FORMULAS EXPRESSING RELATIONS BETWEEN THE CIRCULAR M 65. Let AB and BM represent two arcs, having the common radius 1; denote the first by b, and the second by a. From M draw MP, perpendicular to CA, and MN perpendicular to CB; from N draw NP' perpendicular to CA, and NL parallel to AC. Then, by definition, we shall have, PM sin (a + b), NM sin a, L IN B PP'A C From the figure, we have, PM = PL + LM. (1.) From the right-angled triangle CP'N (Art. 37), we have, P'N = CN sin b; or, since P'N = PL, PL = cos a sin b. Since the triangle MLN is similar to CP'N, the angle LMN is equal to the angle P'CN; hence, from the right-angled triangle MLN, we have, Substituting the values of PM, PL, and LM, in Equation (1), we have, sin (a + b) = sin a cos b + cos a sin b; · (A.) that is, the sine of the sum of two arcs, is equal to the sine of the first into the cosine of the second, plus the cosine of the first into the sine of the second. Since the above formula is true for any values of a and -b, for b; whence, b, we may substitute sin (a - b) = (− that is, the sine of the difference of two arcs, is equal to the sine of the first into the cosine of the second, minus the cosine of the first into the sine of the second. If, in Formula (B), we substitute (90° — a), have, a), for a, we (2.) sin (90°—a—b) = sin (90°— a) cos b—cos (90°—a) sin b ; but (Art. 63), sin (90° — a — b) = sin [90°— (a + b)] = cos (a + b), and, sin (90° — a) = cos ɑ, cos (90° - a) sin a; hence, by substitution in Equation (2), we have, cos (a + b) cos a cos b — sin a sin b; · (☺.) that is, the cosine of the sum of two arcs, is equal to the rectangle of their cosines, minus the rectangle of their sines. 'If, If, in Formula (), we substitute - b, for b, we find, cos (a - b) = cos a cos ( — b) or, sin a sin (b), cos (a - b) = cos a cos b + sin a sin b ; ・ ・ (D.) that is, the cosine of the difference of two arcs, is equal to the rectangle of their cosines, plus the rectangle of their sines. If we divide Formula (A) by Formula), member by member, we have, Dividing both terms of the second member by cos a cos b, sine divided by the cosine is equal to recollecting that the the tangent, we find, tan a + tan b tan (a + b) · (2.) 1 - tan a tan b that is, the tangent of the sum of two arcs, is equal to the sum of their tangents, divided by 1 minus the rectangle of their tangents If, in Formula (), we substitute b, for b, recollecting that we have, tan (— b) tan b, tan a tan b tan (a - b) 1 + tan a tan b (F.) that is, the tangent of the difference of two arcs, is equal to the difference of their tangents, divided by 1 plus the rectangle of their tangents. In like manner, dividing Formula () by Formula (A), member by member, and reducing, we have, cot a cot b 1 cot (a + b) cot a + cot b (G.) and thence, by the substitution of b, for b, 66. If, in Formulas (A), (@), (A), (C), (E), and (G), we make ab, we find, afterwards for sin2a, its value, 1 — cos2α, we have, Dividing Equation (A'), first by Equation (4), and then by Equation (3), member by member, we have, Taking the reciprocals of both members of the last two 67. If Formulas (A) and (B) be first added, member to member, and then subtracted, and the same operations be performed upon (C) and (D), we shall obtain, |