COSX = smx = Similarly 2 2i the other functions may be expressed in terms of c'°, These are the exponential sions for the circular functions of x. if i is omitted from these exponentials, the resulting functions are called the hyperbolic cosine, hyper bolic sine of the angle x. Hyperbolic functions are so called because they have geometric rela tions with the equilateral hyperbola analogous to those between the circular functions and the circle. The common notation for such functions is sinhO, cosh°, tanhO, corresponding to the circular function sin& cos0, tan°. The values of these functions have been tabulated and are of service in analytic trigonometry.
The spherical triangle, like the plane triangle, has six elements, the three sides a, b, c, and the angles A, B, C. But the three sides of the spherical triangle are angular as well as linear magnitudes. The triangle is completely deter mined when any three of its six elements are given, since there exist relations between the given and the sought parts by means of which the latter may be found. In the right-angled or quadrantal triangle, however, as in the case of the right-angled plane triangle, only two ele ments are necessary to determine the remaining parts. Thus, given c, A, in the right-angled triangle, ABC, the remaining parts are given by the formulas sins, = sine, sinA, tanb = tanc, cos A, cotB = cose, tani. The corresponding formulas when any other two parts are given may be obtained by Napier's rules concerning the relations of the five circular parts (q.v.), viz. a, b, complement of A, complement of B, comple meat of c. In the case of oblique triangles no simple rules have been found, but each case is dependent upon the appropriate formula. Thins in the oblique triangle ABC, given a, b, and A, the formulas for the remaining parts are sinB sinA sinb tan sin (A 4- B) sine , ', c=tanl (a br cot 3 C= tan (A B)• sin (a+ b) sin (a— It is evident in spherical trigonometry, as well as in plane, that three elements taken at random may not satisfy the conditions for a triangle, or they may satisfy the conditions for more than one. The treatment of the ambiguous eases in
spherical trigonometry is quite formidable, since every line intersects every other line in two points and multiplies the cases to be considered.
The measurement of spherical polygons may be made to depend upon that of the triangle. For, if, by drawing diagonals, the polygons can be divided into triangles each of which contains three known or obtainable elements, then all the parts of the polygon can be determined. Since the elements of the spherical polygon the elements of the polyhedral angle whose vertex is at the centre of the sphere, the formulas of spherical trigonometry apply to problems involv ing the relations of the parts of such figures. E.g., given two face angles and the included dihedral angle of a trihedral angle, the remaining face and dihedral angle may be determined by the same formulas as apply to the corresponding case of the spherical triangle. By aid of the formu las of spherical trigonometry the theories of transversals, coaxal circles, poles and polars, may he developed for the figures of the sphere. Spherical trigonometry is of great importance also in the theory of power circles, stereographic projection, and geodesy. It is also the basis of the chief calculations of astronomy; e.g. the solution of the so-called astronomical triangle is involved in finding the latitude and longitude of a place, the time of day, the azimuth of a star, and various other data.
Some traces of trigonometry exist in the earli est known writings on mathematics. In the Papy rus of Ahmes (see Atoms) a ratio is mentioned called a sent, and because of its relation to the methods of measuring the pyramids, this ratio seems to correspond to the cosine or the tangent of an angle. But to the Greeks are due the first scientific trigonometric investigations. The sex agesimal division of the circle was known to the Babylonians, but Hipparclms was the first to complete a table of chords. Heron (q.v.) com 2 r pitted the values of cot n , for a = 3, 4, . . .