DERIVATIVES OF SINE AND COSINE. 173 ... the ratio [AB]• of the vanishing arc [AQB] to its vanishing = chord [AB] cannot be less than one, and cannot be greater than one, and therefore must be 1; which proves the proposition (300), observing that the arc is perpendicular to the radius (177). Scholium. It is not stated that the vanishing arc, when employed as an increment, merely may be regarded as a straight line perpendicular to the radius, but it is proved that it must be so regarded. On the other hand, it is to be observed that we do not affirm that the arc will ever actually become a straight line, or that it will not always exceed its chord in length, but that, for the purpose pointed out, it must be so regarded, in order to reduce it to zero, and thereby to eliminate it from the function under investigation. PROPOSITION VI. To find the derivatives of the sine and cosine regarded as functions of the arc. Denoting the arc by x, the sine by y, and the cosine by z, the functions may be represented by which are the same as y = fx, z = $x, y = sinx, z= cost. Fig. 54. Attributing to the incremental vanishing arc h, (311), and to y, z, the corresponding increments, k,-i, (306), the similar triangles, whose homologous sides, taken in order, are k, h, —i ; z, r = 1, y, give us The derivative of the sine regarded as a function of (312) its arc, is equal to the cosine, radius being unity. y Again we have :-=[]= 7 = − y ; i. e., = The derivative of the cosine regarded as a function of (313) its arc, is equal to the sine taken minus. Proceeding to the 2d, 3d, 4th, &c., derivatives, observing that m (294) gives (−ƒ)' = − (ƒ') when =-1, we find, (312), (313), n = =- =+z,, &c,, &c., &c. ; i. e., Cor. The sine and its derivatives are alternately sine, - (314) cosine; sine, cosine; ..., in which the algebraical signs alternate in pairs, +, +; −, −; +, +; and the cosine and its derivatives are alternately cosine, sine; cosine, sine; in which the signs alternate in pairs, also alternating, +, -; −, +; +, PROPOSITION VII. To develope the sine and cosine in terms of the arc. Let y=sin(a + x) and z = cosin(a+x); then are y and z continuous functions of the arc x, y = Fx, z = F2x, which it is required to determine. In (314) substituting a +x for x, we find y=sin(a + x), y' = cos(a + x), y" = — sin(a+x), zo= cosa, zo 11 - sina, z= cosa, z" sina, zỡ = cosa, ... ; Making a = 0, and observing that (306) sin(a_o) = sin0 = 0, and cos(a_o) = r = 1 there results, (315), (316), and these are the developments required. They were discovered by Newton.* If we change x intox, (317) and (318) become (6), (6). Cor. The sine of an arc changes from + to as the arc (319) itself changes from + to but the cosine remains still + while the arc passes through the value zero, which is in accordance with (180). Nothing, however, it is to be observed, has been demonstrated in regard to arcs greater than 90°, or > a quadrant. PROPOSITION VIII. It is required to express the sine and cosine of the sum and difference of two arcs in terms of the sines and cosines of the arcs themselves. Changing x intox, (315) and (316) become with which and (315), (316), combining (317), (318), there results sin(a + c) = sina cosx – cosa sinx, cos( a − 2) = cosa cosx + sina sing. (320) These four forms are constantly recurring in trigonometrical analysis, and should therefore be committed to memory; they may be enunciated as follows: I. The sine of the sum of two arcs is equal to the sine of the first multiplied into the cosine of the second, plus the cosine of the first multiplied into the sine of the second. II. The sine of the difference of two arcs is equal to the sine of the first multiplied into the cosine of the second, minus the cosine of the first multiplied into the sine of the second. III. The cosine of the sum of two arcs is equal to the cosine of the first multiplied into the cosine of the second, minus the sine of the first multiplied into the sine of the second. IV. The cosine of the difference of two arcs is equal to the cosine of the first multiplied into the cosine of the second, plus the sine of the first multiplied into the sine of the second. Consequences. Making a = x, we have (320) Cor. 1. sin2z = 2 sint cost, (321) Cor. 2. cos2x = cos2x — sin2x ; (322) but (305), 1 =cos2x + sin2x; which, combined with (322) and (321), gives (328) Cor. 5. 1+sin2x = (cos2 + sinx), = .. Cor. 7. (1+sin2x)+(1 − sin2x)+ Cor. 8. (1+sin2x)= (1 − sin2x)* = 2 sinx. What is the sine of a double arc? of half an arc? the cosine of a double arc? of half an arc? Enunciate (323), (324), (325), (326), (327), (328). Adding and subtracting forms (320), and making a +x=p, a-x = q, and .. a = \(p + q), x = \(p − q), we have Cor. 9. sinp+ sing = 2 sin(p + q) cost(p-q), (329) (330) SUPPLEMENTARY ARCS. 177 (331) (332) Cor. 11. cosp+cosq = 2 cos(p+q) • cost (p − q), Cor. 12. cosq+cosp = 2 sint(p + q) • sin (p − q). These forms are useful in the application of logarithms, by converting sums and differences into products and quotients. Dividing (329) by (330) we have (307), (308), The sum of the sines of two arcs is to their difference as the tangent of half their sum is to the tangent of half their difference. By similar processes, other forms, occasionally useful, may be developed, as Cor. 14. cosp+cosq=cott(p+q)• cott(p − q), cosq - cosp Cor. 15. cosp+cosq sinp sinq Cor. 16. Making a = 90°, forms (320) become sin(90° + 2) = CO$2, sin(90° — 2) = cos2 ; (334) (335) (336) cos(90° + x) = — sinx, cos(90° — x) = sinx; i. e., Cor. 17. The sines of supplementary arcs are equivalent, (337) being equal to the cosine of what one exceeds and the other falls short of 90°. Cor. 18. The cosines of supplementary arcs are numeri (338) cally equal, but have contrary algebraical signs. Arcs are supplementary when their sum amounts to 180°, as (90°+x)+(90° - x) = 180°. In the above forms, making x = 90°, we have sin 180° sin(90° +90°) = cos90° = 0, = cos180° = cos(90° + 90°) = — sin90° Cor. 19. The sine of 180° is = 0, and the cosine |