diameters AD and BH be perpendicular to each other, they will divide the circle into quadrants. 194. In the first quadrant AB, the sine, cosine, tangent, &c. are considered all positive. In the second quadrant BD, the sine P'S' continues positive; because it is still on the upper side of the diameter AD, from which it is measured. But the cosine, which is measured from BH, becomes negative, as soon as it changes from the right to the left of this line. (Alg. 507.) In the third quadrant, the sine becomes negative, by changing from the upper side to the under side of DA. The cosine continues negative, being still on the left of BH. In the fourth quadrant, the sine continues negative. But the cosine becomes positive, by passing to the right of BH. 195. The signs of the tangents and secants may be derived from those of the sines and cosines. The relations of these several lines to each other must be such, that a uniform method of calculation may extend through the different quad rants. In the first quadrant, (Art. 93. Propor. 1.) Rx sin R: cos:: tan: sin, that is, Tan = COS The sign of the quotient is determined from the signs of the divisor and dividend. (Alg. 123.) The radius is considered as always positive. If then the sine and cosine be both positive, or both negative, the tangent will be positive. But if one of these be positive, while the other is negative, the tangent will be negative. Now by the preceding article, In the 2d quadrant, the sine is positive, and the cosine negative. The tangent must therefore be negative. In the 3d quadrant, the sine and cosine are both negative. In the fourth quadrant, the sine is negative, and the cosine positive. The tangent must therefore be negative. 196. By the 9th, 3d, and 6th proportions in art. 93, R2 1. Tan: R::R: cot, that is, Cot= tan Therefore, as radius is uniformly positive, the cotangent must have the same sign as the tangent. 4th The secant, therefore, must have the same sign as the cosinę. R2 3. Sin: R::R: cosec, that is, Cosec= sin The cosecant, therefore, must have the same sign as the sine. The versed sine, as it is measured from A, in one direction only, is invariably positive. 197. The tangent AT (Fig. 36.) increases, as the arc extends from A. towards B. See also Fig. 11. Near B the increase is very rapid; and when the difference between the arc and 90°, is less than any assignable quantity, the tangent is greater than any assignable quantity, and is said to be infinite. (Alg. 447.) If the arc is exactly 90 degrees, it has, strictly speaking, no tangent. For a tangent is a line, drawn perpendicular to the diameter which passes through one end of the arc, and extended till it meets a line proceeding from the centre through the other end. (Art. 84.) But if the arc is 90 degrees, as AB (Fig. 36.) the angle ACB is a right angle, and therefore AT is parallel to CB; so that, if these lines be extended ever so far, they can never meet. Still, as an arc infinitely near to 90° has a tangent infinitely great, it as frequently said, in concise terms, that the tangent of 90o is infinite. In the second quadrant, the tangent is, at first, infinitely great, and gradually diminishes, till at D it is reduced to nothing. In the third quadrant it increases again, becomes infinite near H, and is reduced to nothing at A. The cotangent is inversely as the tangent. It is therefore nothing at B and H, (Fig. 36.) and infinite near A and D. 198. The secant increases with the tangent, through the first quadrant, and becomes infinite near B; it then diminishes, in the second quadrant, till at D it is equal to the radius CD. In the third quadrant, it increases again, becomes infinite near H, after which it diminishes, till it becomes equal to radius. The cosecant decreases, as the secant increases, and v. v. It is therefore equal to radius at B and H, and infinite near A and D. 199. The sine increases through the first quadrant, till at B (Fig. 36.) it is equal to radius. See also Fig. 13. It then diminishes, and is reduced to nothing at D. In the third quadrant, it increases again, becomes equal to radius at H, and is reduced to nothing at A. The cosine decreases through the first quadrant, and is reduced to nothing at B. In the second quadrant, it increases till it becomes equal to radius at D. It then diminishes again, is reduced to nothing at H, and afterwards increases till it becomes equal to radius at A. In all these cases, the arc is supposed to begin at A, and to extend round in the direction of BDH. 200. The sine and cosine vary from nothing to radius, which they never exceed. The secant and cosecant are never less than radius, but may be greater than any given length. The tangent and cotangent have every value from nothing to infinity. Each of these lines, after reaching its greatest limit, begins to decrease; and as soon as it arrives at its least limit, begins to increase. Thus the sine begins to decrease, after becoming equal to radius, which is its greatest limit. But the secant begins to increase after becoming equal to radius, which is its least limit. 