1000000031

TRI G O N O M E T lt Y.

10 Ihe leg DG

1.93952

lo is

radius

90°,

10 00000

,he fine

or D

37°, 47'

9í. 8723

Then by .caCe

(l. we find ,he leg DG required, ,hos :

R: S, B

:: B

D·:

DG,

as radíus

is to the fi ne of

fo is ,he hvpo,h. DB

,he leg D G

T he Icg D O may al

ner,

u;z.

.)0

2°

13'

14 2

fouod in ,he

10 .00000

9.897 8 ,

2. 15

2.05° 10

following meln-

T o ,he log of ,he C\lm of Ihe hypolhenuCe and

7 •

gil'"n leg.

vi•.

229

2 .,59 4

.dd ,be logari,hm of ,heir diff<fene<,

"iz.

.74°36

and their fum is

Ihe half of Ih.. is

~ . I 0020

2 .0 5°10

,he log. of

2.2

,be leg r<quired.

mAy

be done

taki ng (he

(quare

of Ihe

givc.n

lec

f~onl

Ihe fquare of Ihe hypolheou Ce • •od :he Cqu..e rOOI of

the

r~mainder

is Ihe

It.g

required : thus, in the prefeot caCe,

The Cqqare of Ihe

hypolhen.fe

(1 42)

The fquare of Ihe leg BG (8 7) i.

Their difFerence is

Whof< 10gari,hlO

T he half of whirh is

which aof\Vers

the natural

quired.

1 2

595

4.1 0020

2 . 0 50 10

num~r

11 2 .2

,he leg re-

T hus

h.ve

we gone Ihrough Ihe feveo caC .. o( righl '

angled plane tri,onometry; from which we may

obfen c:,

l . l"'hat to find a fid<, when the

an~les

aTe given, any fide

may be made the radius.

2 .

'ro

flnd an angle, one of tbe

given fides mu(l o( neceOi,y be made ,he radius .

OBLIQ..UE . A HGLED PLAN':

TRICOHONlTI.V.

oblíque.angled plaoe ttigonometry tbere are

fix

ca(es ;

bu, befo¡e we (hew ,hdr Colullonl it will' be proper 'o pre–

.Ill(e

,he (ollowing Iheorem•.

TH50U M

In . ny triangle ABC

(ibid.

6g.

2. )

the fides are proportional tOthe figns of the oppvfite angles:

.hu., in' Ihe ,riangle A B C, A B : B C : : S , C : S, A, aod

B : A

e ::

S, C : S : B : alfo A C : B C : : S, B : S, A.

Dllnonjlrlllion.

Lel Ihe

triangl~

ABC be iftCcribed

•

cirele; ,hen, it is pl.,o (rrom ,he property o( Ihe cirele)

tha, ,he hal( of eaeh fide i,

I~e

fine o(

liS

oppofile angle :

but the fines. of t}:)eCe angles, in tabulH paru , are propor.

tional to the fines of the Carne in any other meafure; thae–

fore, in ,he ,riaogle ABC, ,he fi nes of Ihe angles will be

as the I",.lves of their oppofite Cides

and lince lhe halY't:s

are as the whole.¡

foJlows, th.t.t

che

fines of the angles

are .. Iheir oppofi« Gde.

S, C : S,

A ':

B :.

BC,

&c.

TH5o •.

In any plaoe triaogle, as ABC

(i. id.

o·

2.)

Ihe fum o( , he lide•• AB and BC, i.

,he d,ffaenee o(

,hefe fKl e•• a. ,he tangeol o( hal( Ihe fum

,he

.o~les

}jACo A BC, at Ihe baCe, i. 'o ,he laogenl o( h, I(,he diC·

ft rence of ,h, Ce angles.

D . INM.

Pooduce

B ; and make

equal

BC ;

join HC

. Dd

(ro01 B lel

(. 11

, he perpondiclllar B E ; ,hrough

draw BD pafdl.lcl

AC, and m. ke HF equallo CD,

V OL. l1I. N 0

98.

and join

,¡'Jo';

. Iro "ke B l equal 'o BA, and draw l e;

par.lld

or AC .

