOCHHYCM ce CMalhyjy.

a1 < a2 => cos a1 > cos a2

3a,u;aTaK

Ilopel)aj no Be1111'!11H11 OIIITpe yrnoBe a1, a2, a3, axo je:

1 cos a2 < cos a3 < cos a1;

2) tg a2 < tg a3 < tg a1.

E

E

ctga1

3491

350

A

1 2

1

Bpe~ocm: Tpu:roHoMeTpu:jcKHX «l>YHK~u:ja omTpu:x yrrroea

TpmoH0MeTpi1jcKe ¢ymaurje yrnOBa o)l; 30° H 60° je)l;HOCTaBHo o)l;pel)yjeMo 8HpajynH rrpaBoyrm1 Ipoyrao 'Ilrj:H cy orrnpH yrJIOBH je)l;HaKM OBHM yrnoBHMa H qHja je XJIDOTeH)73a je~e Jzy)KKHe (CJI:HKa )l;OJie JieBO).

1

tg 300 = 2 =-1 =-1 . ../3 = ../3 J3 ../3 ../3 ../3 3

2

J3 ctg 30° = _L = ../3

1

2

1

sin 30°=1.. 1

1 2

J3 cos 30° = 2 = ../3

1 2

AKo ce 8poj obm1Ka /;i, a > 0 nojaBH Kao

b ~ . . HMeHHJiau; pa3JJOMKa ya' youHqaJeHO Je µ;a Ce

Taj pa3JIOMaK pau;HOHaJJHJlle, Tj. IlOMHO)l(Jf ca

~ H npMKa)l(e y obJJHKY KOjJf je norO)l;HHjJf

3a µ;alba H3paqyHamu1>a:

b - b ya_bya Ya- Ya. Ya--a-.

KopHcTelrn: HCTH rpoyrao, je)l;HOCTaBHO H3paqyttaBaMO Bpe)l;HOCTH TpHroHoMeTpHjCKHX ¢ymaurja yrna o.n; 60°.

TpKroHoMeipzjcKe ¢yma::{Hje yrna O)l; 45° )l;ObHjaMo H3paqyttaBalheM pa3Mepa owoBapajyt:m:x crp~a HeKor je)l;HaKoKpaKo-rrpaBoyrnor Tpoyrna, Ha rrpHMep, oHor 'Ilrje cy KaTeTe je~e Jzy)KKHe.

1 tg45°=- = 1,

1

sin 45° = -1- = ../2,

V2 2

0 1 ctg45 =-= 1

1

1 V2 cos45°=-=-

V2 2 ,Uo8HjeHe pes)'JITaTe rrpHKa3yjeMo y Hape)l;Hoj rn8eJIH.

___,__ tg ctg sin

../3 ../3 1

30° 3 2 --+- - +--~

../3 ../3 ../3 60° 3 2 .....___

45° 1 1 V2 2

cos

../3 2

1 -2

V2 2

w rr11mnrm1nmmnwrd««M«««l'.'.'.!.ll»!lmmm

~

:;,,r ....

Y ommeM cnyqajy, Bpep;Hocrn rp1uoH0MerpwjcKmc cpymai;Hja yrnoBa 'llije cy Mepe

rro3Hare op;pel)yjy ce rroMony raK03BaHmc HayqHmc KaJIKY1Iaropa (scientific calculator) .

J1aKo ce KanKynaropw Mel)ycocSHo pa3JIMKyjy, rrocryrraK H3paqyHaBa:Ena Bpep;HOCTif

TpHrOHOMeTpHjCKMX cpymai;wja yrnoBa y rrpHHIJ;Mrry je CByp;a MCTH.

1 IIpHMep I.

J13paqyttajMo cos 58°.

• IlpBo je rrorpecSHo ,,caorrrnrwr11" Kam<yJiaropy p;a je Mepa yrna H3paJKeHa y crenemrna

(~TO Ce, 3aBMCHO Op; KaJIKynaropa, IIOCTIDKe TaCTepMMa r_, \f1,M!G ) mm ~ ~_.,), jep ce Mepe yrnoBa ocwM crerreHMMa (Degrees) Mory H3pa)l(a.Ban1 u

p;pyrwM jep;MHMIJ;aMa (Radians, Grads).

