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180° - A Cosec. ja=sec. (90°— (A)=sec.

2

180°- A Sec. JA=cosec.(90°-1A)=cosec.

2 And as A, in trigonometry, must always be less than 180°, A will always be less than 90°, and consequently its sine, tangent, &c. will have the same sign as in the table (K. 100.) GÈNERAL PROPERTIES OF THE SINES, TANGENTS, CHORDS, &c.

OF SINGLE ARCS.

PROPOSITION VIII. (Plate I. Fig. 1.) (P) The chord of any arc is a mean proportional between the diameter and the versed sine of that arc.

The angle brb being in a semicircle, is a right angle (Euclid 31 of III.) And ry, by the definition of a sine, is perpendicular to UB, therefore the triangles bGF, and BGF, are similar to each other, and to the whole triangle BFB, (Euclid 8 of VI.) Hence,

bB : BF::BF : BG; but bB is the diameter, Bf the chord of the arc Bif, and BG is the versed sine of the arc Bik. Q. E. D. (Q) COROL. The sine of any arc is a mean proportional

between the versed sine of that arc, and the versed sine of its supplement.

For the L FbG = BFG, consequently by : GF::GF : BG. Q. E. D.

PROPOSITION IX. (Plate I. Fig. 1.) (R) The square of the sine of any arc added to the square of its versed sine, is equal to the square of the chord of that arc; or to four times the square of the sine of half that arc.

For GF+ GB?=BF2 (Euclid 47 of 1.)

Draw ci parallel to by, and it will cut BF at right angles in z; but if it cut it at right angles, it will bisect it (Euclid 3 of Ill.), therefore z=zF, and br=2B2, therefore BFo=4Bzo. 8. E. D.

PROPOSITION X. (Plate I. Fig. 1.) (S) The tangent of any arc is a mean proportional between the sum and difference of the radius and the secant ; viz. rad. + sec. : tang. :.tang. : sec.-rad.

Join Fc, and produce it to h and to T, so as to meet the tangent bt of the arc BF. Then ht X FT=TBo (Euclid 36 of III.) that is ht : TB::TB : FT. (Euclid 17 of VI.) But ht=rad. + secant ct, and Ft=secant cr- rad. ; therefore rad. + sec. : tang. :: tang. : sec.-rad. G. E. D.

CG

:

GF

CB

BT.

CF

FG

СТ

TB.

CB

CG

СТ

CF.

CF

СК

СА.

PROPOSITION XI. (Plate I. Fig. 1.) (T) The right angled triangles FGC, TBC, CAK, CEF, are equiangular and similar.

For FGC, TBC are right angles, and the angle FCB= L TCB, therefore the remaining angle CFG=LCTB.

The triangle CEF=triangle CGF, for the L CFG = L FCE (Euclid I. and 29), and 2 FEC= L CGF, being each of them right angles, and the side of is common to both the triangles, therefore they are equal (Euclid 26 of I.) Again, AK is parallel to EF, by the definition of a tangent and sine; therefore the triangle cak is equiangular with the triangle cer; and consequently with cGF and CBT. In the triangles cor and CBT.

:
(U)
cosine : sine :i radius

: tangent.
:
(W)

radius : sine :: secant : tangent.
:

::
(X)

radius : cosine :: secant : radius.
In the triangle CEF=cGF and cka.
: CE=GF::

:
radius : sine :: co-secant : radius.
In the triangles CTB and CKA.
:

:
radius : tangent::co-tangent : radius.
:EF=CG::

:
(A)

co-tangent : cosine :: Co-secant : radius.

(B) By comparison. Cosine : sine :: co-tangent : radius. (U and Z, above.) Secant : tang.::co-secant : radius. (W and Y, above.) Secant : rad. ::co-secant : co-tang. (X and A, above.)

(C) Tangent x co-tangent=radius square. (Z, above.)

Therefore the tangent of any arc x its co-tangent=the tangent of any other arc xits co-tangent.

(D) Sine x co-secant-radius square. (Y, above.)

Therefore the sine of any arc x its co-secant=sine of any other arc xits co-secant.

(E) Co-sine x secant=radius square. (X, above.)

Therefore the co-sine of any arc x its secant=co-sine of any other arc xits secant.

(F) Square radius=square'sine + square co-sine. For, CF?=GF2+cG? (Euclid 47 of I.)

(Y)

CB

BT

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(2) {

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(I) Sine-tangent x co-sine

square rad.

tang A. rad

sec

cosec A

(G) Square radius=square secant-square tangent=square Co-secant-square co-tangent.

For, CB’=CT?— TB? (Euclid 47 of I.), and aco=CK? – AK”.

(H) Square radius=co-sine x secant (X. 103.)=sine x co-secant (Y. 103.)=tangent x co-tangent (Z. 103.)

tang.
(U.103.) x rad(W.103.)=
radius

secant
cosine x rad.
(Y. 103.)=

(B. 103.), &c. Co-sec.

co-tangent (K) And, generally, if a=any arc*, rad=radius, cos=cosine, tang=tangent, cot=cotangent, sec=secant, cosec=co-secant, vers=versed sine, vers sup=versed sine of the supplement, or superversed sine, the following formulæ will be easily deduced, where the sign x is represented by a point (.).