Th"n i, is pJ..in ,h., A H will I>e Ihe film, .nd HI ,he

di ffore nce o( Ihe fides AA aod BC ; .od finee HA is

e~ual

'o BC , .oel BE perpendicular

HC . Iherefore H E

e–

qu. 1 'o EC ; and BD bdng p".lIel 'o ilC aod IG. · and

I\B <qua l

BI Iherd orc e D or HF is equal ' o UD,

.od conrequeo dy H G is equ,.I

YO, aod hal( HO i. e–

qual lO hal( FD or ED . Again, finee H B is equal,o BC,

.no BE perpendienlAC 'o HC. Iherdore Ihe aogle EBC i.

1,.lf ,he angle H BC ; bUI ,he angle HBC is equal

Ihe

fum o( Ihe .ngle, A and C . conCequcndy Ihe anglc EBC

e~u.1

h.l( ,he fum o( Ihe aogles A . od C. Alfo, fioe<

is equ, 1 ,o·RC. and HF equal ,oCD, and Ihe ,oelude.J

angles SHF BC D equ.!. il (ollows Ih.. ,he aogle HBE

is equ . 1

the . ngle- O RC, which

equal ' o SCA; .o,{

finco HSD is eqll:1

the , ogle A . and HS F equal

BCA,

Ihere(ore FBD is Ihe dofr« eoce. •nel EBD half the differ–

ence o( Ihe IIVO

an~les

A and SCA: lo m.king ES ,he

radius,

is plain

is the tangent of half ¡he

fumo

and

ED the tangent o( half the difference of the two 3ftgles

Ihe baCe. N ow IU being l'.rallel

AC. Ihe , riangle.

HIG

.nrl

HAC will be equiangular; confequently AH:

IH : : eH : GH ;

bm the wholcs

2re

as their halves

theTe–

fore AH : lH :

{CH : {CH; and (,oce { CH is equal 'o

EC, and {- CH equal lO {FD=ED, ,herd ore AH: IH!!

EC : ED. N ow AH is Ihe fum, and IH ,he difl'oreoce of

,he fides

al fo EC i. Ihe tangeol o( half Ihe fumo aod E D

che tangent of

half the differenc.:c of (he t\Vo angles at the

bafe; conCequently, in any triangle, as (he fum of tbe GdeS

is to their difference,

is the tangent cf half the fum of

the anglt"s al the baCe to the tangent of

haH

their dlfference.

THEouM

lB .

half ,he fum of IwO quan,ilie. be

added hal( Iheir difference. Ihe fum Ivill be ,he grealer of

,bem; .nd if from hal( ,heir furo be fubtraéled

h.lf

Iheir

differ<nce, ,he remainder will be ,he leaR of

,h~m .

Sup–

pore

{he' treater

qwantity to

x=8j~

and the IclTer

z=6'.

thcCi is thei r [um

1.1.

and difFerc::ncc

2 :

where(ore, adding

.!..i

= 7 'o

get 8 the greacdl of

the

two quantities:

and , in the Carne manncr,

!..1-

-7-

1=6, the lean of (he two quantities .

TH<OR. IV. In aoy right-iioed "iangle, ASO

(ihid.

3.)

Ihe b. Ce ¡\D i.

Ihe f.m of ,he fide. ·AS ond' BD,

as lhe differcnce of the

Cide,

is to the difference of lh.: feg–

mento o( Ihe bare m. de

,he perpendiEular BE,

"iz .

Ihe

difFereftC< belween AE ED .

D EMO' . ProdllC< DB ,ill BGbe equal ' o BA,heldr..

leg; and

0 0

B as a centre, with (he diHance BA or BG . de–

rcribe ,he d rcle AGH

which will cu, S O and A D io

'he

poio..

.nd

F :

,he" il is

pl.io

GD i. Ihe fumo and HD

,he

di ff~reoce

,he

fides ; .Ifo f,oce AD i. equal

' o

EF,

Ihel efore FD i. ,he differeoce of Ihe Cegmenls of ,he b. fe ;

bu, A D : GD :: HD :FD; t1werure ,he

b.Ce

Ihe

fum o(,he ftdes.

élc.

was ' o be proved .

H aviog dlablinlOCl Ihefe preliOlio3ry Iheolr ms . \\Oe

1h.1i

now rroceed

,he fol.,ion o( ,he fix caCe. o(ob lique.• ngled

plane

trigClllometry.

C ASE

In .ny oblique.•

o~led

plane triaocle,

' IVO

fides

Q...

and