IOM 



NAVIGATION. 



[SOLUTION OF TIU ANGLES. 



following rule, when of the three given parti two are 

 opposite to one anntli 



K t LE. To find an angle. At one of the given aides 

 to the other, to it the tine of the angle opposite tu th 

 former, to the sine of the angle op|>osite to the Utter. 



To Jind a tide. The tines of the given angles are to eaol 

 other at the tides opposite to those angles. 



If a, 6, be any two aides, and A, B, their opposite 

 anglet, 



b sin. A 

 .*. a : b : : sin. A : tin. B - i 



a tin. B 

 Or, sin. A : sin. B : : o : 6 " a j n ~X* 



.'.log. tin. B = log. 6 + log. sin. A -log. a; log. b- 



log. a -f- log. sin. B log. sin. A. 

 Whi-n an angle is to be determined by this rule, there 

 is sometimes a choice of (wo angles ; and the case is thei 

 called the ambiguous cote. The ambiguity arises fron 

 the circumstance that the sought angle is to be infn r< 

 from its tine ; and in the absence of all overruling re- 

 strictions, too angles have each equal claim to the same 

 tine : an angle and its supplement. But if it happen thai 

 one of the given tides which is opposite to the given 

 angle, is greater than the other given side, the angle 

 opposite to the hitter must be acnte, otherwise the triangle 

 would have two obtuse angles, which is impossible. Also, 

 if either of two angles are known to be obtuse, the others 

 must, of course, each be acute. Except under these 

 conditions, the angle sought, may be either acute or 

 obtuse ; so that there are two distinct triangles deter- 

 minate from the proposed given parts, these parts be- 

 longing equally to both triangles. 



EXAMPLES. 



1. In a triangle two of whose sides aro o = 95-12, and 

 b = 98, and of which the angle A opposite to the former 

 is 32 15', it is required to find the angle B opposite to 

 the hitter, as also the third side c. 



To find the angle B. 



As o = 95-12 -1-9783 



: 6 = 'J8 1-9912 



:: sin. A, 32 15' 97-7-' 



: sin. B, 33 21' 97401 



To find the side c. 



As sin. A, 32 15' -97272 



: sin. C, 114 24' 9 -'.i:.!i4 



: : a = 95-12 1-9783 



: e - 162-3 2-2105 



This example comes under the ambiguous case noticed 

 above, for the side b, opposite to the sought angle, being 

 greater than the side a, opposite the given angle, the 

 angle B may be either 33 21* or its supplement 146 3!K, 

 at the sine determined above belongs equally to both ; 

 and each of these angles is greater than 32* 15'. The side 

 e is calculated here for the acute angle B = 33 21', for 

 which C - 180 - (32 15' -f 33 21') = 114 24'. If it bo 

 calculated for the obtuse angle B = 146 39', then C will 

 be C - 180 -(32 15' + 14tf" 39') - 1 6'. 



It will be seen that there is an inconvenience in having 

 a tulitrartirr. logarithm in each of the columns ; it may 

 be replaced by an additive quantity as follows : Instead 

 of writing down 1 -9783 from the table, write down what 

 this number wantt of 10, which is 8-0217 ; but instead of 

 subtracting from 10, to get this remainder, in the 

 ordinary way, commence with the leading figure 1 on the 

 left, and proceed from figure to figure towards the right, 

 tubtracting etch from 9, till the last, 3, is reached, which 

 subtract from 10 : this is of course the same, in effect, 

 i subtracting the 3 from 10 ? in the common way, and 

 carrying 1 from every figure in proceeding from right to 

 it it it easier to write down the remainder by the 

 former way than by the latter : thus, pointing to the 1 in 

 the number 1-0783, we writ* down 8; to the 9 we write 



down ; to the 7 we writo down 2 ; to the 8 we write 

 down 1 ; and, lastly, to the 3 we write down 7- The 

 remainder thus written in place of the tabular logarithm, 

 is called the arithmetical complement of that logarithm, 

 The arithmetical complement thould always bo written 

 instead of a subtractive !<>,'. , and att<ltd ; and the 

 10, tluiN committed, is to oe corrected by suppressing 1C 

 in the result of the addition. 



The foregoing work should, therefore, be modified 

 thus : 



To find the angle B. 



At a - 95-12, Arith. Camp. . 8-0217 



: 6 - 98 1-9912 



: : sin. A, 32 J l.V . . . 97-7-' 



: sin B, 33 21' . 



97401 



To find the side c. 



As sin. A, 32 15', Arith. Comp. . -2728 



: sin. C, 114 24' . . .8 

 : : a _ 95-12 .... 1-9783 



102 -3 





2 Given A - 48 C 3', B = 40' 14', and c = 376, to find 

 a and b. 



C = 180 - (48 3" + 40 14') = 91 43'. 



To find a. 



As sin. C, 91 4S 7 , Arith. Comp. . -0002 



: sin. A, 48 3' ... . . 9-8714 



: : c = 376 2-5752 



: a =279-8 



2-44C8 



To find 5. 



As sin. C, 91 43' Arith. Comp. . -0002 



: sin. B, 40 14' . . 9-8102 



; : c - 376 2-5752 



243 

 = 355, c 



. 2-3850 

 336, and A = 49 26', to find 



58'. 



4. Given a 



3. Given a 

 6 and C. 



Ans, 6 =-465-3, C = 46' 

 310, B = 62 9', and C = 41 13'. 

 6 = 281 7, - 210, A = 76 38'. 



5. Given a - 70, 6 = 104, and B = 44 12". 



A - 27 SO 7 , C = 107 49', e - 142. 



6. Given b = 104, c = 142, and B = 44 12. 



There are two triangles having N Q _ -y ^, . ,^ .^. 

 these parts in each; the re- ( A I 63 37' or 27 69'! 



or 70. 



maining parts being as here 

 annexed. 



II. IVTien the given parts are two tidei and the in- 

 anyle. 



Investigation of the Ride. 



Let the two sides AC, BO (Fig. 8), and their included 

 angle A C B, be given, to find the remaining angles A, B, 

 of the triangle ABC. 



From the greater CA, of 



>he two given sides, cut off a 



part CD, equal to the less CB ; 



and also prolong A C, till C D' 



- C B : then C B = C D = C D', 



and consequently, with centre C, 



and radius C B, a circle may be 



iroumsoribed about D B D', so 



hat the angle D B D' is a right 



angle (Euc. 31 of III.), and therefore C DB is the com- 



ilement of D'. 



NowCDB-A+ABD: addCB D=CD B to each ; 

 then 2 C D B = B + A, .'. C D B = (B + A), \ 



.'. sin. ADB = sin. i (B -f A) (l) 



i.nd consequently, since D' is tho couip. of D, I ' ' 

 tin. A L>' B = cos. J (B + A) ; 







