Of the Tangent, Cotangent, Secant, and Cosecant.

(7.) The tangent of an arc, and, therefore, of the angle which it measures, is a line drawn from one extremity of the arc, touching it at that extremity, and terminating in the diameter produced, drawn through the other extremity: thus BT is the tangent of the arc BC.

C 2

B

The cotangent is the tangent of the complement: thus Dt is the cotangent of the arc BC. It is easy to trace the changes which these two lines undergo as the arc BC increases from 0, for which value the tangent is obviously 0, and the cotangent infinite. Observing the same rules here as for the sine and cosine, we see that in the first quadrant the tangent and cotangent are both positive, in the second the tangent BT, and cotangent Dt, re both negative; in the third the tangent BT, and cotangent Dt, are both positive; and in the fourth the tangent BT, and cotangent Dt, are both negative, and so on; but as we shall soon see, the signs of the tangent and cotangent may always be at once inferred from those of the sine and cosine.

Ts

The secant of an arc is that portion of the prolonged diameter, limitting the tangent, which is included between the centre and tangent; and the cosecant is the secant of the complement. Thus in the last figure AT is the secant of the arc BC, and At the cosecant.

In the four trigonometrical lines, sine, cosine, tangent, and cotangent, we have seen that each is posited in one or other of two directly opposite directions, and that, therefore, one or other of the opposite signs + and, prefixed to the numerical value of any such line, served to point out the proper direction for any particular value of the arc or angle. But as the secant and cosecant continually vary in direction, as well as in magnitude with the arc or angle, the geometrical position of either of these lines does not so clearly indicate to us the sign with which it should be represented. The proper sign, however, is always readily ascertained from knowing the signs of the sine and cosine, for upon these two lines all the others depend, as we shall shortly show.

(8.) Besides the six trigonometrical lines now defined there are three others, sometimes, although but seldom, employed; these are the versed sine or sagitta, the coversed sine, and the suversed sine. The versed sine of an arc BC (see fig. to art. 5) is the line BS between the commencement of the arc and the sine; it is always equal to the radius minus, the cosine, and, therefore, is always positive. The coversed sine is the versed sine of the complement, so that the coversed sine of BC is Ds (see fig. to art. 6); also the suversed sine is the versed sine of the supplement. As the versed sine of any arc must be positive, it follows that the coversed sine and suversed sine must always be positive.

(9.) The following is the way in which the trigonometrical lines, connected with any arc or angle w, are expressed in computation;

From knowing the numerical value of any one of these lines, those of all the others may be obtained; thus, let the sine be given, then since the radius sine and cosine always form a right-angled triangle, of which the hypotenuse is the radius = 1, (see the fig. in art. 5), we have

cos. w√sin.2 w.

Again, since the triangle formed by the radius, sine, and cosine, is always similar to that formed by the secant, tangent, and radius, and to that formed by the cosecant, radius, and cotangent, as the student will at once see by sketching these lines for any arc, it follows, from the proportionality of

and, from these expressions, we at once see that the signs of the several lines, as well as their numerical values, are deducible from those of the sine and cosine.

Now the numerical expression for sin. w, for all values of w, from w=0 to w = = 90°, (between which limits every possible value is comprised) are actually computed by methods to be hereafter explained, and thence the values of the other trigonometrical lines are deduced. These values are then arranged as in table 111, at the end, and form a table of natural sines, cosines, &c. By help of such a table we may readily find the values of the same lines, computed to any other radius R; for as observed at (5) we shall merely have to multiply the tabular value by R. Writing; therefore, for distinction sake, the words sin., cos., &c. in capitals, when the value of the radius is other than unity, the foregoing equations are the same as

&c. for sin. w, cos. w, &c. the formula will become generalized so as to hold good for any value of the radius whatever.

(10.) It is obvious that when any trigonometrical formula is thus generalized every term in it will be the same abstract number as in the original formula; whatever powers or roots of the lines enter the formula they will always be divided by the same powers or roots of the radius R. The denominators will all be removed by multiplying each term by the highest power of R, which enters, and the result will ne

cessarily be a homogeneous expression; that is, every term will have the same dimensions, or will involve as factors the same number of lines. Hence, in order to generalize any trigonometrical formula, or to render it independent of any particular value of R, it will be necessary merely to introduce into the several terms such powers of R as will render them all of the same dimension. For example, the following formula, viz.

sin. (A+B)=sin. A cos. B + sin. B cos. A ;

in which the term on the left is of one dimension, and the terms on the right are each of two dimensions, will become homogeneous by introducing the factor R into the left hand number, so that when this is the value of the radius instead of unity, the formala will be

R sin. (A+B)=sin. A cos. B + sin. B cos. A;

each term being the product of two lines.

In like manner the formula

cos. 4 A8 cos.4 A-8 cos.2 A + 1,

becomes when the radius is R instead of unity

R3 cos. 4 A= 8 cos. A-8 R2 cos.2 A + R';

the powers of R being introduced so as to render each term of four dimensions.

From the preceding definitions and remarks the following simple properties are immediately deducible, viz.

1. The sine of an arc is equal to half the chord of twice that arc.

2. The chord of 60° being equal to the radius (Geom. p. 119), therefore, the sine of 30°, or the cosine of 60°, is equal to half the radius.*

3. Hence, from the expression for the secant at the top of the preceding page, the secant of 60° is equal to the diameter of the circle.

4. The tangent of 45° is equal to the cotangent, and, therefore, to the radius, (see fig. to art. 7.)

(11). We shall terminate this introductory chapter with a table exhibiting the correlative values of the trigonometrical lines, situated in different quadrants; it is readily constructed from the values of the sine and cosine, by help of the relations in (9), bearing in mind that an arc and its supplement have the same sine.

Table of the Correlative Values of the Trigonometrical Lines.

This last line is the same as the first; and any line will, obviously, remain unaltered if we add to the corresponding arc a whole circumference or any number of circumferences. If we take a negatively, we may extend the table as follows:

0

tan. w+cosec. w

sec. w

1

90° -w+cos. w + sin. w + cot. w + tan. w+cosec. w+ sec. w

and by continuing this series of arcs the same values of the trigonome

trical lines would obviously recur as before.

It is obvious that the cosine of a negative arc, whether less or greater than a quadrant, is the same as the cosine of the same arc, taken positively; but the sine of a negative arc, although the same in magnitude as that of an equal positive arc, has an opposite sign: hence, by the equations at (9), the sine, tangent, cotangent, and cosecant, will have opposite signs to those of the same arc taken positively; but the cosine and secant will have the same signs.