All the results obtained in Art. 26, as particular cases of formula (1) of this Chapter, may also be deduced as particular cases of formula (8), by supposing the triangle to be equilateral or right-angled, as the case may require. 35. Analogous Formula in Plano.—If we suppose the radius of the sphere to be indefinitely great, we have (Chap. II., Art. 18) cos a = 1. Therefore, cos A+ cos B cos C = sin B sin C, If the internal bisector of the angle C make an angle with the opposite side, we have Adding these equations, we eliminate 0, and deduce the above expression. [This result may be obtained at once from Ex. 1]. [This follows, as in Ex. 1, by drawing the external, instead of the internal, bisector of C]. 4. Prove the relation COS B sin (a + b) sin c = 0. COS A 5. If a great circle passes through the vertex C, making angles a and B with the sides a and b, the angle 0, which it makes with the side c, is given by the equation For cos A sin a cos B sin B = + cos 0 sin C. cos A+ cos e cos B = sin 0 sin ẞ8 cos 8 (Art. 26 (1)). where is the intercept between the vertex and the base; also cos B cos e cos a = sin e sin a cos 8, eliminate & between these equations; therefore, &c. 6. If through any point P on a sphere three great circles be drawn, cutting the sides of a triangle at angles X, Y, Z; X1, Y1, Z1; X2, Y2, Z2, respectively; prove the following determinant* relation :— Let the three concurrent arcs make angles a, B, and a + B with each other. *For a knowledge of Determinants, vide Burnside and Panton's Theory of Equations, Chap. xi. Since the side a is cut at angles X, X1, and X2; by Ex. 5, Eliminating a and B from the three equations, the above result easily follows. π 7. Having given that the sides of a triangle are each find the sides of the supplemental triangle. [ 2π Since the angles of the latter triangle are each 8. Find a relation connecting the angles of a triangle if one side a is a quadrant. Ans. cos A+ cos B cos C = 0. 9. Given A, B, C, find the angle which the bisector of the vertical angle makes with the base. If i be the length of the internal bisector, we have 10. If e' denote the angle made by the external bisector of the vertical angle with the base, show that 11. From any three points A, B, and C, on a great circle, secondaries AA', BB', and CC' are drawn to another great circle; prove that the algebraic sum sin B'C' cos A + sin C'A' cos B + sin A'B' cos C is equal to zero. (Apply Ex. 5.) 12. The extremities of the diameter AB of a small circle are joined with a point C on the circle; prove that the angles subtended at the pole by the T [a = 1⁄2 by (Ex. 8). Hence the locus is a great circle, having the vertex B for pole]. 36. Expression for the side of a Spherical Triangle in terms of the Trigonometrical Functions of the Angles (cf. Art. 33). Again, 2 cos2 Expression for the Side of a Spherical Triangle. 49 a 2 We shall use the symbol N to denote the radical in this expression, so that we have sin C sin A sin b = sin A sin B sin c = 2 N. Remark. The positive sign has been given to the radi a a a a cals in the formula for sin, cos and tan since is 2' 2 2' π necessarily less than 2' See also Chap. 11., Art. 23. E |