The tangent and secant have their greatest values, namely ∞, together; that is, at the top and bottom of the circle. They have also their least values, that of the tangent being 0, and that of the secant R, together, to wit, at the right and left points of the circle.

22. In the first quadrant the secant is estimated from the centre towards the second extremity of the arc. In the second and third quadrants it is estimated in the opposite direction. According to the principle which it is necessary to observe, and of which we have before spoken, the secant must in these quadrants be considered as negative. In the fourth quadrant the secant is again estimated towards the second extremity of the arc, and is therefore positive.

The vertical diameter separates the positive from the negative secants, the positive being in the quadrants on the right of this diameter, and the negative being on the left.

23. We have now exhibited three of the trigonometrical lines. There are three others closely connected with these in character, called the cosine, the cotangent, and the cosecant; the reason for which names will presently appear.

The difference between an arc or angle and a right angle or 90° = 1008, is called the complement of the arc or angle. Thus 40° is the complement of 50°; 60° is the complement of 30°; 75 is the complement of 255, and in general 90°-a, or 100%-a, is the complement of the arc a..

EXERCISES.

1. The complement of 24° 32′ = 65° 28'.

110 15' -(20° 15').

17° 36′ 43′′ 72° 23′ 17′′.

2.

66

3.

2160 45'-126° 45'.

The cosine, cotangent, and cosecant, are the sine, tangent, and secant of the complement. Thus the cosine of 50° is the sine of 40°; the cotangent of 30° is the tangent of 60°; and in general the cosine, cotangent, or cosecant, of the arc a, is the sine, tangent, or secant of 90°-a.

then another definition for the cosine of an arc, viz. the distance from the foot of the sine of the arc to the centre of the circle.

25. If the arc terminate on the right of the vertical diameter, i. e. in the first or fourth quadrant, the foot of the sine will fall on the right of the centre; but if the arc terminate on the left of the vertical diameter, i. e. in the 2d or 3d quadrant, the foot of the sine will fall on the left of the centre. The cosine being estimated in opposite directions in these two cases, must have opposite signs. It is therefore positive in the 1st and 4th quadrants, and negative in the 2d and 3d.

It will be recollected that the positive were separated from the negative secants, as the positive are here seen to be from the negative cosines, by the vertical diameter. The secant and cosine have therefore always the same algebraic sign.

It was shown (Art. 15), that sin (180°—a) = sin a; so also cos. (180°-a) is equal in length to cos. a, since they are both the distance from the foot of the same sine (MP in the diagram of Art. 14) to the centre, i. e. if we suppose one of the ares to originate at A, the other at B, and both to be extended towards м in opposite directions.* But if a < 90°, it follows that 180°-a terminates in the second quadrant, hence its cosine is negative; if a > 90° then cos. a is negative, and 180°—a being in the first quadrant, its cosine is positive; therefore, the cosine of an arc and the cosine of its supplement are equal with contrary signs.

* Both arcs a and 1800-a are now supposed to originate at the same point a, and to be estimated in the same direction.

26. The cosine of 0° (being equal to the sine of the complement of 0° which is 90°) is R. The cosine of 90° is equal to the sine of 00, which is 0. The cosine of 180°, being the distance from the foot of the sine to the centre, and being also on the left of the vertical diameter, is

- R, as may be seen from the preceding diagram. The cosine of 270°, being the distance from the foot of the sine to the centre, since the sine falls on the centre, is 0.

The least value of the cosine is 0; the greatest value is R. When the sine has its least value, the cosine has its greatest; and vice versâ.

The versed sine of an arc, which is seldom employed in Trigonometry, but often in Mechanics, is the distance from the foot of the sine to the origin of the arc, thus PA in the last diagram is the versed sine of the arc

AM.

27. Before noticing the cotangent and cosecant, let us consider the manner of treating negative arcs. Such ares commencing at the point a in the diagram ought evidently, on the general principle already repeatedly mentioned, to be laid off

upon the circumference in

C

the opposite direction from the positive arcs, i. e. downwards.

