1000000031

809 / 868

T R G O N O M E T R ~
and an oogl. o~poli,. lo o•• of ,h.m being ~iYeo. 'o find
lhe angle oppofi'e 'o ,he o,her.
EXAMPLE . In lh. ,riangle ABC (ihid. un 4.) fuppofe
AB=IS 6. BC=34' .nd ,he angle C (oppofi te ' o AD)=S6°
30'; reqoired Ihe angle A, oppofi,e ' o BC.
2 . G~oUJ(lrkally : Draw the line AC, and at any point
of lt, foppofe C. make Ihe .ogle C=S6° 30' ; Ihen I. ke
CH=84, .nd ",i,h Ihe Icng,h of 156 = AB taken in yoor
compafrc:s from the (ame (c¡le of equal parts. fi Jt:ing ane
poio, io B , wi,h ,he o,her crof. AC ro A . L,Oly, join
A .od B ; fo ,be tri.ngle i. cooOroaed, and Ibe requi red
angle A may be Olear. red by. line of chord•..
2 . Bycalcula/io,,: W. , hy ,heor. l.' ,he folJowiDg
proporrioa for 6nding .he angle A, "i• .
A B : B C :: S. C: S, A.
i .•. •• AB IS6 2.' 931 2
To Be 84 1.924,8
So is S, C 560 30' 9.921J 1
To S, A 26° 4" 9.650'1
CASE Il. The angles. and a fide oppofite lo one of ,hem,
being giV:A, to 60d a "de oppofite to another.
EUMPLF.. In ,he triangl. HBG (ihid. nO 5.) ~,ppofe
Ihe .agle H 46° IS', aDd Ihe angle B 54° ,,'. confequeo,–
ly .I,e angle G 79n 23', aad ,he leg HB 125. required
HG.
C.omdrically: Oraw HB 125. from .ay line of equal
paru, and make the ~ngle H 46° lS', aod B 54~ '2',
Ihen produce I~e line. HG 'Dd BG tiJl Ihey meel one .no·
.her in ,he poin. G: fo ,he tri'Dgle is conClroBed , and HG
is·meafured by taking it, length jo your: compa(fes, and ap
plyio~ il 10 Ihe f.Ole lioe of equal parts .ha! HB was takeo
uken from .
2 . B} calrola/io,,: By,he 6rfl of Ihe preeediDg Iheorem.,
we have Ihi, oo. logy for finding HG, "iz.
S, G : HB:: S, B : HG.
i . r. As Ihe fine of G 7'1° 23' 9.992SO
il'o .he leg HB 125 2'°969 1
fo i. Ihe fine or B 54° ,,' 9.9°996
10 ,he leg HG 103-4 2.0143 7
, C.... IIl. T wo flÓe. ond ao . ogle oppoGle.o ooe of .heID
¡ iveo, to find the third fideo
EXAMPLE. In ,he triangle KLM (ibid. o· 6.) fuppofe
Ihe fide CL 126 equal par.. , .nd KM '30 of ,hefe parlS,
.nd Ihe .ngle L (oppofite 'o KM) 63° 20'. requíred Ihe
lid. ML.
J. C,om"rically: The conllruélion of ,hi. cafe is Ihe
wilh ,hat in Cafe l. (Ihere being Ihe ,hing. gíven
in bOlh,) .nd ,he leg ML may be meafured by applyin~
it tO the rame lioe of equal parta that the othef two were
take" {romo
2. By cahu/a/ion: T he folu,ion of Ihis ""fe depends up–
on the r\Vo preceding onel ; aud, Grít, w-e muH fiad the other
tI'O angles lry 1. • hus:
MK:S,L::KL :S, M.
i . •. Asthe firle MK 'lo 2.11394
T o .he of L 63° 20' 9 9 5I 16
So i..he fide KL 126 • . 10°37
To .he fine of M 60° l' 9.93159
Tben by 11. we ha.e Ihe requi red lec ML, ,hus:
S. L:S. K :: MK:ML.
;. ~. A, the fine o{ L 63"" 2 0'
To ,be fine of K 53 o 39'
So is KM 130 2. 11 394
To ML 11 7.2 2.068so
CA" IV. Two fideo Ind Ihe con..ioed angle being , iven,
10 find ,he o,her 'wo abgle•.
EXAMPLE: In Ih. frian,le ACO (ibid. nO 1.) fappoCe
AC=lo). AO=126, and Ihe ingle A=S4 ° 30'; required
.he angle. C and O .
l . C''''/~/rically : Oraw AO=126, lod make .he anglo
A=54° 30; Iheo fel off 10) equal PI'" from A '0 C :
h flll', join C and O ; and fo the triangle il conClruaed,
'Dd Ihe 'Agles C .nd O m,'y be me.fured by. líne of chord•.
2 . By calctlla/ion: The folo,ion of Ibis cafe depend• .up .
on the fecond ,nd thírd of Ihe precediRg Iheorem.; .nd fi,lI
\Ve mull find Ihe fum .nd dilference of .he fides. and halr
Ibe fu m of ,he uDkDOWO aDgle., Ih.s:
The leg AO i. ,,6
The leg AC i. 10)
Their fum is
And their difFerence is
The fum of .he Ihree .o¡le. A, O. aod e, i.
The In¡le A i.
So Ihe fum or Ihe angle. C Ind O will be 12SO30'
And Iheir fum i. 6, ° 4S'
Then by Iheor. 2. we Ihe following-proportion, "iz.
As ,he fum or,he fideo AO aDd AC=229 2.359 84
T o .heir di/l'erence 23 1.361 73
So i. Ihe "ogeol .he fum of ( 6' -, 88 6
Ihe uDkDOWO aogles C . od O S 2 45 10.2 1
To tango of Iheir dilference 11 0 . ' 9.29"0S
Now huiDg Ihe foOl .nd .he dilfereote of ,he
IWO.unknolO••ogle. C aod D; we li.d Ihe q••o,i,y
of .hem by Iheorcm 3. Ibus:
To half ,he funa uf Ihe ••gles C aod-O, "iz.
Add half Iheir difference, "iz.
And Ihe funa i. Ihe gre.,er .ngle e
Agaio froAl h. lf their fwn, "iz.
Take balf Ibei, di/ference, "¡".
And ,here.wiU .be lelfer angle o.
N . B. The gre..er .ogle i• •I",.y. ,h" fubtended by .he
greater fide: tI{u., in the pre(ent care . the greater angle C.
is fualended hy Ihe greater AO ; .04 Ihe lelfer angle
O 'is fub,ended hy ,he lelfer ude AC.
CA" V . Two fides and Ibe contaíned angle b<ingginA,
'o fiod .he Ihird fide-
EX' MPL. . Io Ihe trilogle RCD (ibid. nO 8.) fuppofe
BC=154, Bo.=I33. aod ,be .ngle H=S6° 03'; required
,he fide CO .
l . Cto1llclrica/ly: T he conllrull:ion of this care: ¡s rhe
\Vilh Ihal of .he ••nd ,he lenglh of OC í. found
by taking i.. leng'h in your compotT<s, . nd applying il lO>
the (ame Jine of equal parts thu the: t \VO IC8' were u.kcn
rrom.
2 . By 'trlculaJion: T he folu,ioo oflhi, c. fe dcpcnds u~·
on Ihe fecond aod fourth; 'Dd 6rft IVe lIIull 60d Ihe aogles
by .he laCl cafe ; ,hu. :
A••be fum of ,he fideo DO. and BC 287 2.408g
Is