called the tangent, and CT the secant of the arc AM, or of the angle ACM.

These three lines MP, AT, CT, dependent upon the are AM, and always determined by it and the radius, are thus designated: MP=sin AM, or sin ACM, AT=tang AM, or tang ACM, CT sec AM, or sec ACM.

VI. Having taken the arc AD equal to a quadrant, from the points M and D draw the lines MQ, DS perpendicular to the radius CD, the one terminated by that radius, the other terminated by the radius CM produced; the lines MQ, DS and CS, will, in like manner, be the sine, tangent, and secant of the arc MD, the complement of AM. For the sake of brevity, they are called the cosine, cotangent, and cosecant, of the arc AM, and are thus designated: MQ=cos AM, or cos ACM, DS= cot AM, or cot ACM, CS=cosec AM or cosec ACM. In general, A being any arc or angle, we have cos A=sin (100°—A), cot A=tang (1008-A), cosec A=sec (100°—A).

The triangle MQC is, by construction, equal to the triangle CPM; consequently CP-MQ: hence in the right-angled triangle CMP, whose hypotenuse is equal to the radius, the two sides MP, CP are the sine and cosine of the arc AM. As to the triangles CAT, CDS, they are similar to the equal triangles CPM, CQM; hence they are similar to each other. From these principles, we shall very soon deduce the different relations which exist between the lines now defined: before doing so, however, we must examine the progressive march of those lines, when the arc to which they relate increases from zero to 200°,

VII. Suppose one extremity of the arc remains fixed in A, while the other extremity, marked M, runs successively throughout the whole extent of the semicircumference, from A to B in the direction ADB.

When the point M is at A, or when the arc AM is zero, the three points T, M, P, are confounded with the point A; whence it appears that the sine and tangent of an arc zero are zero, and the cosine and secant of this same arc, are each equal to the radius. Hence if R represent the radius of the circle, we have

sin o=o, tang o=o, cos o=R, sec o=R.

VIII. As the point M advances towards D, the sine increases, and likewise the tangent and the secant; but the cosine, the cotangent, and the cosecant, diminish.

When the point M is at the middle of AD, or when the arc AM is 50°, and also its complement MD, the sine MP is equal

to the cosine MQ or CP; and the triangle CMP, now become isosceles, gives the proportion MP: CM:: 1:√2, or sin 50°:

R:: 1:√2. Hence sin 50° = cos 50° =

this same case, the triangle CAT becomes isosceles and equal to the triangle CDS; whence, the tangent of 50° and its cotangent, are each equal to the radius, and consequently we have tang 50°=cot 50°—R.

IX. The arc AM continuing to increase, the sine increases till M arrives at D; at which point the sine is equal to the radius, and the cosine is zero. Hence we have sin 100°=R, cos 100°=0; and it may be observed, that these values are a consequence of the values already found for the sine and cosine of the arc zero; because, the complement of 100° being zero, we have sin 100° = cos o° = R, and cos 100° = sin o° =0.

As to the tangent, it increases very rapidly as the point M approaches D; and finally when this point reaches D, the tangent properly exists no longer, because the lines AT, CD, being parallel, cannot meet. This is expressed by saying that the tangent of 100° is infinite; and we write tang 100° = The complement of 100° being zero, we have tang o=cot 100°, and cot o= tang 100°. Hence cot o∞, and cot 100° = 0.

X. The point M continuing to advance from D towards B, the sines diminish and the cosines increase. Thus M'P' is the sine of the arc AM', and M'Q or CP' its cosine. But the arc M/B is the supplement of AM', since AM'+M'B is equal to a semicircumference; besides, if M'M is drawn parallel to AB, the arcs AM, BM', which are included between parallels, will evidently be equal, and likewise the perpendiculars or sines MP, M'P'. Hence, the sine of an arc or of an angle is equal to the sine of the supplement of that arc or angle.

The arc or angle A has for its supplement 200°-A: hence generally, we have

sin A=(sin 200°—A.)

The same property might also be expressed by the equation sin (100°+B)=sin (100°-B), B being the arc DM or its equal DM'.

XI. The same arcs AM', AM which are supplements of each other, and which have equal sines, have also equal cosines CP', CP; but it must be observed, that these cosines lie in different directions. This difference of situation is expressed in calculation by a difference in the signs; so that if the cosines of arcs less than 100° are considered as positive or affected with the sign+, the cosines of arcs greater than 100° must be consider

ed as negative or affected with the sign we shall have

cos Acos (200—A°)

Hence, generally,

or cos (100+B)—— cos (100°—B); that is, the cosine of an arc or of an angle greater than 100° is equal to the cosine of its supplement negatively taken.

:

The complement of an arc greater than 100° being negative (Art. 3), it is natural that the sign of that complement should be negative but to render this truth still more palpable, let us seek the expression of the distance from the point A to the perpendicular MP. Making the arc AM=x, we have CP=cos x, and the required distance_AP = R—cos x. The same formula must express the distance from the point A to the straight line MP, whatever be the magnitude of the arc AM originating in the point A. Suppose then that the point M come to M', so that a designates the arc AM'; we have still at this point, AP-R-cos x: hence cos x=R—AP'—AC -AP-CP; which shews that cos a is negative in that case: and because CP: CP = cos (200°-x), we have cos x = ―cos (200°—x), as we found above.

