SINE

, or Right Sine, of an arc, in Trigonometry, a right line drawn from one extremity of the arc, perpendicular to the radius drawn to the other extremity of it: Or, it is half the chord of double the arc. Thus the line DE is the sine of the arc BD; either because it is drawn from one end D of that arc, perpendicular to CB the radius drawn to the other end B of the arc; or also because it is half the chord DF of double the arc DBF. For the same reason also DE is the Sine of the arc AD, which is the supplement of BD to a semicircle or 180 degrees; that is, every Sine is common to two arcs, which are supplements to each other, or whose sum make up a semicircle, or 180 degrees.

Hence the Sines increase always from nothing at B till they become the radius CG, which is the greatest, being the Sine of the quadrant BG. From hence they decrease all the way along the second quadrant from G to A, till they quite vanish at the point A, thereby shewing that the Sine of the semicircle BGA, or 180 degrees, is nothing. After this they are negative all the way along the next semicircle, or 3d and 4th quadrants AFB, being drawn on the opposite side, or downwards from the diameter AB.

Whole Sine, or Sinus Totus, is the Sine of the quadrant BG, or of 90 degrees; that is, the Whole Sine is the same with the radius CG.

Sine-Complement, or Cosine, is the sine of an arc DG, which is the complement of another arc BD, to a quadrant. That is, the line DH is the Cosine of the arc BD; because it is the sine of DG which is the complement of BD. And for the same reason DE is the Cosine of DG. Hence the sine and Cosine and radius, of any arc, form a right-angled triangle CDE or CDH, of which the radius CD is the hypotenuse; and therefore the square of the radius is equal to the sum of the squares of the sine and Cosine of any arc, that is, .

It is evident that the Cosine of o or nothing, is the whole radius CB. From B, where this Cosine is greatest, the Cosine decreases as the arc increases from B along the quadrant BDG, till it become o for the complete quadrant BG. After this, the Cosines, decreasing, become negative more and more all the way to the complete semicircle at A. Then the Cosines increase again all the way from A through I to B; at I the negation is destroyed, and the Cosine is equal to o or nothing; from I to B it is positive, and at B it is again become equal to the radius. So that, in general, the Cosines in the 1st and 4th quadrants are positive, but in the 2d and 3d negative.

Versed Sine, is the part of the diameter between the sine and the arc. So BE is the Versed Sine of the arc BD, and AE the Versed Sine of AD, also GH the Versed Sine of DG, &c. All Versed Sines are affirmative. The sum of the Versed Sine and cosine, of any arc or angle, is equal to the radius, that is, .—The sine, cosine, and Versed Sine, of an arc, are also the same of an angle, or the number of degrees &c, which it measures.

The Sines &c, of every degree and minute in a quadrant, are calculated to the radius 1, and ranged in tables for use. But because operations with these natural Sines require much labour in multiplying and dividing by them, the logarithms of them are taken, and ranged in tables also; and these logarithmic Sines are commonly used in practice, instead of the natural ones, as they require only additions and subtractions, instead of the multiplications and divisions. For the method of constructing the scales of Sines &c, see the article Scale.

The Sines were introduced into trigonometry by the Arabians. And for the etymology of the word Sine see Introduction to my Logarithms, pa. 17 &c. And the various ways of calculating tables of the Sines, may be seen in the same place, pa. 13 &c.

Theorems for the Sines, Cosines, &c, one from another. From the definitions of them, and the common property of right-angled triangles, with that of the circle, viz, that , are easily deduced these following values of the Sines, &c, viz, putting s = the sine DE, c = the cosine CE, v = versed sine BE, v = suppl. versed sine AE, r = radius AC or CB, a = arc BD; then | . See many other curious expressions of this kind in Bougainville's Calcul Integral, and in Bertrand's Mathematics.

From some of the foregoing theorems the Sines of a great variety of angles, or number of degrees, may be computed. Ex. gr. as below.

Radius being 1. Then for multiple arcs: the ;

That is, multiplying any Sine or cosine by 2c, and the next preceding Sine or cosine subtracted from it, it gives the next following Sine or cosine. Hence

And, in general,

In estimating the quantity of the Sines &c, we assume radius for unity; and then compute the quantity of the Sines, tangents, and secants, in fractions of it. From Ptolomy's Almagest we learn, that the ancients divided the radius into 60 parts, which they called degrees, and thence determined the chords in mi- | nutes, seconds, and thirds; that is, in sexagesimal fractions of the radius, which they likewise used in the resolution of triangles. As to the Sines, tangents and secants, they are modern inventions; the Sines being introduced by the Moors or Saracens, and the tangents and secants afterwards by the Europeans. See Introd. to my Logs. pa. 1 to 19.