Let us for simplicity suppose the arc in question to

M

be less than a quadrant; being laid off downwards, such an arc will terminate in the fourth quadrant. Hence we see that the trigonometrical lines of a negative arc must be

an arc in the fourth quadrant. the cosine +, the tangent ·

affected with the same signs as those of Thus the sine of a negative arc will be the secant +.

Secondly, suppose the given negative arc to be greater than a quadrant; were it positive, some of its trigonometrical lines would be negative. The rule given above, which determines the signs of its trigonometrical lines, by those of an arc in the 4th quadrant, will apply with this modification, that when the trigonometrical line is + in the fourth quadrant, the corresponding trigonometrical line of the negative arc has the same sign as that of a positive arc of the same magnitude, and when the trigonometrical line is in the fourth quadrant, a contrary sign.

The truth of this assertion may be seen, by trying negative arcs of

various magnitudes upon the diagram, laying them off downwards from the right point of the circle, and observing in which quadrant their extremities fall. They will be found in every case to give results agreeable to the rule just stated.

THE COTANGENT AND COSECANT.

28. The cotangent of 0° is equal to the tangent of 90° (Art. 23) and is therefore ∞. The cotangent of 90° is equal to the tangent of 0° and is 0. The cotangent of 180° is equal to the tangent of 90° — 180° = the tangent of 90°= ∞, since — 90° is a negative arc, and terminates at the bottom of the circle, or the 270° point. The cotangent of 270° the tangent of 90°-270°=the tangent of When the tangent has its least value, which is 0, the cotangent has its greatest which is ∞, and vice versâ.

=

1800 0.

29. The cosecant of 0° the secant of 90° = ∞. The cosecant of 90° the secant of 0° = R. The cosecant of 180° = the secant of 90° 180°∞. The cosecant of 270° the secant of - 180°

=-R.

When the secant has its least value, which is R, the cosecant has its greatest, which is ∞, and vice versâ. The cotangent and cosecant have their greatest values together and their least values together, viz. that of the one 0, of the other R, at the top and bottom of the circle, and both at the right and left points.

30. With regard to the signs of the cotangent and cosecant in the different quadrants, they will be most conveniently discovered from the analytical expressions for these lines which we shall presently have. We add here, however, which so far as the cotangent and cosecant are concerned must be for a moment taken for granted, that the six trigonometrical lines may be arranged in three pairs, each pair having always the same algebraic sign.

We have seen that the secant and cosine go together in this way; so do also the cosecant and sine; and so do the tangent and cotangent. The positive sines and cosecants are separated from the negative by the horizontal diameter; the positive cosines and secants from the negative, by the vertical diameter; and the tangent and cotangent are together + and alternately in the successive quadrants.

31. The following algebraic notation is employed for the six trigonometrical lines. Let a be the algebraic expression for the number of degrees in any arc, then the trigonometrical lines of the arc a will be expressed thus; sin a, tan a, sec a, cos a, cot a. cosec a.

Cot a tan a = R is read, the cotangent of the arc a multiplied by the tangent of the same arc is equal to the square of the radius of the circle in which these trigonometrical lines are supposed to be drawn. Cot a and tan a are expressions for straight lines, and the equation above expresses that the rectangle formed by the tangent and cotangent of an arc is equivalent to the square formed upon the radius.

The two members of the above equation contain the same number of dimensions, and are therefore homogeneous. This ought to be the case in all trigonometrical equations; because a line cannot be equal to the rectangle of two lines or a surface, nor either of these to a solid.

Sometimes in analytical investigations R is supposed to be equal to 1; R3 and R3 would also be equal to 1. Whether this 1 is a unit of length, of surface, or of solidity, must be determined by what is required to preserve the homogeneity of the equation.

32. The tangent, secant, cotangent, and cosecant may be expressed in terms of the sine and cosine.

The values of the four former in terms of the two latter are derived

whence multiplying the means and dividing by the first term, we obtain the last

RX sin a

tan a=

cos a

that is, the tangent of any arc is equal to radius multiplied by the sine divided by the cosine of the same arc. If R be made equal to 1, then