From this it appears, that an obtuse angle has the same sine and the same cosine as the acute angle which forms its supplement; only with this difference, that the cosine of the obtuse angle must be affected with the sign Thus we have sin 150° sin 50° R2, and cos 150° -R√2.

-

cos 50° =

As to the arc ADB, which is equal to the semicircumference, its sine is zero, and its cosine is equal to the radius taken negatively hence we have sin 2000, and cos 200°――R. This might also be derived from the formulas sin A= sin (200-A), and cos A= cos (200°-A), by making A=200°.

XII. Let us now examine what is the tangent of an arc AM greater than 100°. According to the Definition, this tangent is determined by the concourse of the lines AT, CM. These lines do not meet in the direction AT; but they meet in the opposite direction AV; whence it is obvious that the tangent of an arc greater than 100° must be negative. Also, because AV is the tangent of the arc AN, the supplement of AM' (since NAM is a semicircumference), it follows that the tangent of an arc or of an angle greater than 100° is equal to that of its supplement, taken negatively; so that we have tang Atang (200-A).

The same thing is true of the cotangent represented by DS', which is equal to DS the cotangent of AM, and in a different direction. Hence we have likewise cot A ·cot (200°—A).

The tangents and cotangents are therefore negative, like

the cosines, from 100° to 200°. And at this latter limit, we have tang 200° = o and cot 200° — —

cot o

* XIII. In trigonometry, the sines, cosines, &c. of arcs or angles greater than 200° do not require to be considered; the angles of triangles, rectilineal as well as spherical, and the sides of the latter being always comprehended between 0 and 200°. But in various applications of geometry, there is frequently occasion to reason about arcs greater than the semicircumference, and even about arcs containing several circumferences. It will therefore be necessary to find the expression of the sines and cosines of those arcs whatever be their magnitude.

We observe, in the first place, that two equal arcs AM, AN with contrary signs, have equal sines MP, PN with contrary signs; while the cosine CP is the same for both. Hence we have in general

sin(x)=sin x
cos (-x) = cos x,

formulas which will serve to express the sines and cosines of negative arcs.

From 0° to 200° the sines are always positive, because they always lie on the same side of the diameter AB; from 200° to 400°, the sines are negative, because they lie on the opposite side of their diameter. Suppose ABN an arc greater than 200°; its sine P'N' is equal to PM, the sine of the arc AM= x-200°. Hence we have in general

sin x sin (x-200°)

This formula will give us the sines between 200° and 400°, by means of the sines between 0° and 200°: in particular it gives sin 400°sin 200°-0; and accordingly, if an arc is equal to the whole circumference, its two extremities will evidently be confounded together at the same point, and the sine be reduced to zero.

It is no less evident, that if one or several circumferences were added to any arc AM, it would still terminate exactly at the point M, and the arc thus increased would have the same sine as the arc AM; hence if C represent a whole circumference or 400°, we shall have

sin x= · sin (C+x)=sin (2C+x)=sin (3C+x), &c. The same observation is applicable to the cosine, tangent, &c. Hence it appears, that whatever be the magnitude of a the proposed arc, its sine may always be expressed, with a proper

The sections marked thus (*) are either of greater difficulty, or of less extensive use. They may be omitted without violating the continuity of the reasoning.-ED.

sign, by the sine of an arc less than 100°. For, in the first place, we may subtract 400 from the arc a as often as they are contained in it; and y being the remainder, we shall have sin x— siny. Then if y is greater than 200°, make y=200+%, and we have sin y sin z. Thus all the cases are reduced to that in which the proposed arc is less than 200°; and since we farther have sin (100+x)=sin (100—x), they are likewise ultimately reducible to the case, in which the proposed arc is between zero and 100°.

*XIV. The cosines are always reducible to sines, by means of the formula cos A=sin (100°—A); or if we require it, by means of the formula cos A-sin (100°+A): and thus, if we can find the value of the sines in all possible cases, we can also find that of the cosines. Besides, the figure will easily shew us that the negative cosines are separated from the positive cosines by the diameter DE; all the arcs whose extremities fall on the left side of DE, having a positive cosine, while those whose extremities fall on the right have a negative cosine.

Thus from 0° to 100° the cosines are positive; from 100° to 300° they are negative; from 300° to 400° they again become positive; and after a whole revolution, they assume the same values as in the preceding revolution, for cos (400°+x)=cos x.

From these explanations, it will evidently appear, that the sines and cosines of the various arcs which are multiples of the quadrant, have the following values :

And generally, k designating any whole number, we shall have

sin 2k. 100° 0,

cos (2k+1). 100°=0,

sin (4k+1).100°=R,

sin (k-1). 100°——R,

cos 4k. 100°-R,

cos (4k+2).100°——R

What we have just said concerning the sines and cosines renders it unnecessary for us to enter into any particular detail respecting the tangents, cotangents, &c. of arcs greater than 200°: the values of these quantities are always easily deduced from those of the sines and cosines of the same arcs; as we shall see by the formulas, which we now proceed to explain.