Dividing the formulæ marked () by those marked () BEFORE proceeding to apply the formulæ deduced in the last chapter to the solution of triangles, we shall make a few remarks upon the construction of those tables, by means of which we are enabled to reduce our trigonometrical calculations to numerical results. It is manifest, from definitions 1o, 2o, 3o, &c. that the various trigonometrical quantities, the sine, the cosine, the tangent, &c. are abstract numbers representing the comparative length of certain lines. We have already obtained the numerical value of these quantities in a few particular cases, and we shall now show how the numbers, corresponding to angles of every degree of magnitude, may be obtained by the application of the most simple principles. The numbers corresponding to the sine, cosine, &c. of all angles from 1" up to 90°, when arranged in a table, form what is called the Trigonometrical canon. The first operation to be performed is To compute the numerical value of the sine and cosine of 1'. We have seen, Chap. II. formula (j) that By which formula the sine of any angle is given in terms of the sine of twice that angle. and applying the above formula, we have sin. 15°=√1—sin.' 30° But by Chap. II. sin. 30°= .. sin.2 30°=1 sin. 15°√√√1. = =√2√3 =.2588190 Similarly, sin. 7°30'=√ √ sin.' 15° =√}{√(.2588190) =.1305268 &c.=&c. It is manifest, that, by continuing the process, we shall obtain in succession the sines of 3°45', of 1°52'30", &c. In this way we find 30° Sin. or sin. 1' 45" 28" iv 30.0005113269, &c. 211 30° Sin. or sin. 52" 44" 3v 45 =.0002556634, &c. 212 From which it appears, that, when the operation above mentioned has been repeated so many times, the sine of the arc is halved at the same time that the arc itself is bisected: that is, The sines of very small arcs are nearly proportional to the arcs themselves. Hence we shall have Sin. 52" 44"" 3iv 45o : sin. 1':: 52′′ 44′′ 31 45′′ : l' The sine and cosine of 1' being thus determined, we shall proceed to show in what manner we shall now be enabled to compute the sines and cosines of all superior angles. By formula (m) Chap. II. Sin. (n+1)=2 cos. sin. n sin. (n-1) If we suppose 0=1' and n to be taken to the numbers 1, 2, 3, in succession, we find ... Sin. 2' 2 cos. 1' sin. 1'- sin. 0.0005817764 =cos. 89° 58' Sin. 3' 2 cos. 1' sin. 2'- sin. 1'= .0008726645...=cos. 89° 57′ Sin. 4' 2 cos. 1' sin. 3'- sin. 2'.0011635526 ... cos. 89° 56' &c.=&c. Again, by employing formula (0), Chap. II. Cos. (n+1) 8=2 cos. cos. n 8-cos. (n-1) If, as before, we suppose =1' and n=1, 2, 3, ..... in succession, sin. 89° 58' sin. 89° 57' =sin. 89° 56' It is manifest, that, by continuing the above processes, we shall obtain the numerical values of the sines and cosines of all angles from 1' up to 90°. These being determined, the tangents, cotangents, &c. may be calculated by means of the relations established in table I. The above operations are exceedingly laborious, but require a knowledge of the fundamental rules of arithmetic alone. It is manifest that, in employing this method, an error committed in the sine or cosine of an inferior arc, will entail errors on the sines or cosines of all succeeding arcs. Hence is created the necessity of some check on the computist, and of some independent mode of examining the accuracy of the computation. For this purpose, formulæ, derived immediately from established properties, are employed; if the numerical results from these formulæ agree with the results obtained by a regular process of computation, then it is almost a certain conclusion that the latter process has been rightly conducted. Formulæ employed for this purpose are called formulæ of verification, and of these any number may be obtained; it will be sufficient for our present purpose to give one. sin. 12° 30'=√1+sin. 25° ±√1— sin. 25° cos. 12° 30'√1+sin. 25° 1- sin. 25° tab. I. Hence, if the values of the sine and cosine of 12° 30', and of the sine of 25° obtained by the method already explained, when substituted in these equations, render the two members identical, we conclude that our operations are correct. The values of the sine and cosine of 30°, 45°, 60°, &c. which were obtained in Chap. II., may be employed as formulæ of verification. We can obtain finite expressions, although under an incommensurable form, for the sines of arcs of 3°, and all the multiples of 3°, i. e. for 3o, 6o, 9°, 12°, 15°,18°, 21°, 24°, 27°, 30°, 33°, 36°, 39°, 42°, 45°, 48°, 51°, 54°, 57°, 60°, 63°, 66°, 69°, 72°, 75°, 78°, 81°, 84°, 87°, 90°. We first obtain the values of the sines 30°, 45°, 60°, 18°, and from these we obtain all the others, by means of the formulæ, for Sin. (+6), sin. (B), &c. The numerical value of the trigonometrical functions have been calculated by some to ten places of figures, by others as far as twelve. We must have tables calculated to ten places to have the seconds and tenths of a second with precision, when we make use of the sines of angles which differ but little from 90°, or of the cosines of angles of a few seconds only. Tables in general, however, are calculated as far as seven places only, and these give results sufficiently accurate For all ordinary purposes. |