TRIGONOMETRY. 



Thefe formulae ftew immediately the truth of our firft Hence, when two fides and the contained angle are given, 

 theorem, viz. " the fides of plane triangles have the fame we have 

 ratio as the fines of their oppofite angles." 



Again, fince a : B :: fin. A : fin. B, we have a + b : 

 a— i :: fin. A + fin. B : fin. A — fin. B ; that is, 



a + i fin. A + fin. B 



But 



a-b~ 

 fin. A + fin. B 



fin. A - fin. B 

 tan. i (A + B) 



a= ^ 1*^ + f ' — 2 i c cof. A 

 i = ^ la' + f' — 2 a r cof. B 



f= ^/ -Ja' + ^' — 2a^cof. C 



(IV.1 



fin. A- fin. B tan. 1 (A - B) 



See Arithmetic of Sines ; confequently 



a_^h tan. X (A + B) 



a-*~tan. i(A- B) 



Ora+ i:a-i :: tan. i (A + B) : tan. i (A - B) ; 



that is, " the fum of the fides is to the difference of the 

 fides, as the tangent of half the fum of the oppofite angles 

 is to the tangent of half the difference," which is our fecond 

 theorem : and other forms of folution are readily obtained 

 from the two fundamental equations (I.) and (,11.) iii%. 



a= b cof. C + c cof. B 

 b = a cof. C + <: cof. A 

 c =.c cof. B + i cof. A 



For multiplying the firfl of thefe equations by a, the fecond 

 by b, and the third by c, and each of the equations thus 

 obtained being taken from the fum of the other two, there 

 will arife 



i'' ■=. 2b c cof. A' 



And when the three fides are given, 

 cof.A = ^^ + ^^-'^^ 



cof. B = 



cof. C 



zab 



J 



(v.) 



Thefe formuloe are very convenient for computation, the 

 former when the cofine of the given angle has any real frac- 

 tional value, and the latter, when the three fides are com- 

 pletely integral, and fmall numbers : in other cafes, they will 

 be found more convenient under the following form, iiiz. 



[i^c — a] (b + c- a) 



cof. A 



cof. B = 



cof.C 



K (VI.) 



b' + c-" -i 



a* -)- f * — 4' = 2 a (• CO 



a* + i' — f* = 2a^ CO 



cof. i A 



f.AT 



2ab 



■III.) 



Or v\e may fubflitute for cof. A its equal 2 cof.' \ A 

 (fee Sine), and we have. 



+ b + 



:}{l^a 



+ b+ c) 



be 



l}] 



(VII. } 



which is purely logarithmic : the cof. i B, and cof. i C, being precifely analogous to the above, are omitted. 

 And in a fimilar manner, we find 



fin. I A 



And again, by divifion. 



tan. Jr A = 



= vi {i^'' + ^ + ''>-'] {i(« + * + ')-*} 1 (Vm.) 



Of thefe feveral rules for the determination of the fides 

 and angles of plane triangles, we have before obferved, that 

 the formulae \ IV. ) and (V. ) are befl adapted to fmall in- 

 tegral values of the fides ; and to real fraftional values of the 

 cofine, in other cafes, one or other of the three latter will beft 

 apply. When the angle fought is very finall, it is ufually 

 better to employ N° (VIII.) than N" ^VII.) The 

 method indicated in N° (IX.) is commodious, and very 

 correft, except when A is either very fmall or near 180". 



In fome cafes, where great accuracy is required, the 

 operator may wifh to obviate the uncertainties that would 

 arife from the ufe of fome of thefe formula: ; for which 

 purpofe Dr. Mafkelyne has given, in the IntrodufUon to 

 Taylor's Logarithms, the following rules in reference to 

 the fines and tangents of very fmall arcs. 



1. To find the Sine. — To the log. of the arc reduced into 

 feconds, with the decimal annexed, add the conftant quantity 



4.6855749, and from the fum fubtraft one-third of the 

 arithmetical complement of the log. cofine, and the re- 

 mainder will be the log. fine of the given arc. 



2. To find the Arc from the Sine. — To the given log. fine 

 of a fmall arc, add 5.3144251, and one-third of the arith- 

 metical complement of log-, cofine ; fubtract 10 from the 

 index of the fum, the remainder will be the logarithm of the 

 number of feconds and decimals in the given arc. 



3. To find the Tangent To the log. arc and the conftant 



quantity 4.6855749, add two-thu-ds of the arithmetical 

 complement of the log. cofine, and the fum is the log. tangent 

 of the given arc. 



4. To find the Arc from the Tangent To the log. tangent 



add 5.3 14425 1, and from the fum fubtratt two-thirds of the 

 arithmetical complement of log. cofine ; take 10 from the 

 index, and there will remain the logarithm of the number of 

 feconds and decimals in the given arc. 



Trigono- 