Regiomontanus, at first, with the ancients, divided the radius into 60 degrees; and determined the Sines of the several degrees in decimal fractions of it. But he afterwards found it would be more convenient to assume 1 for radius, or 1 with any number of cyphers, and take the Sines in decimal parts of it; and thus he introduced the present method in trigonometry. In this way, different authors have divided the radius into more or fewer decimal parts; but in the common tables of Sines and tangents, the radius is conceived as divided into 10000000 parts; by which all the Sines are estimated.

An idea of some of the modes of constructing the tables of Sines, may be conceived from what here follows: First, by common geometry the sides of some of the regular polygons inscribed in the circle are computed, from the given radius, which will be the chords of certain portions of the circumference, denoted by the number of the sides; viz, the side of the triangle the chord of the 3d part, or 120 degrees; the side of the pentagon the chord of the 5th part, or 72 degrees; the side of the hexagon the chord of the 6th part, or 60 degrees; the side of the octagon the chord of the 8th part, or 45 degrees; and so on. By this means there are obtained the chords of several of such arcs; and the halves of these chords will be the Sines of the halves of the same arcs. Then the theorem will give the cosines of the same half arcs. Next, by bisecting these arcs continually, there will be found the Sines and cosines of a continued series as far as we please by these two theorems, . Then, by the theorems for the sums and differences of arcs, from the foregoing series, will be derived the Sines and cosines of various other arcs, till we arrive at length at the arc of 1′, or 1″, &c, whose Sine and cosine thus become known.

Or, rather, the sine of 1 minute will be much more easily found from the series , because the arc is equal to its Sine in small arcs; whence s = a only in such small arcs. But the length of the arc of 180° or 10800′ is known to be 3.14159265, &c; therefore, by proportion, as 10800′ : 1′ :: 3.14159265 : 0.0002908882 = a the arc or s the sine of 1′, which number is true to the last place of decimals. Then, for the cosine of 1′, it is the cosine of the same 1′.

Hence we shall readily obtain the Sines and cosines of all the multiples of 1′, as of 2′, 3′ 4′ 5′, &c, by the application of these two theorems, ; for supposing a = the arc of 1, then c = 0.9999999577, and taking n successively, equal to 1, 2, 3, 4, &c, the theorems for the Sines and cosines give severally the Sines and cosines of 1′, 2′, 3′, 4′, &c; viz, the Sines thus: .

In this manner then all the Sines and cosines are made, by only one constant multiplication and a subtraction, up to 30 degrees, forming thus the Sines of the first and last 30 degrees of the quadrant, or from 0 to 30° and from 60° to 90°; or, which will be much the same thing, the Sines only may be thus computed all the way up to 60°.

Then the Sines of the remaining 30°, from 60 to 90, will be found by one addition only for each of them, by means of this theorem, viz, ; that is, to the sine of any arc below 60°, add the Sine of its defect below 60, and the sum will be the Sine of another arc which is just as much above 60.

The Sines of all arcs being thus found, they give also very easily the versed sines, the tangents, and the secants. The versed sines are only the arithmetical complements to 1, that is, each cosine taken from the radius 1.

The tangents are found by these three theorems:

1. As cosine to sine, so is radius to tangent.

2. Radius is a mean proportional between the tangent and cotangent.

3. Half the difference between the tangent and cotangent, is equal to the tangent of the difference between the arc and its complement. Or, the sum arising from the addition of double the tangent of an arc with the tangent of half its complement, is equal to the tangent of the sum of that arc and the said half complement.

By the 1st and 2d of these theorems, the tangents are to be found for one half of the quadrant: then the other half of them will be found by one single addition, or subtraction, for each, by the 3d theorem.

This done, the secants will be all found by addition or subtraction only, by these two theorems: 1st. The secant of an arc, is equal to the sum of its tangent and the tangent of half its complement. 2nd. The secant | of an arc, is equal to the difference between the tangent of that arc and the tangent of the arc added to half its complement.

Artificial Sines, are the logarithmic Sines, or the logarithms of the Sines.

Curve or Figure of the Sines. See Figure of the Sines, &c. To what is there said of the figure of the Sines, may be here added as follows, from a property just given above, viz, if x denote the absciss of this curve, or the corresponding circular arc, and y its ordinate, or the Sine of that arc; then the equation of the curve will be this, ; where h = 2.718281828, &c, the number whose hyp. log. is 1.

Line of Sines, is a line on the sector, or Gunter's scale, &c, divided according to the Sines, or expressing the Sines. See those articles.

Sine of Incidence, or of Refraction, is used for the Sine of the angle of incidence, &c.

SINICAL Quadrant, is a quadrant, made of wood or metal, with lines drawn from each side intersecting one another, with an index, divided by sines, also with 90 degrees on the limb, and two sights at the edge. Its use is to take the altitude of the sun. Instead of the sines, it is sometimes divided all into equal parts; and then it is used by seamen to resolve, by inspection, any problem of plane sailing.