III.

Spherical Trigonometry

Spherical trigonometry, which is used principally in navigation and astronomy, is concerned with spherical triangles, that is, figures that are arcs of great circles (see Navigation) on the surface of a sphere. 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, being arcs of great circles on the surface of a sphere and measured by the angle subtended at the center. The triangle is completely determined when any three of its six elements are given, since relations exist between the various parts by means of which unknown elements may be found.

In the right-angled or quadrantal triangle, however, as in the case of the right-angled plane triangle, only two elements are needed to determine all of the remaining parts. Thus, given c, A in the right-angled triangle, ABC, with C = 90°, the remaining parts are given by the formula as sin a = sin c sin A; tan b = tan c cos A; cot B = cos c tan A. When any other two parts are given the corresponding formulas may be obtained by Napier's rules concerning the relations of the five circular parts, a, b, complement of c, complement of A, complement of B. With respect to any particular part, the remaining parts are classified as adjacent and opposite; the sine of any part is equal to the product of the tangents of the adjacent parts and also to the product of the cosines of the opposite parts.

In the case of oblique triangles no simple rules have been found, but each case depends on the appropriate formula. Thus in the oblique triangle ABC, given a, b, and A, the formulas for the remaining parts are

In spherical trigonometry, as well as in plane, 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 certain cases in spherical trigonometry is quite formidable, because 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. If, by drawing diagonals, one can divide the polygons into triangles, each of which contains three known or obtainable elements, then all the parts of the polygon can be determined.

Spherical trigonometry is of great importance in the theory of stereographic projection and in geodesy. It is also the basis of the chief calculations of astronomy; for example, the solution of the so-called astronomical triangle is involved in finding the latitude and longitude of a place, the time of day, the position of a star, and various other data.