ain co. All which will appear from an inlast diagrams.

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

rst quadrant the secant is estimated from the the second extremity of the arc; in the second rants it is estimated in the opposite direction. he principle which it is necessary to observe, we have before spoken, the secant must in these considered as negative. In the fourth quadrant again estimated towards the second extremity of s therefore positive.

al diameter separates the positive from the negthe positive being in the quadrants on the right eter, and the negative being on the left.

ave now exhibited three of the trigonometrical re are three others closely connected with these , called the cosine, the cotangent and the coseason for which names will presently appear. erence between an arc or angle and 90° or a right lled the complement of the arc or angle. Thus complement of 50°; 60° is the complement of 30°; eral 90°-a is the complement of the arc a. The angent and cosccant, are the sine, tangent, and the complement. Thus the cosine of 50° is the ; the cotangent of 30° is the tangent of 60°; and the cosine, cotangent or cosecant of the arc a is tangent or secant of 90°-a.

The tangent of an arc in the fourth quadrant is negative, as may be seen from the annexed diagram.

20. The least value of the tangent is 0. The greatest value is oo. So that the tangent has all possible values. But these it

has if we do not re

B

A

gard the sign, in the first quadrant; and the same rule applies to finding the length of the tangent belonging to any given arc, from that of an arc in the first quadrant, as was given for the sine.

The tangent changes its sign in every quadrant, that is four times in going round the circle. It is positive in the first and third, two diagonal quadrants, and negative in the second and fourth, the other two diagonal quadrants.

The tangent is ∞ at the top and bottom of the circle, and 0 on the right and left.

THE SECANT.

21. The secant of an arc is a line drawn from the centre of the circle to the extremity of the tangent.

In the preceding diagrams, cr is the secant of the arc AM It is also the secant of the angle measured by the arc.

As the arc with its tangent diminishes, the secant diminishes; and when the arc and tangent are 0, the secant is equal to R. The secant can never be less than radius, because the tangent cannot pass within the circumference, and consequently the line from the centre to the extremity of the tangent, must extend at least to the circumference. When the arc is 90° the secant is co When the arc is 180° the secant is 0 again. And when the arc is 270° or three quadrants,

All which will appear from an in

the secant is again ∞. spection of the last diagrams.

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 90° or a right angle, is called the complement of the arc or angle. Thus 40° is the complement of 50°; 60° is the complement of 30°; and in general 90°—a is the complement of the arc a. 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.

4

CP.

But мQ Hence Cp is We have then another defini

it is the cosine of the arc AM. also the cosine of the arc AM. tion 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 terminates 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 terminates 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; hence 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. 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 90°--a being in the first quadrant, its cosine is positive; therefore,