• 3aTMM je IIOTpec5HO ytteTM MepHH c5poj BeJIMqMHe yrna; y HaIIIeM CJI)"iajy Opoj 58.

• Haj3ap;, rrpHTMCKOM Ha racrep cos Ha eKpaHy ce MCIIMcyje KOHaqttO MHOro

p;eu;wMaJia Bpep;HOCTM cos 58°: 0,52991926423320495404678115181609 . . . •

Moryne je op;pep;MTM M yrao YKOJIMKO je II03Hara Bpep;Hocr HeKe rpwroHoMerpHj cKe

cpyttKu;wje 3a OBaj yrao.

IIpHMep 2.

Op;pep;MMO Mepy yrna a y crerreHMMa aKo je cos a= 0,53.

• IlpBo je rrorpecSHo ,,caoIIIIITHTw" KanKynaropy p;a Hae 3a:HHMa Mepa yrna y crerremrna

• 3arwM cSw rpeoano ytterw p;ary Bpep;Hocr rpwroHoMerpajcKe cpymcquj e; y HameM

cnyqajy 0,53. • Haj3ap;, aKTMBHparw KOMaHp;y Koja je yrna:BHOM 03HaqeHa ca cos -1 (IIITo ce rroCTJDKe

racrepwMa •v ""T •.• ,,

ne c5HTM MCIIMCaHO

GQS _, MJIH " 2ndP _ J' l.--~ j ). Y HameM c,-ryqajy Ha e1qnray

57,994545172235756228065410195401 rnro 3Haqw p;a je a~ 58°. +

3aµ;aTaK

Kopwcren:M KanKynarnp

1) H3paqyttaj cos 24°, sin 15,2°, tg 78°, ctg 53°15 '. 2) op;pep;w Benwq:MHY ornrpor yrna a a:Ko je sin a= 0,84.

Pe3y;rraTe 3aOk'J>~"TihyjeMo IIO IIpaBIUU.W:a x:oj a cy

;:i:aTa Ha crpam1 T2 ..

& Ha Ka;'IK}"Iaropnlota

j e cpyma:mja TaHTeHc yraasHo~ 03Ha"'leHa ca

TA.~'\' .

<l>ymmuj e KOTaHreHc Hajqem.he Helot.a., jep je oHa

KOMII03KU.Kja <Prmaurje 1 "' . x H - 11 't' )'HKillfJe x TaHTeHC.

351

- E

~ JiHe Be 11 l:loleb)

rpm o eipnit. au 4>,mmma

sin:a = 'sin ar co~a = cos a)2 tia=1tg a)l

ctg=a=lctgaF

~: .-l b c

. a Slil a=-

c b cosa=-c a

tga=-b b

ctga=-a

Axo ·e a,b > 0, mua je:

a 'a • b = 'b;

· N=a.

352

OcHOBHH TpHroHoMeTpujcKH up;eHTHTeTH

A.Ko HaM je II03HaTa spewiocT jewie TpurottoMeTpujcKe cp)'HKQHje HeKor yrna, je.:raOCTa.BHO je Om>eAJf.Tif H CBe OCTaJie.

3a CBaKH OIIITap yrao a Ta'lHe cy jeAHaKOCTH: I sin:a +cos2a = 1,

2 tg a · ctg a= 1, sin a

tga=--, cos a

1 cos a ctga=--=--.

tg a sin a

P.01rn3. HeKa je ABC rrpoH3BOJbaH rrpasoyrJIH Tpoyrao ca rrpaBHM yrnoM y TeMetty CH OIIITpHM yrno.M a y TeMetty A.

Je.ri;HaKOCT no.n: (1 .n:o8HjaMO H3 .n:ecp.IIBH~Hja CHttyCa H KOCHHyca, rrpHMeHOM IIrrraropHHe TeopeMe.