201. The substance of several of the preceding articles, is comprised in the following tables. The first shows the signs of the trigonometrical lines, in each of the quadrants of the circle. The other gives the values of these lines, at the extremity of each quadrant. Here r is put for radius, and ∞ for infinite. 202. By comparing these two tables, it will be seen, that sach of the trigonometrical lines changes from positive to negative, or from negative to positive, in that part of the circle in which the line is either nothing or infinite. Thus the tangent changes from positive to negative, in passing from the first quadrant to the second, through the place where it is infinite. It becomes positive again, in passing from the second quadrant to the third, through the point in which it is nothing. 203. There can be no more than 360 degrees in any circle. But a body may have a number of successive revolutions, in the same circle; as the earth moves round the sun, nearly in the same orbit, year after year. In astronomical calculations, it is frequently necessary to add together parts of different revolutions. The sum may be more than 360°. But a body which has made more than a complete revolution in a circle, is only brought back to a point which it had passed over before. So the sine, tangent, &c. of an arc greater than 360°, is the same as the sine, tangent, &c. of some arc less than 360°. If an entire circumference, or a number of circumferences be added to any arc, it will terminate in the same point as before. So that, if C be put for a whole circumference or 360°, and a be any arc what ever; sin x=sin (C+x)=sin (2C+x)=sin (3C+x), &c. tan x=tan (C+x)=tan (2C+x)=tan (3C+x), &c. 204. It is evident also, that, in a number of successive revolutions, in the same circle; The first quadrant must coincide with the 5th, 9th, 13th, 17th, The second, with the The third, with the The fourth, with the 6th, 10th, 14th, 18th, &c. 7th, 11th, 15th, 19th, &c. 8th, 12th, 16th, 20th, &c. 205. If an arc extending in a certain direction from a given point, be considered positive; an arc extending from the same point, in an opposite direction, is to be considered negative. (Alg. 507.) Thus, if the arc extending from A to S (Fig. 36.) be positive; an arc extending from A to S"" will be negative. The latter will not terminate in the same quadrant as the other; and the signs of the tabular lines must be accommodated to this circumstance. Thus the sine of AS will be positive, while that of AS"" will be negative. (Art. 194.) When a greater arc is subtracted from a less, if the latter be positive, the remainder must be negative. (Alg. 58, 9.) TRIGONOMETRICAL FORMULE. 206. From the view which has here been taken of the changes in the trigonometrical lines, it will be easy to see, in what parts of the circle each of them increases or decreases. But this does not determine their exact values, except at the extremities of the several quadrants. In the analytical investigations which are carried on by means of these lines, it is necessary to calculate the changes produced in them, by a given increase or diminution of the arcs to which they belong. In this there would be no difficulty, if the sines, tangents, &c. were proportioned to their arcs. But this is far from being the case. If an arc is doubled, its sine is not exactly doubled. Neither is its tangent or secant. We have to inquire, then, in what manner, the sine, tangent, &c. of one arc may be obtained, from those of other arcs already known. The problem on which almost the whole of this branch of analysis depends, consists in deriving, from the sines and cosines of two given arcs, expressions for the sine and cosine of their sum and difference. For, by addition and subtraction, a few arcs may be so combined and varied, as to produce others of almost every dimension. And the expressions for the tangents and secants may be deduced from those of the sines and cosines. Expressions for the SINE and COSINE of the SUM and DIFFERENCE of arcs. 207. Let a=AH, the greater of the given arcs, Then a+b=AH+HL=AL, the sum of the two arcs, Draw the chord DL, and the radius CH, which may be represented by R. As DH is, by construction, equal to HL; DQ is equal to QL, and therefore DL is perpendicular to CH. (Euc. 3. 3.) Draw DO, HN, QP, and LM, each perpendicular to AC; and DS and QB parallel to AC. From the definitions of the sine and cosine, (Art. 82, 9,) it is evident, that N |