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THE FIRST PRINCIPLES OF TRIGONOMETRICAL ANALYSIS.
Art. 191. In the preceding sections, sines, tangents, and secants have been employed in calculating the sides and angles of triangles. But the use of these lines is not confined to this object. Important assistance is derived from them, in conducting many of the investigations in the higher branches of analysis, particularly in physical astronomy. It does not belong to an elementary treatise of trigonometry, to prosecute these inquiries to any considerable extent. But this is the proper place for preparing the formula, the applications of which are to be made elsewhere.
Positive and negative signs in trigonometry.
192. Before entering on a particular consideration of the algebraic expressions which are produced by combinations of the several trigonometrical lines, it will be necessary to attend to the positive and negative signs in the different quarters of the circle. The sines, tangents, &c. in the tables, are calculated for a single quadrant only. But these are made to answer for the whole circle. For they are of the same length in each of the four quadrants. (Art. 90.) Some of them, however are positive; while others are negative. In algebraic processes, this distinction must not be neglected.
193. For the purpose of tracing the changes of the signs, in different parts of the circle, let it be supposed that a straight line CT (Fig. 36.) is fixed at one end C, while the other end is carried round, like a rod moving on a pivot; so that the point S shall describe the circle ABDH. If the two diameters AD and BH be perpendicular to each other, they will divide the circle into quadrants.
* Euler's Analysis of Infinites, Hutton's Mathematics, Lacroix's Differential Calculus, Mansfield's Essays, Legendre's, Lacroix's, Playfair's, Cagnoli's, and Woodhouse's Trigonometry.
1 4. 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 nega. tive, 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 quadrants. In the first quadrant, (Art. 93. Propor. 1.)
R X sin
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 neg. ative.
The tangent must therefore be negative. In the 3d quadrant, the sine and cosine are both negative.
The tangent must therefore be positive. In the 4th 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.
R? 1. Tan : R::R : cot, that is Cot=
Therefore, as radius is uniformly positive, the cotangent must have the same sign as the tangent.
R? 2. Cos : R::R: sec, that is, Sec= The secant, therefore, must have the same sign as the cosine.
R? 3. Sin :R::R : cosec, that is, Cosec= The cosecant, therefore, must have the same sign as the sine.
The versed sine, as it is measured from A, 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 exacily 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 center through the other end. (Art. 34.) 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 never can meet. Still, as an arc infinitely near to 90° has a tangent infinitely great, it is frequently said, in concise terms, that the tangent of 90° 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 Dit 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. 1'. 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 radiùs 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 equalto 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 extrernity of each quadrant.
Quadrant 1st 2d sd 4th Sine and cosecant
+ + Cosine and secant
+ Tangent and cotangent +
0 90° 180° 270° 360° Sine Cosine
202. By comparing these two tables, it will be seen, that each;of the trigonometrical lines changes from positive to ne. gative, or from negative to positive, in that part of the circle
OSO 8 8
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. T'here 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 x be any arc whatever ;
sin x=sin (C+x)=sin (2C+x)=sin (3C+x), &c.
tanx=tan(C+x)=tan(2C+x)=tan(3C+x), &c. 204. It is evident also, that, in a number of successive tevolutions, in the same circle ; The first quadrant must coincide with the 5th, 9th, 13th, 17th, The second, with the
6th, 10th, 14th, 18th, &c. The third, with the
7th, 11th, 15th, 19th, &c. The fourth, with the
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 oth
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 FORMULÆ. 206. From the view which has here been taken of the changes in the trigonometrical lines, it will be easy to see, in