, (a )2

( b )2

a2 + b2 c2 sin-a+cos2a= ~ + ~ =- c2 = c2 =l

Je.ri;HaKOCTif rro.n: (2) H (3) cy AHpeKTHe IIOCJie)\HQe AecpHHHQHja TpHrOHOMeTpHjCKHX cpyma01j a.

a b a b

tg a ·ctg a= b·-;; = l; sin a c a cos a c b -- = - = - = tg a; -- = - = - = ctg a. • cos a b b sin a a a

c c

H3 ,n:oKa3airn::x jeAHaKocnt, y3 Many noMoh anre8pe, H3BOAHMO H cne,n:ehe je,n:ttaKOCTH (Koje cy, ttapastto, Ta<rne 3a CBaKH omTap yrao a).

sin a sin a sin a sin a tga cos a cos a cos a cos a

~l + tg2a sin2 a cos2a + sin2a = ~=-1 =sina

l+--cos2 a cos2a cos a

1 1 1 1 1

~l-tg2a sin2a cos2a + sin2a = ~=-1 =cosa

l+--cos2a cos 2a cos a

1 ctg a sin a = H cos a = .

~l + ctg2a ~l + ctg2a

--

f

Y3i1Majytrn: y ofo11p CBe HaBe,n;ette je,n;irnKocT11, cpopMJ'lpaMo rn8erry cpopI1-ryna KojJ!Ma ce ycnocTaBJbajy Be3e 113Mel)y tteKe ,n;Be TPITTOHOMeTp11jCKe cpyttKI.J;11je.

O,n;pe,n;11 sin a cos a tga

aKo je ,n;aTO

sin a cos a = .J 1 - sin 2 a tga = sin a

.J1 - sin 2a

cos a sin a = .J1-cos 2a tga = .J1 - cos2a

cos a

tga 1 tga sin a=

cos a = ~l + tg2a ~l + tg2a

1 ctga 1 sin a= cos a= tga=--ctga ~l + ctg2a ~l + ctg2a ctga

-, IIpnMep 1.

AKo je tg a= 2,4, np11MeHoM ,n;oKa3aHHX cpopM)'lla ,n;o811jaMo 11 ocrnne Bpe,n;HOCTJ1 Tp11rOHOMeTp11jcK11X cpyttKI.J;Hja.

1 1 1 5 ctga =-=-=- =- =0,41(6)

tg a 2,4 12 12

5 tg a 2,4 2,4 12

sin a= = = - = - =0,(923076) ~l+ tg2a J1 + (2,4)2 2,6 13

cos a= .J1 -sm' a = ~1-( 12 )2

= ~ = ~ =O, (384615) • 13 ~169 13

ctga - --

.J1- sin:a ctga =

sin a

cos a ctga =

.J1 -cos2a

1 ctga =-

tga

3a CBaKl1 OIIITap yrao a TaqHe cy je,n;HaKOCTH: sin (90° -a) =cos a, cos (90° -a) =sin a ,

tg (90° -a) =ctg a, ctg (90° -a) =tg a.

YpUTOHO~eTpui.:Ke <l>YHKQUJe KOMIL1eM<OHT3P""X yr'IOBa

,I(oKa3. HeKa je ABC npo113BOJbaH npaBoyrID1 Tpoyrao ca IIpaBJfM yrJioM y TeMeHY C H ornTp11M yrJioM a y TeMetty A. Ta,n;a je yrao ~ (Ko,n; TeMeHa B) je,ll;liaK 90° - a , na TPIDKeHe je,n;ttaKocrn ,n;o811jaMo .n;HpeKTHO 113 ,n;ecp11Hm_i;Mja TpMroHoMeTJnrjcKJfX cpyma:mja.

sin (90° - a)= sin ~=.!!._=cos a c

b tg (90° - a) =tg ~ =-=ctg a

a

cos (90° - a ) =cos ~ =~=sin a c a

ctg (90° - a ) =ctg ~ =- =tg a • b

~~ a

A b c

o a

353

A O:::ipel)1raaa.e Mepa

OCHOBHJ£X eaeMeHara -rpoyu ra Ha3IIBa ce n pewaBClJbe wpofi<a..

