Page:Collier's New Encyclopedia v. 09.djvu/578

LEFT TBIGONOMETRY 508 TRILOaY of 0. In other words, the cosine becomes negative, and continues so till OP has completed three right angles. In the same way, as AOP passes through the value of two right angles and becomes re-entrant, the sine becomes negative, be- ing thenceforward measured downward till OP has made one complete revolu- tion. After one complete revolution both sine and cosine, and also secant and co- secant, begin to go through exactly the same cycle of changes in magnitude and sign as at first. There are, therefore, periodic functions, and their period is 2ir or four right angles. The tangent and cotangent, however, go through their cycle of changes in half this period or two right angles. All possible numerical values of the functions are obtained in the first quadrant. It is therefore suf- ficient in constructing tables of the trig- onometrical functions to tabulate for an- gles from 0° to 90° inclusive. For exam- ple, the angle 130° (90°+40°) has the same sine as the angle of 50 (90°-40°); and its cosine differs only by being nega- tive. Of greater practical importance than the tables of the functions them- selves are the table of their logarithms. These are generally tabulated for every degree and minute of angle from 0° to 90° ; and proportional parts are added by which is readily calculated the num- ber corresponding to an angle involving seconds of arc. The calculation of the functions and their logarithms is a sufficiently laborious task. It is generally effected by means of series, though the values for certain particular angles can be found by_ the simplest of arithmetical operations. Thus, the cosine of 60° is evidently Vz', sine 60° is therefore %V3; tangent 60° is V3; and so on. What might be called the fundamental series for the sine and cosine in terms of the arc are: gin. 9z=e - 1 ■ +. . . 1.2.3 1.2.S.4.5 1.2.3.4.5.6.7 e* 0* ff" COS. 9=1 1 1- . . . 1.2 1.2.3.4 1.2.3.4.5.6 If we make all the signs in these two series positive we get two other functions of 0, which are called the hyperbolic sine and cosine of 0, and are written sinh. 6 and cosh, 6 respectively. Related to these functions are the hyperbolic tangent, co- tangent, secant, and cosecant; and they are connected by relations similar to, though not quite identical with, the ordi- nary circular functions. We may see, by adding the series with signs all positive, that the sum of the hyperbolic sine and cosine is the exponential of 0, Demoivre's theorem gives the corresponding equa- tion for the circular sine and cnsine. The reason for the names circular and hyperbolic may be partially indicated thus: The relation cos.^ 6 -{- sin.^ ^ = 1 may be put in the form x^ + 2/* = a^ which is the equation of a circle of radius a, referred to rectangular axes. The equation of the rectangular hyper- bola is «> — 2/^ = ct^ to which there cor- responds the relation cosh.s 9 — sinh.' 0= 1. The hyperbolic sines and cosines are really exponential functions, and are not periodic. They are of constant oc- currence both in the higher analysis and in mathematical physics. To facilitate their use in calculation, tables have re- cently been constructed. Besides the series given above, there are many others, some of which are par- ticularly serviceable for calculating the values of the functions or the values of their logarithms. There are also the con- verse series, by which an angle is found in terms of one of its circular functions. One of the simplest, and at the same time most historically famous of these is Greg- ory's series, which expresses an angle in ascending powers of its tangent. It is as follows: e = tan. e — Jtan.» 6 +lUn? 6 — ten.'^fl +. Of great importance are the addition formulae which express any required function of the sum or difference of two given angles in terms of the trigonometri- cal functions of these angles. They are readily established for the circular func- tions by application of the elementary theorems of orthogonal projection. Sim- ilar formulae hold for the hyperbolic func- tions. As plane trigonometry has to do chiefly with the solution of plane trian- gles, so spherical trigonometry is devo- ted to the discussion of spherical trian- gles. In navigation, geodesy, and astron- omy the formulae of spherical trigonome- try are in constant use. The ordinary text-books on trigonometry do little more than present the subject in its practical bearings. It is impossible to make any progress in the higher mathematics with- out a thorough knowledge of the proper- ties of the trigonometrical functions. TRILOGY, a series of three dramas, which, though complete each in itself, bear a certain relation to each other, and form one historical and poetical picture. The term belongs more particularly to the Greek drama. In Athens it was cus- tomary to exhibit on the same occasion three serious dramas, or a trilogy, at first connected by a sequence of subject, but afterward unconnected, and on dis- tinct subjects, a fourth or satyric drama being also added, the characters of which were satyrs. Shakespeare's "Henry VI." may be called a trilogy.