MATHEMATICS. TRIGONOMET11Y. 



[LOGARITHMIC TABLRS. 



may use the number of minutes in 0, omitting the odd 

 eooiuls, anil tlieu, tho former formula gives us L. sin. 0, 

 when or n* is given, and the latter gives us the n* 

 L. tin. is ghvn. 



Knul 0, having given that L. sin. - 7.2777613. On 

 looking into Tables we find that is between 6' and 7', 

 hence in this case cos. cos. 6' 



L.sin.0 -7-2777013 



I L. cos. 6' - 0- 6006608-10. 



+ 1-3522416 - 8-6477684-10. 



89LOO 



B 



a 



2-6921805 

 6921768 



97 

 90 



7. 



391.0086 = n. '.0 = 6', 31" .0086 



If we find n" by more refined tables 



L. sin. = T -2777613 

 L. sin. 6'. 31' = 7 "2777514 



99,00 



75,21 1 



23,790 

 22,563 



a 



.'. = 6'.31'.013. 



.'. by this method n = 391-013 



which does not differ materially from the result we ob- 

 tained by the less refined table and the formula. 



Find L. sin. e when = 7'.31".37. 



Here n = 451.37 L. cos. 6 = L. cos. 7' 

 L. sin. - log. n = 2 '6545327 



+ 3- L. cos. 7' 



33333330 

 + 1-3522416 = 1-3522416 



73401073 

 The more refined tables give us at once 



L. sin. 7' 31" = 7-3397511 



Diff. 1" = 9C19 x -3 = 2885,7 



07 = 673,33 



7-3401070 



In like manner if we have so small an angle that 

 0* 6* . . . may be omitted, and if 6 be an angle of * 

 seconds, 



o 3 







tan. 



/.log. tan. log. n ain. 1* ^- log. cos.0 



o 



L. tan. - log. n+ L. sin. 1" + ? . (10 L. cos. 0). 



3 



L. tan. - log. n + 11 -3522416 | L. cos. 0. 



This formula can be used in the same manner as the 

 one for L. sin. 0. 



(72). Delambrt'i method f or L. tines of small Angles. 



There is another method of treating tho L. sines of 

 mall angles, of which the following is an account. 



Sin. <?- O-J + ^-Ac. 



sin. ( . (0* 0*\\ 



. . log. A log. jl JT- - . r- } Omitting 0* 

 \ iM /> 



- 



26 156 



Iff* 0* 

 ~ M l6" + 180 



From this a table for log. 



a. 0. sin. 1* 







calculated. Kow suppose to contain n" 

 Then n sin. 1" = 



. sin. n* sin. B. sin. 1* 



can easily be 







L. sin. n* = log. n -f- L. 



sin. 0. sin. 1* 



and since L. 





is given by the Tables, if we 



know n we can at once find L. sin. n". And conversely 

 if we have given L. siu. 0, we can find n ; for sin. = 

 = n sin. 1" nearly. .'. log. n = L. sin. L. sin. 1* 

 nearly, whence we know n approximately. This will 



enable us to obtain the value of log. from 



the Tables with sufficient accuracy ; which being known, 



we have log. n = L. sin. log. 



sin. sin. 1* 



, which 



gives us n accurately. 

 It can easily be shown, that in calculating tables for log. 



siu - Q from the formula, 







, sin. 



log- TJ - AI 



e "^o '" ifco 



that for tables of seven places of decimals up to 6% 0* 



01 



can be omitted. 



STT 



For when is an angle of 5, = 



itr 







180 (180)* " -0000001,39927 . . . 



Let us now look back ami consider what wo have done 

 in the previous pages. (1). We have investigated tho 

 various general relations existing between the trigono- 

 metrical functions. (2). We nave investigated the 

 general relations existing between the sides and angles of 

 triangles. (3). We have fully discussed the mode of 

 constructing trigonometrical tables, i. e., the means of 

 obtaining numerical results from our trigonometrical 

 formulas. 



We now proceed to the actual numerical calculation of 

 the sides and angles of triangles, which we shall tind will 

 invariably consist in reducing formulas to numbers. For 

 instance, 



Given sin. (m -J- x) sin. (m r) = cos. (m n) 

 cos. (m + n) 



Find x when m = 12- 13' and n = 7- 37'. 



Since 

 and 



and 



sin. fm -f- x) sin. (m x) = 2 sin. x cos. m 



cos. (m n) cos. (m -f- n) = 2 sin. m sin. n. 



.'. 2 sin. x cos. tn = 2 sin. m. sin. n. 



.'. sin. x tan. m. sin. n. 



.'. log. sin. x = log. tan. m + log. sin. n. 



.'. log. sin. x -f 10 = log. tan. m. + 10 + log. 



sin. n + 10 10 



.'. L. sin. x = L. tan. m. + L. sin, n 10. 

 L. tan. m = 9-3354823 

 L. sin. n = 9-1223624 10 



8 '4578447 

 L. sin. 1 38'- 40* 8-4678369 



diff. 1* = 732,8 X 'I 

 Ans. ar = 1. 38'. < 



78 . 



73-28- 



