which we now adopt as the definitions of the circular sine which we adopt as the definitions of the hyperbolic sine and cosine, and write sinh wsin, w=1/csch w, cosh w=cos, w=1/sech w, By comparing these definitions, for the arguments w and iw respectively, remembering that i= 1 and 1/i=— i, we find easily the following important relations : 91. Formulæ. From the foregoing definitions of the modocyclic functions are deduced, by the processes indicated, the following formula: From (a) and (b) by addition and subtraction: (1). COS ̧ W + « ̄1 sin ̧ w= By dividing (3) successively by its first and second From (a) and (b) by carrying out the indicated multiplications, additions and subtractions: From (3) and (9) by addition and subtraction: From (12) and (13) by making w'=w: (15). cot, 2 = (cot, w× ̄tan ̧ w). K From the two forms of (6) by addition and subtraction: (16). sin (w+w') + sin, (w — w') K = 2 sin ̧ w⋅ cos w', =2 cos w⚫ sink w’. (17). sin (w+w') — sin, (w — w') K From the two forms of (7) by addition and subtraction : (18). cos (w+w') + cos (w — w') (19). cos (w+w') — cos (w — w') K In (16), (17), (18), (19) write z for w+w', z′ for w-w'; they then take the respective forms: From (12) and (13) by transposition of terms and a generalized form of Demoivre's theorem. (Cf. Art. 74.) All of the foregoing formulæ pass into the corresponding circular and hyperbolic special forms, by the respective substitutions κi and κ=I. = Making the proper substitutions from (g), (h), (i), (j), (k), (l) in (6) and (7), after assigning to the values i and I successively, we obtain : = = cosh w⋅ cos w'i sinh w・ sin w'. By definitions (a) and (b) and the definitions of circular and hyperbolic sines and cosines (Art. 90): |