354

Ilo,t1; oCHoBmru eJieMeHTifMa rrpaBoyrnor rpoyrna no,u;pa3)'MeBajy ce lberoBe cTpamm;e H orrrrpu )'TJIOBH. AKo cy II03HaTa He.Ka ABa ocHOBHa eneMeHTa rrpaBoyrnor Tpoyrna, ocTan:e op;pe})yje~m KopJicTetrn IIHraropmry TeopeMJ, TeopeMY o 3c511py yrnoBa y rpo)TJIY H geqnnun~Hje rpuroHOMeTpHjCirnX cpyHICT-\Hja. Pa3MOTp11heMo CBe KapaKTepncTifqi{e crryqajeBe.

IIpD1epl.

O.u;pe;:µIMo XJmoTeHJ3Y H omTpe yrnoBe rrpaBoyrnor Tpoyrna ABC q11je cy KaTeTe a=8,4cmH b=lOcm.

XJmoTeey3y o,u;pe})yjeMo npHMeHoM IlHTarop11He TeopeMe:

c =Va2 +b2=../8,42 +102 = ../170,56:::: 13,lcm .

.U,a bKCMO op;pewun1 yrao a, o,u;pe):lMMo Hajnpe H>eroB Tattrettc:

a tg a=-= 0,84.

b

yfa IIOCJieW:I>e jep;HaKOCTK .n;oc5Hja.Mo ):la je a:::: 40° (BH):IH IIpHMep 2. Ha CTpaHH 351). Haj3a;:i:, KopHcretrn je.D;HaKOCT a+~= 90°, .r1oc5Hja.Mo .r1a je ~:::: 50°. +

maTaK

O~ew1 Heno3HaTy KaTery H orrrrpe yrnOBe rrpaBoyrnor Tpoyrna ABC qHja je je.r1Ha KaTeTa a = 8,4cm H nm0Teey3a c = 1 Ocm.

IIpD1ep2.

O,u;pe]{l~fO x:m:roTeHJ3y, Herro3HaTy KaTeTy 11 Herro3HaT OIIITap yrao rrpaBoyrnor rpoyrna ABC ~tja je je,D;Ha KaTeTa a= 12,5cm H OIIITap yrao HacrrpaM lbe a= 36°.

Jifa je,u:HaKOCHi a+~ =90°, ):IObHjaMO HeII03HaTJ1 OIIITap yrao ~ =54°.

KaKo je sin 36° ::::: 0,5878, H3 je):IHaKOCTH !!_=sin a, .r1oc511jaMo .r1a je c

12,5 c= r:::: 12,5: 0,5878:::: 21,27cm.

sin 36° ~a HeII03Hary KaTery M02KeMO o,u;pe):IHTH Ha BHIIIe HaqHHa. Ha rrpHMep, rrpnMeHOM IlHraropHHe TeopeMe .r1oc511jaMo .r1a je

b =../c2-a2:::: ../21,272 -12,52

:::: 17,2lcm.

TaKo})e, M02KeMo HCKopHCTHTH je):IHaKocT tg a=!!_. KaKo je tg 36° r:::: 0,7265, HMaMo .r1a je a b

b =- r:::: 12,5: 0,7265:::: 17,2lcm. + tga

3a;laTaK

O,u;pe/:IH nm0Teey3y, HeII03HaTy KaTeTy H tterro3HaT onnap yrao rrpaBoyrnor Tpoyrna ABC 'Il1.ja je je,D;Ha Karera b = 12,5cm 11 OIIITap yrao Koj11 Ha IhY HaJie2Ke a= 36°.

G 0

"'"'"'"""''"

~====:=::;;;....;;;::;::;::;.....;=:=::=:::::::::::::::::::::::::::::::~::::::::::::::!.~~~~~~~~~~~-~ ~ -- ;;; ~

Ha cTpaHH 334 cMo BH.n;e1m .n;a ce pacToja1na .n;o He.n;ocTyrrma: TaqaKa Mory

H3paqy1rnBaTJ1 MepelbeM o.n;roBapajytrnx yrnoBa. 3HaqajHa rrpHMeHa TpJITOHoMe-rpaje

orne.n;a ce yrrpaBo rrpmmKOM OBaKBHX H3paqyttaBalha.

