2 creases from 1 when 0 = 0, to infinity when 0=: it then changes its sign and decreases from infinity to 1 when 07: from = to 0=2π, it decreases from infinity to 1: the cosecant of 0 is π infinity when 0 = 0, and 1 when 0 =: it becomes infinity again 2 795. The relations of the tangent and cotangent with the The relasecant and cosecant are expressed by the equations (Art. 787, tangent and No. 10), tions of the cotangent with the secant and cosecant. √(1 + tan3 0) = sec 0, √(1 + cot3 0) = cosec 0. 796. The versed sine of an angle is the abbreviated sion for 1 cos 0: it is written vers 0. If BAP be an angle, and if PM be drawn perpendicular BM to AB (Fig. in Note, Art. 785), then the ratio is the versed sine of the angle BAP*. It never changes its sign, and the limits of its values are 0 when = 0, and 2 when 0 = = it increases whilst increases from 0 to я, and it diminishes whilst 0 increases from π to 2π. The use of this term is now almost entirely abandoned. The versed sine was formerly defined as the line BM, intercepted between the beginning B of the arc and the perpendicular PM drawn through its extremity: it was sometimes called the sagitta of an arc, for if the arc PB was doubled, and PM produced to make a complete chord, then BM would be the position of the arrow upon the stretched bow: it is chiefly with a view to make the writings of the older mathematicians intelligible, that it is expedient to refer to terms and definitions which have now fallen into disuse. CHAPTER XXIX. What is meant by a canon of sines, &c. ON THE CONSTRUCTION OF A CANON OF SINES AND COSINES, 797. WE have hitherto considered sines and cosines, tangents and cotangents as possessing determinate values for determinate sines, co- angles, without attempting to assign them, except in the cases of angles of 45o, 30° and 18°, and their successive binary multiples and submultiples (Arts. 766, 709, 777): in the present Chapter we shall proceed to shew in what manner their numerical values may be determined generally for every minute and degree of the quadrant, with a view to the construction of a Table or Canon, in which those successive values may be registered: for in the ordinary applications of Trigonometry, the sine or cosine, corresponding to a given angle, and conversely, the angle corresponding to a given sine or cosine, are not found, by the actual calculation of their values, but always by reference to such a Table. To find the sine and cosine of 1'. 798. As the basis of our enquiries, we shall begin with the calculation of the numerical values of the sine and cosine of 1'. If, in the formula (Art. 776), we replace successively by 30° {whose sine is, (Art. 766)}, 211, we shall find, and so on as far as 30o fore the first of the successive binary submultiples of 30° which is less than 1': and inasmuch as the sines of very small angles increase and diminish very nearly in the same proportion with the angles themselves*, it follows that The corresponding value of the cosine of 1' derived from the equation gives us cos 1'=√(1-sin2 1') cos 1'= .9999999577. sine and 799. The knowledge of the sine and cosine of 1' of a degree, Given the will form the basis of our calculation of the sines and cosines of cosine of 1', all angles differing from each other by 1', between an angle of 1' and 90°. For this purpose we make use of the formula, (Art. 774), * The truth of this proposition may be inferred from the equation to construct a canon of sines and cosines. sin 20 = 2 cos 0 sin or sin 20 2 cos 0, where, when is very small, cos is very nearly equal to 1, and therefore the sine is almost exactly doubled when the angle is doubled, and conversely: .9999999674, which differs from 1 by .0000000326 only again, 211 from whence it will be seen, for very small angles, how nearly the sines are bisected, when the angles are so. Not neces ceed be If we replace by 1', and substitute successively the natural numbers 1, 2, 3, 4, &c. for n, we shall get In this manner we might proceed as far as the sine and cosine sary to pro- of 45°, in proceeding beyond which we shall find the same values of yond 45o. sines and cosines which have been previously determined: for inasmuch as sin (45° + 0) = cos (45o — 0) and cos (45o + 0) = sin (45° — 0), it will follow that the sines of the angles in the series ascending from 45° will be severally equal to the cosines of the complements of angles in the descending series which have been already determined, and conversely: it thus becomes necessary to construct the canon of sines and cosines as far as 45°, and no further. Beyond 30° the process of may be 800. But it will be observed that the processes of calculation, which are founded upon the formulæ in the last Article, calculation involve the formation of a very laborious product, which it is greatly sim- expedient to supersede, when practicable, by others which require the more simple and expeditious operations of addition and subtraction only: one of the most convenient of these is the formula plified. sin (30° + 0) = cos 0 - sin (30° - 0)*, by which the sines of the angles from 30° to 60° may be calculated by the mere subtraction of the sines of (30°-0) (where may have every value from 1' to 30°), from the cosine of : and since sin (30° + 0) = cos (60° – 0), the determination of the series of sines from 30° to 60° will give us the corresponding series of cosines in an inverse order. * For sin (300 + 0) + sin (30o - 0) = 2 sin 30o cos 0 = cos 0, since 2 sin 30o = 1. results ob 801. In the formula which we have given, we have cal- Process of verifying culated the values of the sines and cosines, at least as far as the correct30°, independently of each other, from the ascertained values ness of the of the sine and cosine of 1', and consequently we may use the tained. values of the sine and cosine of any assigned angle 0, to test the accuracy of the calculation, by substituting them in the equation cos 0+ sin3 0 = 1: for if this equation be not satisfied, the values of one or both of them are necessarily incorrect: thus if we take the values of the sine and cosine of 5', which are found in Art. 799, we shall find and cos2 5'.99999788460111872929, sin 5'.00000211539745892836, cos2 5' + sin2 5′ = .99999999999857765765, a number which differs from 1 by a quantity less than .0000000000015, a discrepancy which is referrible to the influence of terms in the calculated values of cos 5' and sin 5', which are necessarily omitted, as being beyond the 10th place of decimals, to which the registered values are limited. verification 802. The methods of calculating the successive terms of a Process of canon of sines and cosines, are methods of continuation, where by the inan error committed in the determination of any one of them terposition of stops or is transmitted to all those which succeed it: and it is in order of values calculated to arrest the continued propagation of errors, as well as to verify by other the correctness of the calculations, at different points of their methods. progress, when no such errors exist, that it is usual to interpose, as stops, the values of any such terms in the series as can be determined by independent methods. Such are the sines and cosines of 45°, 30° and 18°, and their binary submultiples, or of the sum and difference of any angles in the series thus formed*. of a canon 803. A canon of tangents may be formed from a canon of Formation sines and cosines, by dividing the sines by the cosines: and a of tangents canon of cotangents may be similarly formed by dividing the and cotancosines by the sines. If however we have found the tangents We thus get sin (18° 15°) sin 30= sin 18° cos 15° - cos 18° sin 15o = .0523360. = gents. |