+ + hence, the tangents of all arcs which terminate in the third quadrant are positive. At E the tangent becomes infinite: that is, tang 270°=0. 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°= -00; cot 270°=0. Let q stand for a quadrant; then the following table will show 9 the signs of the trigonometrical lines in the different quadrants. 19 29 3q 49 Sine + + Cosine + Tangent + Cotangent 구 + XIII. In trigonometry, the sines, cosines, &c. of arcs or angles greater than 180° do not require to be considered; the 1 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 x the arc, we have in general, sin (-2)=—sin x -x)=cos a + COS ΜΛΤ 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) cot (an arc)=-cot (its supplement). It is no less evident, that S' S 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 P' would have the same sine as B the arc AM; hence if C represent a whole circumference or 360°, we shall have N R sin x=sin (C+x)=sin x=sin V (2C+x), &c. E The same observation is applicable to the cosine, tan PA gent, &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° +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°-2), 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 (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, 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)=cos X. COS COS &c. 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 0°=0 sin 90°=R 0°-R 90°=0 sin 180°=0 sin 270o=-R cos 180°=-R cos 270o=0 sin 360°=0 sin 450°=R cos 360°=R cos 450°=0 sin 540°=0 sin 630°=-R cos 540°=-R cos 630°=0 sin 720o=0 sin 810°=R cos 720o=R cos 810°=0 &c. &c. &c. And generally, k designating any whole number we shall have sin 2k. 90'=0, cos (2k+1).90°=0, sin (4k+1).90o=R, cos 4k . 90°=R, sin (4k-1).90°=_R, cos (4k+2).90o=4R. 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. بة For the radius CA, perpen S S dicular to the chord MN, bisects this chord, and likewise My ΜΛΤ the arc MAN; hence MP, the sine of the arc MA, is half the chord MN which subtends P' P the arc MAN, the double of A B MA. The chord which subtends the sixth part of the circum- N R ference is equal to the radius i V hence E 360° sin or sin 30°=R, 12 in other words, the sine of a third part of the right angle is equal to the half of the radius T* XVI. The square of the sine S' D of an arc, together with the square of the cosine, is equal M MT to the square of the radius ; so that in general terms we have . sin 2A +cos? A=R”. P This property results im A B N R E sine may be found, and reciprocally, by means of the formulas cos X=IV (R2_sin’A), and sin A=V (R?-cos’A). The sign of these formulas is +, or —, because the same sine 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°= V(R?—R”)=VARP=RV 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 : R sin A CP : PM::CA: AT; cr cos A : sin A ::R:tang A : cos A CP : CM ::CA : CT; or cos A :R::R: sec A: sin A PM:CP::CD:DS; or sin A : cos A ::R:cot A Rcos A sin A R? PM : CM :: CD: CS; or sin A:R::R: cosec 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. COS sin By means of these formulas, several results, which have already been obtained concerning the trigonometrical lines, may be confirmed. If, for example, we make A=90°, we shall have sin A=R, cos A=0; and consequently tang 90°= R? an expression which designates an infinite quantity; for, the quotient of radius divided by a very small quantity, is very great, and increases as the divisor diminishes ; hence, the quotient of the radius divided by zero is greater than any finite quantity. The tangent being equal to R.sin; and cotangent to R.COS; ; it follows that tangent and cotangent will both be positive when the sine and cosine have like algebraic signs, and both negative, when the sine and cosine have contrary algebraic signs. Hence, the tangent and cotangent have the same sign in the diagonal quadrants : that is, positive in the 1st and 3d, and negative in the 2d and 4th ; results agreeing with those of Art. XII. In regard to the secants, they will be positive for all arcs of the circumference. For we have secant equal to radius square divided by cosine, and since the secant always falls on the right of the vertical diameter DE, the cosine will be estimated from C towards A, and will consequently be positive. The cosecant is equal to radius square divided by the sine, which will always be above the diameter BA: hence the cosecants are also positive. XVIII. The formulas of the preceding. Article, combined with each other and with the equation, sin "A+cos ?A=R”, furnish some others worthy of attention. Rsin? A First we have R? + tang? A = R + = cos? A R? (sin? A+cos? A). R ; hence R2+tang? A=sec? A, a cos? A formula which might be immediately deduced from the rightangled triangle CAT. By these formulas, or by the right-angled triangle CDS, we have also RP+cot? A=cosec? A. Lastly, by taking the product of the two formulas tang A= R sin A Rcos A and cot A= we have tang A x cot A=R”, a sin A R2 formula which gives cot A=R* and cot A. R? We likewise have cot B= cos ?A tang Ā: tang A= tang B |