IlpHMep 3.

lifapaqyttajMo y.n;albeHOCT 5po.n;a o.n; cBeTHOHHKa BHCOKor Sm:

(a) aKo ceca Bpxa CBeTHOHHKa

5po.n; BH.n;H rro.n; yrnoM o.n; 85°; (5) aKo ce Bpx CBeT.HOHHKa ca

5po.n;a Bl1,ll;.J1 rro.n; yrnoM o.n; 7°.

(, Jfa tg 85° =~ J1 tg 85°::::: 11,43 rne.n;H .n;a je x::::: 91,44m. 8

(5) lifa tg 7° =! J1 tg 7°::::: 0,12 cne.n;H .n;a je x::::: 66,67m. + x

IlpHMep4.

Pa3MOTpHMO HeKOJIHKO KapaKTepHCT.J1qH.J1X CHTyan;Hja y KojHMa Tpe5a

o.n;pe.n;HTH BHCHHY o5jeKTa 5e3 .n;HpeKTHor Mepelba. HapaBHO, rro.n;pa3JMeBa

ce .n;a rrocMaTpaq rroce.n;yje ypel)aj 3a Mepelhe yrnoBa.

1) IIpeTrrocTaBHMO .n;a rrocMaTpaq B.l1,ll;.l1 Bpx .n;pBeTa rro.n; yrnoM o.n; 12° y

o.n;Hocy Ha T3B. JIHHHjy xopH30HTa Koja je y rrpHKa3aHOM CJI)"lajy o.n; ma

y.n;albeHa l,5m. AKO je pacrnjalbe J13Mel)y rrocMaTpaqa J1 .n;pBeTa lOm, oH.n;a ce

BHCHHa .n;pBeTa h MO:lKe o.n;pe.n;HTH Ha cne.n;etm HaqJ1H:

x -=tgl2°:::::0,21, x:::::2,lm, h=x+l,5:::::3,6m. 10

2) IIpeTrrocTaBHMO ca.n;a .n;a ce .n;pBo HanasH Ha HB.J11.l;l1 Y3BHIIIelha Kao Ha

CJIJ1D;J1 .n;ecHO, TaKO .n;a IIOCMaTpaq BJ1,ll;J1 IIO,ll;HO:lKje .n;pBeTa rro.n; yrnoM o.n; 15° y o.n;Hocy Ha JIHHHjy xopH30HTa, a Bpx .n;pBeTa rro.n; yrnoM o.n; 27° y o;mocy

Ha HCTY JIHHHjy. AKo je pacrnjalbe l13Mel)y rrocMaTpaqa l1 Y3BJ11Ilelba lOm, BHCHHa .n;pBeTa h ce MO:lKe o.n;pe.n;HTH Ha cne.n;etm Ha'l.l1H:

x x-h -=tg27°:::::0,51, x:::::5,lm, --=tgl5°:::::0,27, x-h:::::2,7, h:::::2,4m. 10 10

3) YKoJIHKO mi:je II03HaTO pacTojalhe H3Mel)y rrocMaTpaqa H .n;pBeTa, Bpx

.n;pBeTa 5J1 Tpe5ano IIOCMaTpaTH ca .n;Ba pa3JI.J1q.J1Ta MeCTa H3Mel)y KOjHX ce

y.n;albeHOCT MO:lKe o.n;pe.n;J1TJ1. y CHTyan;Hjl1 Kao Ha c~ .n;eCHO, BHCJIBa h ce

o.n;pel)yje Ha cne.n;etm HaqHH:

x 0 x 0 -=tg12 :::::0,21, --=tg7 :::::0,12, y y+lO

o.n;aKJie .n;o5HjaMo .n;a je x::::: 2,8 rra H h =x + 1,5::::: 4,3m. +

lOm

l,Sm

l,5m -

l,5m

355

A

c

360°

AKo je P noBpmirna omTpoyrnor Tpoyrna 'Dlje cy cTpatt:iu~e a H b H lhHMa 3ax:BaheH yrao y, ott.n;a je

P =_!_ab sin y. 2

f/,OKG.3 HeKa je h0

BHCHHa Koja o,n;roBapa CTpaIDuvi a. Ta.n;a je h0

= b sin y, rra 113

¢op~f)'Jle P = ~ ah 0

cneAM .n;a je

3aaa ... "K

Yfapaqyttaj rroBpm.mry Tpoyrna <rnje cy p;Be CTpamni;e 5,2cm 11 12,4cm, a yrao Koj11 otte 3ax:Barnjy je 28°.

