SPHERICAL TRIGONOMETRY. DEFINITIONS AND FIRST PRINCIPLES. ART. 1. SPHERICAL Trigonometry is that part of mathematical science which investigates the relations that exist between the sides and angles of triangles, formed by the intersections of three planes with the surface of a sphere. It may also be defined (from the mode of investigation employed) to be the science which investigates the relations between the several parts of a solid angle, formed by three planes. 2. Let O be a solid angle formed by the three planes, A O B, BO C, and AO C. With O as centre, describe a sphere, intersecting the three planes in A B, BC, AC; then the portion of the surface of the sphere intercepted by the planes A OB, BOC, AOC, is called a spherical triangle. The arcs A B, B C, A C, are the sides of this spherical triangle, and the three dihedral angles formed by the planes, taken two and two, are its angles. Thus, the dihedral angle formed by the planes BA O, CAO, is the angle A; and so on. B Cor. The sides of a spherical triangle are arcs of great circles (Geo. of the Sphere). 3. Since A B, BC, AC, are arcs of circles whose radii are equal, they are measures of the angles A OB, BOC, A OC, at the centre, and hence when the side of a spherical triangle is spoken of, the angle which that side subtends at the centre of the sphere is meant. 4. The sides of a spherical triangle are usually denoted by the letters a, b, c, and the opposite angles by A, B, C, as in plane trigonometry. 5. The inclination of two great circles is the angle made by their tangents at the point of intersection. Since each of these tangents is perpendicular to the radius in which the planes of the circles intersect; the same angle measures the inclination of the planes of the circles (Geo. of Planes). Other definitions and principles referred to in the subsequent investigations are given in the Geometry of the Sphere. THE RIGHT-ANGLED SPHERICAL TRIANGLE. 6. Properties of the right-angled spherical triangle, Let A B C be a spherical triangle, right-angled at C, so that the planes (0 being the centre of the sphere) AOC, BO C, are perpendicular to each other (Art. 2). Take any point E in O B, and draw E F perpen- o dicular to OC; from F draw FD perpendicular to O A, and join E D. Then (Geo. of Planes) E F is perpendicular to the plane A O C, and DE B to OA. Hence the plane angle ED F, which measures the inclination (Geo. of Planes) of the two planes A O B, A O C, is equal to A (Art. 5), and the plane angles EOF, DOF, DO E, are respectively equal to a, b, and c, (Art. 3); the notation being as in Art. 4. Now (Plane Trig. Art. 14), Each of the last three has a similar one deduced from the other side, Similarly, (9). These equations comprehend every case of right-angled spherical triangles. They are all expressed by the two following general rules, which are calle i Napier's Rules for Circular Parts, viz. : Sine middle part = product of tangents of adjacent parts; Sine middle part product of cosines of opposite parts. The circular parts are five in number (the right angle being excluded), viz., the two sides, the complements of their opposite angles, and the complement of the hypothenuse. Any one of these is called a middle part; the two next to it are then called the adjacent parts, and the other two the opposite parts. Thus if a be the middle part, 90° - B and b are the adjacent parts, and 90° A and 90°- e the opposite parts. Hence by the preceding rules, sin a = tan (90° - B) tan b = cot B tan b, These are formulæ (2) and (7); and similarly for the others. THE OBLIQUE-ANGLED SPHERICAL TRIANGLE. 7. To find a relation between the three sides and any angle of a spherical triangle. Let ABC be a spherical triangle, O the centre of the sphere, and O A the radius. Denote of this Draw in the the sides and angles Also b. o A B c, are the measures of the angles BOC, A O C, and A O B. Now (Plane Trig. Art. 23), A D2+A E2 - 2 A D.A E. cos A = DE2 = DO2 + OE - 2 DO.OE. cos a, E or 2 DO.OE. cos a DO'A D2 + O E-AE+ 2 AD. AE. cos A =2A0+2 AD.A E. cos A. Hence, (3). These are the most important formula of Spherical Trigonometry. They are transformed to forms adapted to logarithms in the following Put a+b+c= 2s; then (Plane Trig., Art. 23), (a + b −c) And similarly for any other angle. a) b; hence c) sin (s—b) sin (s—c) sin s sin (s-a) Cor. 1. If a = b; then cos A = cos B, that is, the angles at the base of an isosceles spherical triangle are equal. Cor. 2. If a = = b = c; then cos A = cos B = cos C, and therefore every equilateral spherical triangle is also equiangular. 8. To prove that the sines of the sides of a spherical triangle are to one another as the sines of the opposite angles. By Art. 16, Plane Trig., and the preceding expressions (4) and (5), sin A = 2 sin A cos A sin A sin B: sin C:: sin a sin b: sin c; or, the sines of the sides of a spherical triangle are to one another as the sines of the opposite angles. 9. To find expressions for the sum and difference of two angles of a spherical triangle in terms of the other angle and the sum and difference of the opposite sides. By Art. 7, tan A = Hence, The formulæ (6) and (7) constitute what are called Napier's First Analogies. They are employed in the solution of a spherical triangle when two sides and the included angle are given. When two sides and the included angle are given, it is sometimes desirable, as in Plane Trigonometry, to find the third side without finding the opposite angles. The formulæ of solution in this case are the following: Let a and b be the given sides, and C the included angle; then by Art. 7, vers (a b), etc.; hence vers c, cos (a - b) vers c vers (a - b) + sin a sin b vers C; or, since (Plane Trig., Art. 16) vers sin 2 c = sin 2 (a = b) { 1 + c = 2 sinc, etc., sin a sin b sin 2 C sin(a b) b) (S); The third side e may therefore be found from (8) and (9). 10. To find expressions for the sum and difference of two sides of a spherical triangle in terms of the other side and the sum and difference of the opposite angles. Applying the formulæ (6) and (7) of last Art. to the polar triangle, or, which is the same thing (Geo. of the Sphere), replacing A, B, C, a, b, c, by a, π- b, -C, T- A. - В, π- C, we get |