... 9 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 - sin(a+x), y=sin(a + x), y' = cos(a + x), y' + sin(a+2) = sina(1 − +1.2.3.4 1.2.3.4 5.6 x5 In like manner (314) we have x2 x4 1.2 (315) + +cosa (1-1.2.3 3 Making a = 0, and observing that (306) sin(aa_o) = sin0 = 0, and cos(a-o) = r = 1 + 1.2 and these are the developments required. They were discovered by Newton.* If we change r into — x, (317) and (318) become (6), (6) 2.3 +, -,... 9 ... 1 . 2 = 1 x2 Cor. The sine of an arc changes from + to 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 into — x, (315) and (316) become sin(a-x)=sina(1- ; +, -,...) – cosa (≈ 1.2 with which and (315), (316), combining (317), (318), there results sin(a + 2) = sina cosx – cosa sinx, 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) which, combined with (322) and (321), gives (321) (322) .. Cor. 7. (1+sin2x)*±(1 − sin2x) = 2 cost, Cor. 8. (1+sin2x)(1 − sin2x) = 2 sinx. (323) (324) (325) (326) (327) (328) 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-I = q, and .. a = 3(p+q), x = (p −q), we have Cor. 9. sinp+sing = 2 sint(p+q) cost(p-q), (329) sin (p-q), (330) SUPPLEMENTARY ARCS. Cor. 11. cosp+cosq = 2 cos(p+q)• cost (p − q), 177 (331) (332) Cor. 12. cosq+cosp = 2 sint(p + q)• sint(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) cot (p − q), (334) cosq-cosp Cor. 15. cosp+cosq 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, Cor. 19. The sine of 180° is = 0, and the cosine =sin(180° + z)=−sinữ, cos(180° + x) cos180° sin370° Cor. 20. Cor. 21. Cor. 22. Cor. 23. cos 370° Cor. 24. Cor. 25. - 1, 0; - cos90° sin(370° + x)= = cos(370° + x) = + sinx. Scholium. The consequences evolved by these last forms (337) ... (345), are in accordance with the principle enounced in (180). Indeed, if we suppose the arc to in crease from 0° to through the value 0 360°, the sine will pass at 180° and again at 360°, while the cosine will reduce to zero at 90° and 270°; and it is obvious that the same correlation of values will be repeated in a 2d, 3d, &c., circumFig. 55. ference. The algebraical sign of the tangent will be determined from the relation (307). PROPOSITION IX. It is required to develope the tangent and cotangent of the sum and difference of two arcs in terms of the tangents and cotangents of the arcs themselves. Consulting (307) and (320), we obtain Resolving equation (349) in reference to tan a and consulting (309), (310), we find |