Ilp1111epS.

Ilpamuratt n-rnyrao, n > 4, yrrn:catt je y Kpyr rr011yrrpeqHJ1Ka 1. 0Baj n-TOyrao cacTOjH ce O.ll n je,!J;HaKOKpaJGIX TpoyrnoBa 'DljH cy Kp~H jeAlfHJ1qHe ,ll;}')f(HHe 11 yrao H3Mel)y

360° IMfX --. CBH OBJ1 TpoyrJIOBH cy Mel)yco8Ho rro,n;y,n;apHH l1 IIOBpnrnHa CBaKOf op; lbl1X

n je p~ =_!_. 1 . 1 . sin 3600 =_!_sin 3600.

2 n 2 n ,il,aKJie, noBpmHHa n-TOyrna je

p =~sin 3600. n 2 n

y ra8emi cy ,n;aTe IIOBpIIll1He MHOroyrna 3a HeKOJIHKO KOHKpeTHJ1X Bpep;HOCTH n.

n 6 23 100 1000

P,. 2,598076... 3,102662. .. 3,139525... 3,141571...

Huje TeIIIKo yo<rnrn .n;a he noBenalbeM 8poja n, 8poj P" 8irT11 CBe 8JIH)f(l1 8pojy n, Tj. noBprrrnrut: Kpyra jep;mnf'Illor nonynpe'rnHKa. +

3a;:!aTa -

IIpammaH n-TOyrao, n > 4, y1uicaH je y Kpyr rronynpe'IHHKa 1. AKO je on 0811M OBOf q n-rnyrna, AOKaJIGi .n;a je

p

180° 0 =2n ·sin--.

n n

YnoTpe8oM KaJIKyn.aTOpa o,n;pep;H on 3a n =6, 23, 100, 1000, 10 000. lIITa rrpHMehyjern?

3a,n;aTaK

IlpaBe pH q cery ce no.n; omTpK.'1 yrnoM <p. AKo je A'B' opTOroHanHa rrpojeKIJ;Hja Ha npaBy p A)')fG{ AB Koja Jie)l(H Ha npaBoj q, AOKa)f(J1 .n;a je

A 'B' =AB cos <p.

3a;i:i;an;11

Tp11roH0MeTp11jcKe <l>YHKD;Hje omTpHX yrrroBa

G)=>;n;pe;rt;11Tl1 Bpe;rt;HOCTl1 Tp11rOHOMeTp11jCKl1X cpyHKn;11ja yrnOBa rrp11Ka3aHHX Ha cn11KaMa 11crro;n;.

0,4

~7 o,~ n 24 3

A

/;\vr3ac5ep11 je;n;11H11qHy ;n;y)I(, a 3aT11M KOHCTpym1rn ;rt;y)l(J1 qJ1je cy ;rt;y)l(J1He pe;n;oM je;n;ttaKe: Lfi) tg 75°, ctg 75°, sin 75°, cos 75°; 2' tg 15°, ctg 15°, sin 15°, cos 15°.

0 0HCTPYl1IIIl1 yrao a aKO je: tg a= 1,5; 2) ctg a =2; 3 sin a =0,75; ':t cos a =0,6.

@ rropet)aj no BeJI11q11H11 OIIITpe yrnoBe a1, a

2, a

3 aKo je:

) sin a3

< sina2

<sin a1; 2) ctg a

3 < ctg a

2 < ctg a

1•

Bpep;HOCTH Tp11roH0MeTpHjCKHX <l>YHKD;Hja omTpHx yrrroBa

~J13paqyttaj: 1) tg 30° + ctg 30°;

3) sin2 30° + cos245°;

2) sin 60° (1-cos 60°);

4) tg 45° +ctg 45° sin 30° +cos 30°

G YrrOTpecSoM KanKynaTOpa o;n;pe;n;11 Bpe;rt;HOCTH TpHroH0MeTp11jcKMX cpytt~ja · rne;n;en11X yrnoBa:

1) 17°; 2) 38°30'; 3) 86,25°.

