MATHEMATICS. TRIGONOMETRY. [EKLATIOK OF AJJQLM AJTD SIDES. 



. si - A 

 ' ' a. B 



CN CB 



"AC" CN 



CX CN CN CB CB 



AC^CB 

 , Sin. A a 

 " Sin. B ~ 6 " 



Sin. B _ b^ 



N.B. These relations manifestly can be written in 

 the form 



Sin. A Sin. B Sin. ,^ 



a 5 c 



(37). To proee the Formula. 



a 1 - 6 J + c 1 - 2 tc. cos. A. 



If A be an acute angle. Let A B be the triangle, 

 draw C N perpendicular to the base of the triangle A B. 

 Then (Euclid ii. 12); 



BC"= AC + AB -2 BA-AN. 



Now, AX=AC. cos. CAB =6 cos. A. 



.'. a'- = 6 2 +c* 26c. cos. A. 



Again, if A be an obtuse angle. Then drawing C N 

 perpendicular to AB produced, we have by Euclid, ii. 13, 



BC*=AB* + AC* + 2 BA.AN" 



and AN=CA cos. CAX=6 cos. (180- A) 



= - 6 cos. A. 



A. 



(32). 



Hence whether A be acute or obtuse 

 o*6 + e- -2bc cos. A. . . : 



Similarly, 



6s=e s 4. o s -2ca. cos. B. 

 e* = a 11 + 6 1 -2a6. cos. 0. 



(38) . To deduce the Formulas of Article* 37 from 

 those of 36. 



These formulas can be immediately deduced from the 

 formula 



sin. A _ sin. B ^ sin. C 



o 6 c 



without reference to Euclid's demonstration. Thus, 

 since the three angles of a triangle are together equal to 

 two right angles, we have 



A + B+C=180. 



.'. sin. (A+- H)= sin. (180 -C)=- sin. C. 

 .'. sin. A cos. B + sin. B cos. A= sin C. 



sin. C 



cos. B + 



.sin. U' 



cos. A=l; or 



oos. B + - cos. A - 1. 

 c c 



.'. a cos. B + 6 cos. A = C. 

 .*. o 1 oos.* B + 6 cos. 1 + 2 o 6 cos. A cos. B = c 1 . 



... a sin. A 



Now -j- = : 5- 



6 sin. B 



.'. a sin. B 6 sin. A=0 



.'. o 1 sin. 1 B + 6* sin. 1 A - 2 a 6 sin. A sin.B = 

 adding these two equations together, and remembering 

 that sm. 1 A + cos.' A - 1, we have 



a' + I/ 1 + 2 a& (cos. A cos. B sin. A sin. B)=c* 

 But cos. A oos. B sin. A sin. B = cos. (A + B). 

 Now, cos. (A + B) - cos. (180 C) - cos. C. 



.' . o' + 6' 2 06 cos. C = s * 

 The other formulas can be derived in the same manner. 



(39). Certain Derived Formulas. 

 (a) Again, since 



o' - 6' + c' 26c oos. A. 

 ' o' 



. u- 1- c- a' 

 cos. A 1 + 



2be 



And 1 cos. A 1 

 /. 1 + cos. A- 



fc' + c 1 o 



2bc 



' a 1 





1 oos. A- 



e 





/. Scos. 



,A_ (6 + c)'-o' 



2bc. 



. . A (6 c)' 



2sm - T- k-^ 



A (o 6 + c) (a + 6 c) 

 "'T" ~467- 



Now suppose 2s = a + 6 + c 



.'. 2 ( a) = 6 + c o 



2? 6) = o 6 + c 



cos 4= / " (*~ a ) 

 . . cos. - \S bc 



(33) 



A A 

 row, sin. A= 2 sin. -g cos. -g 



s , A 

 . . sin. A 



) ..... (35) 



In regard to these formulas it will be observed that 

 the angle and the denominators are always the three 

 letters, i.e., if the angle is A the denominator is bc ; if B, 

 the denominator is co, and so on. Again, in (33) it will 

 be seen that the sides in the numerator correspond to 

 the angle, and in (34) the sides in the numerator cor- 

 respond to those in the denominator. Hence we can 

 always adapt these formulas to any angle, thus : 



Q 



sin. B => 

 ac 



..(-a)(.-6)(.-e) 



cos. ^ 



(i>) The following relations are important ^ 



,,. a sin. A 



Since -j- = -r ... 



o siii. x> 



a 6 



sin. A. -f- sin, j 



Bin.B 

 sin. A sin. B 



6 sin. B 



. a 6 sin. A sin. B 

 * ' a + 6 = sin. A + sin. B 



~~ A B . A + B 

 2 oos. g sin. I 



Now 



A + B-180 C. 

 ^~ - cotan. 



f , !, 

 0+6" 



. . (36) 



(40). To obtain Formula (32) in o form adapted for 

 Logarithmic calculation. 



The above formulas, expressing the relation between 

 the sides and angles of a triangle, are in a form adapted 

 for logarithmic computation, except 



