unless we suppose the cosine AM to become negative as soon as the arc AM becomes greater than a quadrant. At the point B the cosine becomes equal to -R; that is, cos 180°=—R. For all arcs, such as ADBN', which terminate in the third quadrant, the cosine is estimated from C towards B, and is consequently negative. At E the cosine becomes zero, and for all arcs which terminate in the fourth quadrant the cosines are estimated from C towards A, and are consequently positive. The sines of all the arcs which terminate in the first and second quadrants, are estimated above the diameter BA, while the sines of those arcs which terminate in the third and fourth quadrants are estimated below it. Hence, considering the former as positive, we must regard the latter as negative. XII. Let us now see what sign is to be given to the tangent of an arc. The tangent of the arc AM falls above the line BA, and we have already regarded the lines estimated in the direc tion AT as positive: therefore the tangents of all arcs which terminate in the first quadrant will be positive. But the tangent of the arc AM', greater than 90°, is determined by the intersection of the two lines M'C and AT. These lines, however, do not meet in the direction AT; but they meet in the opposite direction AV. But since the tangents estimated in the direction AT are positive, those estimated in the direction AV must be negative: therefore, the tangents of all arcs which terminate in the second quadrant will be negative. When the point M' reaches the point B the tangent AV will become equal to zero: that is, tang 180°=0. When the point M' passes the point B, and comes into the position N', the tangent of the arc ADN' will be the line AT: hence, the tangents of all arcs which terminate in the third quadrant are positive. At E the tangent becomes infinite: that is, tang 270° = ∞ When the point has passed along into the fourth quadrant to N, the tangent of the arc ADN'N will be the line AV: hence, the tangents of all arcs which terminate in the fourth quadrant are negative. The cotangents are estimated from the line ED. Those which lie on the side DS are regarded as positive, and those which lie on the side DS' as negative. (Hence, the cotangents are positive in the first quadrant, negative in the second, positive in the third, and negative in the fourth. When the point M is at B the cotangent is infinite; when at E it is zero: hence, cot 180°= =-; cot 270°=0. Let 9 stand for a quadrant; then the following table will show the signs of the trigonometrical lines in the different quadrants. 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 trigonometry, 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 generally consider the arcs as positive which are estimated from A in the direction ADB, and then those arcs must be regarded as negative which are estimated in the contrary direction AEB. We observe, in the first place, that two equal arcs AM, AN with contrary algebraic signs, have equal sines MP, PN, with contrary algebraic signs; while the cosine CP is the same for both. The equal tangents AT, AV, as well as the equal cotangents DS, DS', have also contrary algebraic signs. Hence, calling a the arc, we have in general, By considering the arc AM, and its supplement AM', and recollecting what has been said, we readily see that, sin (an arc)=sin (its supplement) 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 x=sin (2C+x), &c. The same observation is applicable to the cosine, tangent, &c. S' D S M Q MAT P' B P A N N R E 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 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° +z, 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 A=sin (903+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, as has already been shown, that the negative cosines are separated from the positive cosines by the diameter DE; all the arcs whose extremities fall on the right side of DE, having a positive cosine, while those whose extremities fall on the left 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 cos (360°+x)=cosx. 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: sin 2k. 90° 0, sin (4k+1). 90°=R, cos 720° R any whole number we shall cos (2k+1). 90°=0, sin (4k-1). 90°——R, cos (4k+2). 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 value 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. in other words, the sine of a third part of the right angle is equal to the half of the radius. T* sine may be found, and reciprocally, by means of the because the same sine formulas cos A=√(R2-sin A), and sin A=√(R2-cos2A). The sign of these formulas is +, or MP answers to the two arcs AM, AM', whose cosines CP, CP', are equal and have contrary signs; and the same cosine CP answers to the two arcs AM, AN, whose sines MP, PN, are also equal, and have contrary signs. Thus, for example, having found sin 30'=R, we may deduce from it cos 30°, or sin 60° = √ (R2—¿R3) = √ ?R2={R√3, XVII. The sine and cosine of an arc A being given, it is required to find the tangent, secant, cotangent, and cosecant of the same arc. The triangles CPM, CAT, CDS, being similar, we have the proportions: CP : PM :: CA: AT; or cos A: sin A::R: tang A CP : CM :: CA: CT; or cos A: R:: R: sec A= = PM: CP:: CD: DS; or sin A cos A:: R: cot A= PM: CM:: CD: CS; or sin A: R::R: cosec A R sin A cos A R2 cos A R cos A sin A R sin A which are the four formulas required. It may also be observed, that the two last formulas might be deduced from the first two, by simply putting 90°-A instead of A. From these formulas, may be deduced the values, with their proper signs, of the tangents, secants, &c. belonging to any arc whose sine and cosine are known; and since the progressive law of the sines and cosines, according to the different arcs to which they relate, has been developed already, it is unnecessary to say more of the law which regulates the tangents and secants. |