Let B=1, then sin 30° l'cos 1'-sin 29° 59′= .99999996-.4997481.5002519. B=2' .Sin 30' 2'=cos 2'-sin 29° 58'.9999998-.4994961 .5005037. B=3' .Sin 30° 3' cos 3'-sin 29° 57'-.5007556. = B=4' .Sin 30° 4'=cos 4'-sin 29° 56′.5010073. B=5' Sin 30° 5' cos 5'-sin 29° 55'-.5012591. &c. &c. &c. 53. Having computed the sines in this manner as far as 60°, the co-sines from 30° to 60° will likewise be known; the co-sine of any arc above 30° being the same as the sine of an arc as much below 60o. Thus, cos 30° l'=sin 59° 59′=.8658799. cos 30° 2'==sin 59° 58′=.8657344. cos 30° 3' sin 59° 57′.8655887. 54. To find the sines and co-sines from 60° to 90o. The sine of any arc above 60° is the same as the co-sine of an arc at the same distance below 30°; and in like manner, the co-sine of an arc above 60° is the same as the sine of an arc equally below 30°: thus, Sin 60° l'=cos 29° 59′.8661708. cos 60° 1′=sin 29° 59' .4997481. = Sin 60° 2'=cos 29. 58' Sin 60° 3′=cos 22° 57′ &c. cos 60° 2' sin 29° 58′ 55. To find the versed sines and co-versed sines of the quadrant. In any arc less than 90° the versed sine is found by subtracting the co-sine from radius (cor. Art. 14.); and in arcs greater than 90°, it is found by adding the co-sine to radius: thus, • The learner is supposed (in this and the following articles,) to have computed all the preceding sines, co-sines, tangents, &c.; if he has not, he must, in order to work the examples, take them from a table. By means of the formulæ here given, any natural sine, tangent, secant, &c. in the table, which is supected to be wrong, may be examined, and if necessary, corrected. ver. sin l'=1—cos l'=(1-.99999996=).00000004 ver. sin 2′=1—cos 2′=(1—.9999998=).0000002 ver. sin 3'=1-cos 3′-(1-.9999996=).0000004. ver. sin 4'-1-cos 4'-0000007 ver. sin 5'=1-cos 5=.0000011 ver. sin 90° l'=1+cos 89° 69′1.0002909 ver. sin 90° 2'=1+cos 89° 58′=1.0005818 Versed sines for arcs greater than 90, do not occur in the common tables. 56. The co-versed sine is found by subtracting the sine from the radius (cor. Art. 15,); thus, co-versed sin l'=1-sin l'=(1—.0002909=) .9997091 co-versed sin 2′'=1—sin 2′=(1—.0005818=) .9994182 co-versed sin 3'=1—sin 3′-(1-.0008727=) .9991273 57. To find the tangents and co-tangents from 0° to 45o. 58. To find the tangents and co-tangents from 45° to 90°. Because (formula 26. Art. 49.) the tangents of 45°+B= tan. 45°-B+2 tan. 2 B; therefore if And in this manner the tangent of every succeeding minute of the remainder of the quadrant, must be found. 59. To find the secants and co-secants of the quadrant. By the second analogy Art. 35. we have sec A radius being unity; whence if 1 the COS A 60. By this method the secants and co-secants of every minute of the quadrant may be computed, but it is necessary to employ it only for the odd minutes; the secants and co-secants of the even minutes may be obtained by a process which is somewhat more easy; as follows By art. 41. tan A+ sec A-co-tan + 90~ A. =) co-tan 44° 59′—tan 2′-(1.0005819—.0005818=)1.0000001. 61. The numbers thus computed are called natural sines, tangents, &c. they are computed for every degree and minute of the quadrant, and arranged in eight columns, titled at the top and bottom; these together constitute the table of natural sines, tangents, &c. directions for the use of which are given in the introduction to every system of trigonometrical tables P. OF THE TABLE OF LOGARITHMIC SINES, TANGENTS, &c. 62. The logarithmic or artificial sines, tangents, &c. are the logarithms of the sines, tangents, &c. computed to the radius 101010000000000; for since the sines, co-sines, and many of the versed sines and tangents computed to the radius 1 are proper fractions, their logarithms will have a negative index, (vol. 1. page 287.) but by assuming the above number for radius, these fractions become whole numbers, their logarithms affirmative, and the figures expressing any sine, tangent, &c. will be the same in both cases, as likewise their logarithms, excepting the indices, which (as we have observed) will be frequently negative in the former case, but always affirmative in the latter; therefore, in order to find the logarithm of the sine of an arc, calculated to the radius 100, we must add 10 to the index of the logarithm of the same sine to the radius 1: for, let r=the radius, s=the sine of any arc to rad. r, R= a different radius, S=the sine of an arc (to rad. R) similar to the former, then (Art. 38.) P For an account of the tables of sines, tangents, &c. with ample directions to assist the learner in their use, see Dr. Hutton's Mathematical Tables, 2 edit. p. 151, 152. r: R::s: S; which if r=1 and R=101o, becomes 1 : 1010 :: s: S, S=1010 xs, log. S=10x log. 10+ log. s=(since log. S,. ·.· 10=1) 10+ log. s. Q. E. D. 10+log. EXAMPLES.-1. To find the logarithmic sine of L'. To log. of .0002909 (=sin 1′) =−4.4637437 The sum is of 1' to radius 10000000000.a 10 6.4637437 the log. sine 2. To find the logarithmic tangent of 2o. 35′= The sum tangent of 2, 35'. 10 8.6543527 is the log. 3. To find the logarithmic secant of 7°. 5'. The log. of 1.0076908 (=sec 7° 5′)=0.0033273 The log. secant of 7o 5′= 10 10.0033273 4. To find the logarithmic versed sine of 20° 12'. To log. of .0615070 (=ver. s. of 20° 12′) =—2.7889245 Add.. ..... The log. versed sine of 20° 12'= 10 8.7889245 In this manner the logarithmic sines, co-sines, tangents, &c. are computed; viz. by adding 10 to the index of the logarithm of the natural sine, co-sine, tangent, &c. respectively corresponding to the radius 1 '. Having shewn the method of computing the trigonometrical canon, both in natural numbers and logarithms, the next thing to be done is to demonstrate the propositions on which the practical part of trigonometry is founded. THE FUNDAMENTAL THEOREMS OF PLANE TRIGONOMETRY. 63. In a right angled triangle the hypothenuse: is to either of the sides as radius: to the sine of the angle opposite to that side. By the preceding rules any logarithmic sine, tangent, secant, &c. in the table, suspected to be inaccurate, may be examined, and the error (if any should be found) corrected. The log. sine of 1' (as here given) exceeds the truth by .0000176 because the sine of 1' is only .000290888 and not .0002909. See Art. 51. |