middle part, and a and b adjacent parts. Applying Napier's rules to this case, with the parts (a) we have sin (C-90°) = tan (90° — a) tan (90° . which gives cos C cot a cot b, an equation identical with the fifth of the above list. - b); EXERCISES. 1. Let the student deduce the six equations (15) by applying Napier's rules to the parts (a). 2. Through the same point there pass two lines intersecting at right angles, and a plane P making the angle a with one of the lines, and the angle with the other. Express the angle which the plane P forms with the plane of the lines. Ans. Sin A = Vsin' a + sin2 ß. 3. The sides of an obelisk have a slope of 8° from the perpendicular. What is the face-angle at the base of the obelisk, the slope of the edges, and the dihedral angle between two adjacent lateral faces? Ans. Face-angle at base, 82° 4'.6; slope, 11° 14'.5; dihedral angle, 91° 6′.6. In this problem, to reduce to a spherical triangle, consider the centre of the sphere to be at a corner of the obelisk. The slope of the edge will not be represented by either of the six parts of the triangle, but by the complement of the perpendicular from the vertex upon the base. 4. A mason cuts a stone with a rectangular base and four lateral edges, each making an angle of 60° with the base at its corners. What is the inclination of each lateral face to the base, and the dihedral angle between the faces, supposing such inclinations and dihedral angles all equal? Ans. 67° 47'.5 and 98° 12'.8. 5. In another stone the base is rectangular; one lateral face makes an angle of 68° 29' with the base, and the lateral edge bounding this face makes an angle of 52° 15' with the base. What angles does the adjacent lateral face make with the first face and with the base? 6. When the angular distance of the sun from the south point of the horizon is 75°, and from the west point 60°, what is its altitude above the horizon? CHAPTER III. RICAL TRANSFORMATION OF THE FORMULÆ OF SPHERICAL TRIGONOMETRY. 118. Although the formulæ already given suffice for the solution of every spherical triangle, there are many transformations which will facilitate the applications of spherical trigonometry, and render the solutions of triangles more accurate and convenient. Let us first take the fundamental equation (1) of Chapter I., cos a = cos b cos c + sin b sin c cos A. c) - 2 sin b sin c sin' A. (843) (2) 2 sin b sin c = cos (b + c) — cos (b — c), we have, by substituting, cos a = cos (bc) (1 sin' A) + cos (b+c) sin2 A = cos (bc) cos2 4 + cos (b + c) sin2 A. This last equation may also be derived by the following elegant process. The original fundamental equation may be written cos a = cos b cos c (cos2 A + sin2 A) sin b sin c (cos' Asin' A) (because cos' A + sin' A = 1, and cos2 sin' A cos A). 4, the A and of sin' By conjoining the coefficients of cos' equation (2) follows by the addition theorem. By a similar process, from the equation we obtain cos A = cos (B+ C) 2 sin B sin C sin' ta; cos Acos (B+ C) costa · - cos (BC) sin' a. (4) By a slight modification of the process employed in forming the equations (1) and (3) we may find cos acos (b+c) + 2 sin b sin e cos2 4; cos A = cos (BC) + 2 sin B sin C cos' ta; which equations the student may prove as an exercise. (5) (6) 119. Expressions when three sides or three angles are given. From the last equation (1) we find by which any angle is expressed in terms of the three sides. By § 44, 13 we have cos (bc) cos a = 2 sin (a + c − b) sin (a+b−c). (a) If we put s for half the sum of the sides, namely, Substituting these values in (a), the expression for sin' A To find similar expressions for the cosines we take equation (5), which gives cos2 A = cos a ― cos (b+c) 2 sin b sin c But cos acos (b+c) = 2 sin (b+c+a) sin (b+c—a) Since an angle near 90° cannot be accurately determined by its sine, nor one near 0° by its cosine, neither of the formulæ (7) or (8) can be advantageously used in all cases. But by taking the quotient of each equation (7) by the corresponding one of (8) we 120. Treating the equations (3) and (6) in the same way, and we find the following expressions for the sides, in terms of the three angles: For the solution of a triangle in which all three sides or all three angles are given, the equations (9) and (12) are preferable. |