MATHEMATICS. TRIGONOMETRY. 



[OAtCUlATIOX OP TABT.HI. 



.' tan. 

 



or tan. 070 



0* 

 (2). To show that sin. 7 9 f 







for in. 6 2 sin. y eo. y 



. 

 - 2 tan. -j- co*.-5 











v v\ 



-2 tan. 2 (l-sm. J 2 ) 



9 



tan. 



-0. 



f'-JTL * 



T-1 1 r- 





 sin. -v, 



ton. 



I 



sm. 



I 



Now .5 being 7 1 and < - - > being 

 ^. 



a I 2 J 



,0 s 



I -v" must be Z 1 



0* 





 sin. -g 



.*. sin. must be 7 ( 1 -j-) 



3 

 .". sin. > - 



(60). To calculate the value of sin. 1' an<i cos. 1'. 

 Let be an angle of 1' measured by the circular 

 measure. Then, 



= TT -' = T 



180 180 X 60 



. '. Taking the value of IT previously given, 

 = -00029088820. 



f = 000000000006 



Hence, if we only take in the first ten decimal places, 



sin. 1' = 0002908882. 

 And since 



cos. 1' - V' 1 sin.* 1' = 1 4- sin-* * -,rsin. 4 1'.. . 



& O 



= . 9999999577. 



COE. Henoe, it is plain that if our approximation do 

 not extend beyond ten places of decimals, 



Circular measure of angle of 1 = sin. 1' . 

 Similarly, Circular measure of angle of 1* = sin. 1*. 



.'. Circular measure of angle of n" = n sin. 1*, 

 provided n /_ 60. 



(61). To calculate the Sines o/2 / 3' 4'. ... 



Since sin. (A + B) -f- sin. (A B) = 2 cos. B, sin. A. 

 wo have sin. (n + 1)' + tn. (n 1)' 2 cos. 1' sin. n'. 



Now cos. 1' _ 1 0000000423 . - 1 Jfe. 



i. (u + 1)'-}- sin. (n l)' = 2in n' 2k sin. n'. 

 .'.sin. (n-j- 1)' sin.7i'=> sinV sin. (n 1)' 2 A- sin n'. 



This formula is very convenient for calculating the sines 

 and cosines of successive angles. Thus, 



Sin. V Hin. 1' - sin. 1' 2 k sin, 1'. 

 Sin 3' sin. '2' = sin. 2' sin. 1' 2 k sin. V 

 Sin. 4' in. 3' =- sin. 3' sin. 2' 2 k sin. 3'; 

 and so on. It will be observed that the first member of 

 the right-hand side of each equation i.sgivon by the former 

 equation. So that th" only tenn n-quiriin; multipli- 

 cation in 2*. sin. 1', 2A-in. 2', 2k sin. '.'.' in f.-ach equation. 

 Thin multiplication cim be greatly facilitated l.y forming 

 ft Ublo in which 2 k in multiplied by each digit, thus : 



2k _ 



3 2538 



4 



5 



8384 



4230 



6 



- 



8 6708 



9 7614 



where the seven leros in front of the significant digits of 

 2 k are suppressed. By means of this table, which re- 

 sembles that of the proportional parts in the table of 

 logarithms, the multiplication can be performed by 

 means of addition only. It will also be observed that in 

 the case of the sines of the first few minutes the products 

 of /; will have more than ten zeros, and therefore can be 

 omitted, and that under all circumstances they will have 

 at the least seven, so that the multiplication is soon per- 

 formed. Thus, suppose 



Sin. p' = 3759264827 

 2k. sin. p' 0000000- 



59,3 



.0000000318 



By this means we can successively obtain the sines 

 1' y 3' 4', and so on for every minute up to 45. 



(62). To obtain the Cosines of the Angles V 3' 4', <tc. 



Since cos. (A + B) + cos. (A B) = 2 cos. A cos. B. 

 .". Cos. ( + 1)' -f cos. (ji 1)' = 2 cos. 1 cos. n'. 

 Hence as before 

 Cos. (n + 1)' cos. n' = cos. n' cos. n 1' 2k 



cos. n'. 



or, since the cosine continually decreases, 

 Cos. n' cos. n-)-l' = cos. n 1' + cos. n' + 2fccos. n'. 

 Hence 



Cos. 1' cos. 2' = cos. 0' cos. 1' -f 2fc cos. 1'. 

 Cos. 2' cos. 3' = cos. 1' cos. 2' + 2k cos. 2'. 

 Cos. 3' cos. 4' = cos. y cos. 3' + 2k cos. 3'. 

 The method of calculation is precisely similar to that 

 of the sines. It will be further observed, that since sin. 

 (90 A) = cos. A, that we need not to continue the cal- 

 culations of the sines and cosines of the ancles beyond 

 45. For example, if we know sin. 23 15' and cosin. 

 23 J5', these are respectively cosin. 66 45' and sin. 

 00 45'. 



(63) . Simplification of calculation in case of certain Angles. 



Again, the calculation of sines and cosines of angles 

 greater than 30 can be very much simplified ; for 

 Sin. X30 + 0) + sin. (30 0) =- 2 sin . 30 cos. = cos. 



Since sin. 30= i- 



1 



." . sin. (30 + 0) = cos. sin. (30 0) ; 

 now if 30 + 9 be less than 60, then 0, and 30 are 

 each less than 30. Hence by our previous calculations 

 we know both cos. and sin. (30 9), and therefore 

 obtain sin. (30 + 0) by subtraction. 



Thus sin. (41. 15') = cos. (11, 150 sin. (18. 450- 

 It is plain that by this formula we can calculate the 

 sines of angles from 30 to 60. 



And since cos. (60 0) - sin. (30 + 0) this calcula- 

 tion of the sines from 30 to 60 gives the sines from 

 30 to 45, and the cosines from 45 to 30 ; which is what 

 we want to complete the tables from to 45. Thus if 

 we calculate by the preceding formula, sin. (61. 3:!'), 

 this is the same thing as cos. (38. 27')- 



(64). Method of checking the calculation. 



It will be observed, that according to the method 

 above given, tin- sine of a given number of degrees or 

 minutes is infi-rred from the sine of the nun 



ceding. Hence, if an error be made at any < ] 



say in sin. 3 15', it will bo propagated in of 



every succeeding angle ; to arrest the progress of any 



