042 



MATHEMATICS TRIGONOMETRY. 



[SOLWTION OP TI: 



Instead of using the formula which given us tan. 7p 



we my ne those which give sin. 2 ~ OT coe - f 



Example given, o - 490 b - 9CO e - 660 we shall 

 find that 



A - 28 . iff . 12" 



B - 112 .20 .40 

 O- 39 . 29 . 8 



,,r A + B + 0- 180 . .0 



which circumstance is a test of the accuracy of the 



calculation. 



(2). Given ttco ride* and the included Angle. 



e. g., given a b C, find A B r. 

 We have 



A-BZ* A_+B. 



2 



.'. L tan. 



A ~ B 



a-|-b 

 , log. (a - b) + ar. : comp. : log (a + b). 



2 



+ L.tan. A +J*-10. (a) 



It must be observed that --" = 90 - ^ and is 



A-B 



therefore known. Wherefore (o) gives us g = a 



known angle. Then, 



2 



A-B 



andB = 90-a- . 



I 



whence A and B are known. 

 To find c we have 



c = a 



sin. C 

 sin. A 



.'. log. e = log. a + L. sin. C + ar. : comp. : L. sin. A 

 10, which will give us c. 



We may employ a subsidiary angle 6 in the following 

 manner (see Art. 40). 



o b C 



Assume tan. = - f cotan. -^ 

 a -f- o & 



.-. c = (a + b) sin. -g- 



cos. 

 log. tan. = log. (a b) + ar. : comp. : log. (a + b) 



C 

 + log. cotan. -_- 10 (c) 





 log. e log. (o -f- b) + log. sin. -g- 4- ar. : comp. : 



log. cos. 10. (d) 



By using (c) and (d) we can obtain c without first finding 

 A an 



If we solve by the former methods we require six 



logarithms, viz. : those of (o b), (a + b), tan. 



in. C., and sin A. 

 If by the latter we require five logarithms (a b), 



(o + b), cotan. - -> "> -^> and cos- " ! therefore the 

 2 2 



latter method possesses a slight advantage over the 

 former. 



A R 

 The in formula (c) is evidently the same as _- 



in formula (a). 



If 6 7 a the formulas become 



and 



tan. = fr-^ cotan. . 



b + o 



Example : Given a 562. b - 320. C 128 4' 

 We shall find A - 33 34' 40* 

 B - 18 21' 20" 

 c- 800 O08. 

 If we take the second method we shall find 



log. tan. 0-9-1258960 

 .'. log. cos. 6 = 9-9961568 

 And, as before, we shall find, 



= 800-008. 



(3). Given too side*, and an Angle opposite to one of them t 

 e. g., given B. b. e. find A. C. a. 



We have sin. C = j- sin. B. 



.*. log. sin. = log. e + log. sin. B + ar. : comp. 



log. b 10 (o) 



which gives C. Then 



A = 180 (B + C) 

 which gives A. Then 



, sin. A 



o= b -: = 



sin. B 



or, log. a = log. b + log. sin. A + ar. : comp. : 

 log. sin. B 10 (b) 



which gives n. 



N.B. This is sometimes called the ambiguous case of a 

 triangle ; for we have sin. C given us, whence we know 

 C = o, where a Z. 90. Now, sin. a = sin. (180 n) 

 .'. may also equal 180 a. Hence this equation 

 may give us t\vo triangles, one which has C = a, the 

 other having C = 180 a. In many cases, however, 

 this ambiguity does not exist. 



(1). B = or 7 90. Then C = 180 o is inadmissible, 

 since C would then be 7 90, and B + C > 180, 

 which is impossible. 

 (2). If B < 90. Then 



(a). If c sin. B = b then sin. C = 1 or C = 90, and 



the other value of C = ISO 3 90 = 90, or in 



this case there is no ambiguity. 

 03). If c < b. Then C < B, hence if C = a. a < 



B, and 180 o + B > 180, or in this case the 



angle 180 a is inadmissible, and there is no 



ambiguity. 

 (~t). But if c > b. Then both c = a, and c = 180 



a are admissible, and there are two triangles 



determinable from the given values, b. c. B. 

 This can be illustrated geometrically as follows : 



AB = e 

 ABc = B. 



Fig. 25. 



with centre A and radius b describe a circle. Then, 



(a). If b c sin. B, the circle will touch Be in e, 



and the triangle will bo A Be. 



(/3). If c < b the circle will cut Be, in two points 

 C C., on different sides B, and the triangle is 

 ABO,. 



("/). If c > 6 the circle cuts Be in two points e, c 3 on 

 the same side of B, and the triangle may be either 

 ABc, 2 or A Be, ; which is the reason of this case 

 being called ambiguous. 

 Example : Given B- 45 b 305 c - 219-5 

 Then 0-30. 3G'. 22" 

 A - 104. 23'. 38" 

 a =410.8344. 