COS A . tanga (L) I. Sine A=rado - cos? A=

rad rad. cos A

rado
tang A . rad

rad
cot A
Vrado + cot? A

rado+tango A sec A. COS A tang A . cot a rad

V sec? A-rad? cosec A ✓vers sup A. vers A = N(2 rad . vers A) - vers’ A. (M) II. Cos A = rad vers A = vers sup A

rad rad , sine a

rado

rado Nrade- sine? A=

tang A
rado+tango A

sec A tang 1 . cot a 'sine A cosec A

sine

rad. cot A sec A

rad rad. cot A

rad

✓ cosec? A rad? W rad? + cot? A cosec A

rad
rad . sine a

rad (N) III. Tang A=

COS A

✓rado rad?

rad? •» rada – cos? A =

cot A

✓ cosec? A rad

sine A . cosec A sec? A - rado

cot A

cosec A

sine A. cot? A rad (2rad. vers A) - vers’a_radv(2rad. vers supa) - vers sup?a. rad- vers A

vers sup A - rad
rado

sine A. coseck (O) IV. Cot a=coseco A-rado =

tang A

tang A

cosec A

sec A

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cot A

А

[ocr errors]
[ocr errors]

sec A

cosec A

sine a

sine? A

COS A

COS A . sec A

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* Emerson’s Trigonometry, 2d edit. Prop. I. Scholium.

rad , tang A

sine A . sine A

COSA

.

cosec A

COS A

COS A . sec A

COSA

COS A. sec a
rad.cosec A
rad.coS A

rad

W rad' - sine? tang A

sec A

sine A

sine a rad . cos A

rado

rad3 , sine A W rada – cos? A ✓ secA-rad? COS A . tang A (rad - vers A). rad

(vers sup A-rad). rad ✓(2 rad . vers A)-versA „(2rad . vers sup A) - vers sup? A

rade (P) V. Sec A = x/rad+ tango A= rad3 rad. cosec A

tang A . cot A sine A . cot A cot A

COS A
rad?
rad

rad . cosec A

W rad? + cot? A= W rad - sine? A cot A

cosec? A rad tang A . cosec A rad

rad
rad rad vers A vers sup A - rad

rad

rad . sec A (Q) VI. Cosec A = rad +cotA=

sine

tang A rad . cot A

rad

tang A.

cot A COS A . tang A

sine A

sine A
rad
rad

rad

• sec A rado + tango A=" rado - Cos? A tang A

sec? A-rad cot A . sec A

rado rad

✓ 2rad. vers A - vers’ A (R) VII. Vers A=2 rad - vers sup a=rad-Cos A=rad

rad

rad . cot a W rad - sine A =rad rad + tang A

W rad+ cot? A rad

rad =rad

rad

W cosec2 A-rad?

cosec A (S) VIII. Vers sup A

= 2 rad

= rad +

rad ✓rado -- sine? A = rad +cos Arad +

rad +

Mrado + tango A
cot A
rad

sine A + tanga rad +

cosec? A - rad Vrado + cota cosecA

tang A sine A. cot a

rad2 cot A + cosec A rad = rad +

rad. rad (T) If the co-versed sine be wanted, it may be found by subtracting the sine from radius ; that is, co-vers A = rad sine A; also the chord = vers? A

+ sine A

:=rad

sec A

vers A

rad.

rad +

[ocr errors]

seca

cosec A

COS A

cos? A

(U) Besides the preceding formulæ, others may be deduced, thus, sine A +cosA=rad (F. 103)rado + tangA=sec A, and rad +cot A = coseco A (G. 104.) Now because tang A = rad.sine a

rado. sine? A (N. 104.) tange A=

hence rada + tang A rad.sine? A rad. (cos? A+sine? A) rad 4 =rad +

&c. COS? A

cos? A

cos2 A GENERAL PROPERTIES OF SINES, TANGENTS, &c. OF DOUBLE

ARCS AND OF HALF ARCS.

PROPOSITION XII. (Plate I. Fig. 1.) (W) The right-angled triangles bGF, BGF, brb, and OZB, are equiangular and similar ; and cz, the co-sine of the arc bi, is equal to the half of by, the chord of the supplement of double the

arc Bi.

For bor, BGF, and bFB have already been shewn to be equiangular (P. 102.), and the triangles c2B, FGB, have the angle at B common to both of them; also the angle bFB, being an angle in a semicircle, is a right angle; and since cz is parallel to bF, by construction (R. 102.), the angle czB is likewise a right angle.

Now BZ ; cz::BF : rb, but B2=4BF (R. 102.), therefore cz=LF. Hence the following proportions.

CB : BZ (sine arc Bi):: BF (=2B2) : BG (X) radius : sine of an arc::double that sine : versed sine

of double that arc.

CB : Bz::bF (=2cz) : GF (Y) radius: sine of an arc:: double its cosine: sine double

the arc.

(Z)

(A)

(B)

CB : cz::bF (=2cz) : bg radius : cosine of an arc:: double its cosine : versed sine supplement of double the arc.

CB : cz::BF (=2BZ) : GF radius : cosine of an arc:: double the sine : sine of double the arc.

BF (=2B2) : BG::BF (=2cz) : GF double the sine of an arc : versed sine double arc:: double cosine : sine of double the arc.

BF (=2B2) : GF::bf (=2cz): bg double the sine of an arc : sine double the arc:: double the cosine : versed sine of the supplement of double the arc.

BZ ; cz::GB : GF sine of an arc ; its cosine:: versed sine double arc : sine double arc.

(C)

(D)

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