G) YrrOTpecSoM KanKynarnpa o;n;pe;n;11 Mepy yrna a aKo je: sina=0,45; 2) cosa=0,94; 3) tga=l,74; '* ctga=0,25.

(tppon;eH11 Mepy OIIITpor yrna a aKo je:

l) cosa<0,5; 2) sina:s;0,5; 3 l:s;tga; .../3<ctga:s;l. J 3

0cHOBHH TpHrOHOMeTpHjCKH H,IJ;eHTHTeTH

f;.. A . 2 . \..:? KO Je cos a =3, o;n;pe;n;11 Sill a, tg a 11 ctg a.

©KO cy a, ~ 11 y yrJIOBl1 Tpoyrna, ;rt;OKa)l(H je;rt;HaKOCTlf:

. a+A y a+A . y a+A y a+~ y Sill __ I-' =COS -, COS __ I-' =Sill-, tg __ I-' = ctg -, ctg __ t' = tg -. 2 2 2 2 2 2 2 2

A

E AKo je a 1 < ~· mua je:

tg a,< tg a2; ctg ~ < ctg a ;;

sin a1 < sin a,; cos a

2 < cos a .

E

357

B

A

,UoKaJKH .n;a cy 3a CBaKH oilITap yrao a Taqtte je,11;ttaKocT11: ~ (sin a+ cos a)1 sin a -cos a)= 1 - 2cos2 a; 2 tg a(l -ctg a) =tg a-1.

@oKaJKH .n;a cy 3a CBaKH omTap yrao a Taqtte je)l;HaKOCT11:

l

l cos a 1-sin a

1 +sin a

3 ctga-1

ctga+l

cos a

1-tg a.

1 +tg a'

2) (1-cos a)(tg a +ctg a)(l +cos a) =tg a;

4) sin a+tg a ----'~ = tg a.

1 +cos a

Ee3 yrroipe8e Kam<yn:aTOpa 113paqyttaj Bpe,11;ttocT11 113pasa tg 20° · tg 40° . tg 50° . tg 70°; 2 sin2 40° + sin2 50°.

;weWt Heno3HaTe CTpatt~e 11 omTpe yrnoBe npaBoyrnor Tpoyrna ABC ca rrpaB

CBaKH o;:i: 3a.AaTaxa ce

oxe penmTif Ha BKIIIe Ha'UtRa.

OM y TeMeHY c aKO je: ' ~=7,32cm,b=3,64cm;

3 b=2,62cm, c=S,OScm; ~ c=7,45cm, a= 17,2°;

]) a= 12,4cm, c = 24cm; 8) c=6,75cm, p =32°;

6 a = 22,4cm, a = 22,4 °; @ b=6,2cm, a=37°30'; u; a=6,04cm, P=54°;

q b=9,lcm, P=23°.

,11;peWt yrao Koj11 ,11;11jarottana Kou;Ke 3aKnarra caje)l;HOM lbeHoM CTpattoM (cn11Ka neBo).

\)· 0,11;peWt yrnoBe poM8a q11je cy ,11;11jarottane d1=6cm11 d2 =8cm.

~,UaTa je Kpparnu;a rronyrrpeqtt11Ka 4cm. 0,11;pe,11;11 u;ettTpanH11 yrao Koj11 o,11;pel)yje ....., TeTIIBa A)')IGfHe 3cm. ·

G :Vfapaqyttaj A)')KIDfY KpaKoBa je)l;HaKOKpaKor Tpoyrna aKo je lheroBa octtoB11u;a Scm J1 yrao IIpl1 Bpxy 34°.

