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 180°—A : hence generally, we have sin A =(sin 180°—A.) The same property might also be expressed by the equation sin (90°+B) sin (90°-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 90° are considered as positive or affected with the sign +, the cosines of arcs greater than 90° must be considered as negative or affected with the sign Hence, generally, we shall have cos A=cos (180°—A°) or cos (90°+B)=—cos (90°—B); that is, the cosine of an arc or of an angle greater than 90° is equal to the cosine of its supplement taken negatively. COS X. The complement of an arc greater than 90° 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 · 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-cosx: hence cos x=R— AP-AC-AP=-CP; which shews that cos x is negative in that case and because CP CP= cos (180°-x), we have COS X = cos (180°-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 135° = sin 45o = § R√2, and cos 135° = — cos 45° = R√2. 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 180o=0, and cos 180° R. This might also be derived from the formulas sin A sin (180°-A), and cos A= cos (180° making A=180°. = A), by XII. Let us now examine what is the tangent of an arc AM greater than 90°. 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 90° 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 90° is equal to that of its supplement, taken negatively; so that we have tang A-tang (180°-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 (180°—A). =- cot The tangents and cotangents are therefore negative, like the cosines, from 90° to 180°. And at this latter limit, we have tang 180° =o and cot 180° : cot o=∞. XIII. In trigonometry, the sines, cosines. &c. of arcs or angles greater than 180° 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 180°. 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 signs MP, PN with contrary algebraic signs; while the cosine CP is the same for both. Hence we have in general formulas which will serve to express the sines and cosines of negative arcs. From 0° to 180° the sines are always positive, because they always lie on the same side of the diameter AB; from 180° to 360°, the sines are negative, because they lie on the opposite side of their diameter. Suppose ABN=x an arc greater than 180°; its sine P'N' is equal to PM, the sine of the arc AM= -180°. Hence we have in general sin x=sin(x-180°) This formula will give us the sines between 180° and 360°, by means of the sines between 0° and 180°: in particular it gives sin 360°=—sin 180=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 360°, 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 x the proposed arc, its sine may always be expressed, with a proper sign, by the sine of an arc less than 180°. For, in the first place, we may subtract 360° from the arc x as often as they are contained in it; and y being the remainder, we shall have sin x=sin y. Then if y is greater than 180°, make y=180°+ 2, and we have sin y――sin z. Thus all the cases are reduced to that in which the proposed arc is less than 180°; and since we farther have sin (90°+x)= sin (90°—x), they are likewise ultimately reducible to the case, in which the proposed arc is between zero and 90°. XIV. The cosines are always reducible to sines, by means of the formula cos A=sin (90°-A); or if we require it, by means of the formula cos À=sin (90°+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 posi tive 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 90° the cosines are positive; from 90° to 270° they are negative; from 270° to 360° they again become positive; and after a whole revolution, they assume the same values as in the preceding revolution, for eos (360° +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. 90°=0, sin (4k+1). 90°=R, sin (k-1). 90°—R, cos (2k+1). 90°=0, cos 4k. 90° 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 180° 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. THEOREMS AND FORMULAS RELATING TO SINES, COSINES, TANGENTS, &c. XV. The sine of an arc is half the chord which subtends a double arc. 360° sin.30° 12 R, in other words, the sine of a third part of the right angle is equal to the radius. XVI. The square of the sine of an arc, together with the square of the cosine, is equal to the square of the radius; so that in general terms we have sin 'A+cos' A-R2* This property results immediately from the right-angled triangle CMP, in which MP+CP2=CM2. It follows that when the sine of an arc is given, its cosine may be found, and vice versa, by means of the formulas cos A=±√(R23—sin3A), and sin A=±√(R2—cos 2A). The sign of these formulæ is ambiguous, because the same sine MP answers to the two arcs AM, AM', whose cosines CP, CP' are equal and have contrary signs; as the same cosine CP answers to the two arcs AM, AN, whose signs MP, PN are also equal, and have contrary signs. Thus, for example, having found sin 30°=4R, we may deduce from it cos 30°, or sin 60°= √(R2—}R2)=√{R2= 3R √3. XVII. The sine and cosine of the arc A being given, the By sin 2A is here meant the square of sin A; and, in like manner, by ros 2A is meant the square of cos A. |