. Elements of plane and spherical trigonometry . ha theoretical solution of a tri- Fig. Str- angle when two sides and theincluded angle are given, orwhen the three sides are given;but they are not well adaptedfor logarithmic this reason they are seldomused in this form. Other for-mulas derived more or lessdirectly from them are foundto be more useful in practice. The most convenient methods for the complete solution oftriangles are given in Chap. XI. When, however, two sidesof a triangle and the included angle are given, and the thirdside only is required, we can use (125) by m
. Elements of plane and spherical trigonometry . ha theoretical solution of a tri- Fig. Str- angle when two sides and theincluded angle are given, orwhen the three sides are given;but they are not well adaptedfor logarithmic this reason they are seldomused in this form. Other for-mulas derived more or lessdirectly from them are foundto be more useful in practice. The most convenient methods for the complete solution oftriangles are given in Chap. XI. When, however, two sidesof a triangle and the included angle are given, and the thirdside only is required, we can use (125) by means of the deviceshown in the following exercise. Given A = 73° 25, b = 38° 12, c = 49° 37. To find a. In the formula cos a = cos b cos c + sin b sin c cos A,let k sin 0 = cos &, (a) and h cos 6 = sin b cos A. (/?) Substituting these values of cos b and sin 5 cos A,we have cos a = & sin 6 cos c + k cos 6 sin c, or, by (48) cos a = k sin (0 + c) ; where the auxiliary quantities k and 6 are found from (a) and(P) as in § tan 6. cos 6 cot b sin 6 cos A cos .4 GENERAL FORMULAE. 125 and sin 0 sin b cos A cos 0 The logarithmic work may be arranged as follows : loa; cot b + c + c = .10407= .64860= 77° 20 31= 257° 20 31= 126° 57 31= 306° 57 31 log cos blog sin 6log & (±) = (±) sin (0 + c) = (±) cos a a = 49° 5628 It should be noted that while 8 has two values, and hence bothk and sin (0 + c) have two equal values, positive and negative, ahas only one value, since the upper signs of k and sin (6 -f c) mustbe taken together, and also the lower signs. Given B = 51° 16, a = 38°, c = 74°. Find Given (7= 102°, a = 85°, 6 = 63°. Find c. 91. Polar Triangles. We know from geometry that ifABC and ABC be two polar triangles, A = 180° —a,a= 180° — A, B = 180° — 6, 6 = 180° — B: C = 180° — c, c = 180° — ( taking a formula that appliesto any spherical triangle and sub-stituting th
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