2( Harn8 nyra 113pCl)l(aBa ce y npou;ettTHMa 11 o,11;pel)yje Kon11Ko je CTpMa HeKa J38pwfl.J;a (o)l;Hoctto HH38p,11;~a). IIpemocTaBJbajyilli ,11;a je ycrroH paBHOMepatt, Harn8 J38p)l;J1u;e npe)l;CTaBJba KOIDf'IHJ1K TaK03BaHOf BepTHKanHor J1 xop1130HTanttor ycrrotta.

aan16~---rr-7m Bepn1Kamm ..-.-.... zz=:11....~i'...._ ______ .JJ~ ycnoH

,UpyrHM peq11Ma, Har118 y38p,11;11u;e rrpe,11;cTaBJba TaHrea ttar118ttor yrna q> 113pCl)l(eH y rrpou;eHTHMa . ' 0,11;pe,11;11 ttar118tt11 yrao y38p,11;~e q11j11 je Harn8 7%. 2 0,11;pe,11;M ttarn8 y38p,11;Mu;e aKo je ttarn8tt11 yrao 3°.

358

lOOm xopH30HTaJ1m1 ycnoH

2: <l>M3MKa noKasyje ,11;a je 8p3H1Ia v Tena o8n.HKa KBa,11;pa Koje Knl1311 HJ13 CTPMY paBaB v=gt sine.

Y3 3aHeMapJ.IBalbe IpeH>a, rrp11 qeMY je g rpaB11Tau;110tta KOHCTaHTa (y8p3alhe Ko· e --~ p;aje rpaBHTau;Mja), t je BpeMe J1 0 je Harn8HJ1 yrao CTpMe paBHJ1.

KBa,11;ap ce KnM3a HM3 CTPMY paBaH q11j11 je ttar116tt11 yrao 8°. 0,11;pe,11;11 rrp116nIDIC1r"" BpeWJOCT rpaB11Tau;110He KOHCTaHTe aKO je J13MepeHO )l;a KBa,11;ap HaKOH 3 ceKya:re .n;ocTIDKe 8p311HY 0,11; 4,2 MeTapa y ceKJH)l;H.

22 Ypel)aj11 Ha cSpo.u;y ,u;eTeKTOBan11 cy rro,u;MopH11Qy Koja je o.u; cSpo.u;a y.u;aJbeHa 700m 11 Koja ce B11,u;11 rro.u; yrnoM o.u; 47° y o,u;Hocy Ha Hl1BO Mopa. O.u;pe,I1;J1.u;yc511Hy Ha Kojoj ce HaJia311

IIO,Il;MOpH11Qa.

23. EanoH ce Hana311 750m 113Ha,u; je3epa. O.u;pe.u;11 pacTOjalbe 113Mel)y Hajy.u;aJbeH11jmc TaqaKa je3epa aKo ce oHe BH,u;e

rro.u; yrnoB11Ma o.u; 44° 11 25° y o,u;Hocy Ha xopH30HT IIOCMaTpaqa.

24. CBeM11pcKa neTeJI11Qa ce Hana311 3km 113Ha,u; rroBpmHHe MeceQa. O.u;pe,u;H rrpeqH11K KpaTepa KPY)l(HOr oc5n11Ka aKo ce TaqKa HBMQe KpaTepa Koja je Hajc5n11)1(a neTeJI11Q11 113 lhe B11,u;11 rro.u; yrnoM o.u; 25° y o.u;ttocy Ha opfofTy neTeJI11Qe, a lhOj .u;11jaMeTpanHo cyrrpoTHa TaqKa rro.u; yrnoM o.u; 18°.

25. TattreHTe t111 t

2 KOHCTPYJ1CaHe cy 113 TaqKe A Ha Kpy)l(Hl1QY k(O, 4cm).

AKo je IOAI = lOcm, o.u;pe.u;11 yrao rro.u; Koj11M ce ceKy TaHreHTe t111 t

2•

26. AKo cy a 11 ~ OIIIT11 yrnoBH Tpoyrna ABC Koj11 HaJie)l(y Ha CTpaH11Qy c 11 h, je B11c11Ha Koja o,u;roBapa cTpaHHQH c, ,u;0Ka)l(11 .u;a je

h c

c ctg a +ctg ~

27. O.u;pe.u;11 rronyrrpeqH11K Kpyra Koj11 je yrr11caH y je,u;HaxmcpaKu Tpoyrao q11ja je

ornoB11Qa 2cm 11 yrao rrp11 Bpxy 30°.

E

3)'

MATEMATll1KA