Tan. (45 + 0)-tan. (45-0) = 



sin. (45 0) 

 cos. (45 0) 



sin. (45 + ff) cos. (45- 0)- sin. ( l.V- ft)cos. (45 + 0) 

 cos. (45 + 0) cos. (45 0) 



_ sin. 2 e _ 

 = cos. (45-S) cos. (90-46- 6) 

 ___ 2 sin. 20 _ 

 = 2cos. (45 6) sin. (46 9) 



_ 2sin. 20 _ L> sin. 2 

 "cos. (90 -2S) = cos. 29 

 . 2 tan. 2 



.-. tan. (45 + ff) = 2 tan. 2 + tan. (45 0) 



Since is less than 45. 2 is less than 45" + 0. and 

 hence, whatever be the value of 0, we shall have already 

 calculated tan. 2 0. before we need tan. 2 0. for the de- 

 termination of tan. (45 J + 0). 



Thus tan. 81 = 2 tan. 72+ tan. 9 



where before we calculate tan. 81 we shall already have 

 calculated tan. 72^. 



It is plain that if we know the tangents of angles from 

 to 90, we also know the cotangents of the angles 

 from 90 to 0. 



(66). Formulas of verification. 



There are many formulas by which the accuracy of the 

 tables, when calculated, can be tested. The following 

 are some of them : 



(1). Cos. 6 = sin. (30' + 0) + sin. (30 0). 



(2). Sin. + sin. (72 + 0) sin. (72 0) = sin. 

 (36 + 0) -sin. (36 0). 



(3). Cos. (90 0) + cos. (18 - 0) cos. (18 + 0) 

 = cos. (54 0) - cos. (54 + ff). 



The student will readily verify these formulas if he 

 remember the numerical values found in previous articles 

 for the sines, <Src., of 18. t 



(67). Another method of calculating Tablet of natural Sines 

 and Cosines. 



Besides the method already given for calculating the 

 natural sines and cosines of angles, there is another more 

 convenient than that. The following is an account of 

 this second method. 



We have 



Sin ' X = *~ 



1.2.3.4.5 



-*- 



Now suppose x to be an angle = . ^. Then 



n 2 



See ante, p. 621. 



t See ante, p. 621. 



m Tr 



Sin. -. -5 

 n 2 



m ir 

 ~n 2 



II i v I 



n I \ 2 I * 120 T 



362880 



\/r\ 



)(*) 



_ 5 /Y f^Y l i fY /^Y 

 H n / \ 2 J 6 v'v \ 2 / 504 



5040 



Now we ! 



TABLES OF TANGENTS, FTC.] MATHEMATICS. TRIGONOMETRY. 637 



such error, as well as to verify the correctness of the ' 

 calculations, at different points of their progress, when 

 no such errors exist, it is usual to interpose the values of 

 any such terms as can be calculated by independent 

 methods. Thus we have already seen that the sines, <fec. , 

 of the angles 15. 30 3 . 45 ... can be readily expressed, as 

 well as those of 18. 36. 54...* And that hence we 

 can obtain the sines, cfcc., of an angle of 3. These and 

 others will act as stops in the series, and also serve to 

 verify the accuracy of the calculation up to the point 

 where they are inserted. 



(65). The calculation of Tangents and Cotangents. 



By the methods now explained we can calculate the 



natural sines and cosines of all angles from to 90 for 



every minute of a degree. The natural tangents of 



angles between and 45 can be obtained by simple 



division, since tan. = , From 45 up to 90 we can 

 obtain the tangent by the formula, 



Tan. (45 + ff) = 2 tan. 2 + tan. (45- 0.) 

 To prove this formula. 



have already seen that 5- = 1-570796326794897. 



771 / TJt \ * 



Hence if we reduce the coefficients of . I . . . . to 



n \nj 



numbers, we have 



Sin.-- 90 

 n 



= - X 1.570796326794897 (-) X 0.645964097506246 

 n * n' 



+(*)* X 0.079692626246167 ()'x 0.004681754135319 

 +(") X 0.000160441184787 (-) X 0.000003598843235 

 +(-) X 0.000000056921729 (-) X 0.000000000668804 



W' ^ 71 



+(-) X 0.000000000006067 (-) X 0.000000000000044 



A similar formula can be calculated for cos. . 90 It 

 will be observed that is always a fraction less than 



2~, since we only require the sines and cosines of angles 



less than 45. Hence these series converge very rapidly, 

 and from them we can easily calculate the sines and 

 cosines for each degree from 1 up to 45. When these 

 are known, the nines and cosines of the angles for inter- 

 vals of 1', or if necessary 10" or of 1", can be found by 

 the "method of interpolations." 



The method of interpolations involves mathematics of 

 a higher order than is admissible in an elementary course. 

 The advanced reader will find an account of the appli- 

 cation of this method to the calculation of Tables of 

 natural sines and cosines in Airy's Treatise on Trigo- 

 nometry, in the Encyclopedia Metropolitans. 



Calculation of Logarithmic Sines, <tc. 



We now proceed to give an account of the method of 

 constructing Tables of the logarithms of sines, cosines, 

 \-c., of angles, 



N. B. The sines and cosines of angles are never greater 

 than 1. Consequently their logarithms are negative. 

 The numbers printed in the tables are always the loga- 

 rithms of the sines, <bc., with 10 added to them ; thus if L 

 denote the tabular logarithm, and log. the ordinary 

 logarithm 



L. sin. = log. sin. + 10. 



This notation will be observed throughout the following 

 pages. 



The Tables may be constructed by calculating the 

 logarithms of the natural sines and cosines. And then 

 as we have already found the natural sines and cosines 

 of angles for intervals of 1' from to 45, this table will 

 give us the log. sines, and cosines for intervals of 1' from 

 to 45. 



They are, however, more generally calculated by an in- 

 dependent process, of which the following is an account : 



(68). To obtain an expression for log. sin.. 90 ma form 



adapted for calculation. 

 We have already seen (page G34) that 



Sin. x 



