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 1/2; 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: 0' 0' 0'— 1.2.3 1.2.3.4.5 1.2.3.4.5.6.7 0' cos. 8=1 — 1.2 1.2.3.4 1.2.3.4.5.6If 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. 0 and cosh, 0 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 cosine.
The reason for the names circular and hyperbolic may be partially indicated thus: The relation cos' 0 + sin' 0 = 1 may be put in the form x° = a', which is the equation of a circle of radius a, referred to rectangular axes. The equation of the rectangular hyper bola is x, — = a', to which there cor responds the relation cosh., B — 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: = tan. 0 0 — tan' 4 8 f .
Of great importance are the addition formula= 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 formulm 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 formula of spherical trigonome try are in constant use. The ordinary text-hooks 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.