M

T

bility. The value of the tangent of 90°, is expressed algebraically thus, tan 90° = 60.

The tangent of an arc, terminating in the second quadrant, will be cut off below the origin* of the arc. Thus A is the tangent of AM; and according to the principle adopted when treating of the sine, this tangent being in the opposite direction to that of the tangent of an

A arc in the first quadrant, is negative.

When the arc is 180°, the negative tangent which became shorter and shorter, as the second extremity of the arc approached this point, again reduces to 0. Beyond 180°, or in the third quadrant, the tangent is cut off above the origin again. Thus at in the annexed diagram, is the tangent of the arc A B M. The tangent of an arc in the third quadrant is, therefore, positive. When the arc is 270° or 3 quadrants, the tangent becomes paral- B lel to the radius which produced ought to terminate it, and the tangentis again 0.

A

M

* A term applied to the point where the arc commencea.

The tangent of an arc in the fourth quad

в. rant is negative, as may be seen from the annexed diagram. 20. The least value

A. of the tangent is 0. The greatest value is 00.

So that the tangent has all possible values. But these it

T has if we do not regard 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 oo 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, ct 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 tan. gent, must extend at least to the circumference. When the arc is 900 the secant is co When the arc is 180° the secant is Q again. And when the arc is 270° or three quadrants,

the secant is again 0. All which will appear from an inspection of the last diagrams.

The tangent and secant have their greatest values, namely 60, 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 500 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

THE COSINE.

D

P

24. In the annexed

M diagram DM is the complement of the arc AM; and me being a perpendicular from one extremity m of of the arc DM upon the

A diameter which passes through the other extremity p, is the sine of the arc DM. Therefore by the definition it is the cosine of the arc AM. But MQ=CP.

Hence cp is also the cosine of the arc AM.

We have 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 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 1800--a terminates in the second quadrant, hence its cosine is negative; if a > 90° then cos a is negative, and 900--a being in the first quadrant, its cosine is positive; therefore,

the cosine of an arc and its supplement are equal with the contrary sign.

26. The cosine of 0° (being equal to the sine of the complement of 0° which is 90°) is R. The cosine of 900 is equal to the sine of 0° 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 versa.

27. Before noticing the cotangent and cosecant, let us consider the manner of treating negative arcs.

Such arcs commencing at the point A in the diagram ought evidently, on the general principle already repeatedly mentioned, to be laid

с

P

A off upon the circumference in the opposite direction from the positive arcs, i. e., downwards.

M

T Let us for simplicity suppose the arc in question to be less than a quadrant; being laid off downward, such an arc will terminate in the fourth quadrant. Hence we see that the trigonometrical lines of a negative arc must be affected with the same signs as those of an arc in the fourth quadrant. Thus the sine of a negative arc will be the cosine +, the tangent, 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 sines of its trigonometrical lines by those of an arc in the 4th quadrant will apply with this modification,