giving two roots, real and unequal, equal or imaginary, according as a>, =, or <bsin A. A discussion of these two values of c gives the same results as are found in the above four cases. We leave the discussion as an exercise for the student. NOTE. When two sides and the angle opposite the greater are given, there can be no ambiguity, for the angle opposite the less must be acute. When the given angle is a right angle or obtuse, the other two angles are both acute, and there can be no ambiguity. In the solution of triangles there can be no ambiguity, except when an angle is determined by the sine or cosecant, and in no case whatever when the triangle has a right angle. Thus, there are two solutions. See Case III. Ex. 2. Given a = 31.239, b = 49.5053, A B, C, c. Ans. B 56° 56' 56".3, = or C 90° 45' 3".7, or C= 58.456, or 24.382. 32° 18'; find 123° 3' 3".7; 24° 38′ 56′′.3; and thus c is found and the triangle solved. In simple cases the third side c may be found directly by the formula or the formula may be adapted to logarithmic calculation by the use of a subsidiary angle (Art. 90). Ex. 1. Given a = 234.7, b = 185.4, C= 84° 36'; find A, Ex. 2. Given a = .062387, b = .023475, C = 110° 32'; find A, B, c. Ans. A = 52° 10′ 33′′; B = 17° 17' 27"; c = .0739635. 118. Case IV. Given the three sides, as a, b, c; find A, B, C. The solution in this case may be performed by the formulæ of Art. 99. By means of these formulæ we may compute two of the angles, and find the third by subtracting their sum from 180°. But in practice it is better to compute the three angles independently, and check the accuracy of the work by taking their sum. If only one angle is to be found, the formulæ for the sines or cosines may be used. If all the angles are to be found, the tangent formulæ are the most convenient, because then we require only the logarithms of the same four quantities, s, sa, s―b, s — c, to find all the angles; whereas the sine and cosine formulæ require in addition the logs of a, b, c. The tangent formulæ (Art. 99) may be reduced as follows: NOTE, The quantity r is the radius of the inscribed circle (Art. 102). Without the use of logarithms, the angles may be found by the cosine formulæ (Art. 96). These may sometimes be used with advantage, when the given lengths of a, b, c each contain less than three digits. Ex. 2. Find the greatest angle in the triangle whose sides are 13, 14, 15. Let a = 15, b = 14, c=13. Then A is the greatest angle. 1. Given a = EXAMPLES. 254, B = 16°, C = 64°; find b = 71.0919. c = 338.65, A = 53° 24', B = 66° 27'; find 2. Given c = a = 313.46. . 3. Given c = 38, A = c38, A 48°, B = 54°; find a = 28.87, b = 31.43. 4. Given a = 7012.5, B = 38° 12′ 48′′, C = 60°; find b and c. Ans. b = 4382.82; c= 6135.94. 5. Given a 528, b = 252, A = 124° 34'; find B and C. 6. Given a = 170.6, b = 140.5, B = 40°; find A and C. Ans. A = 51° 18′ 21′′, or 128° 41′ 39′′; 7. Given a = 97, b=119, A = 50°; find B and C. Ans. B = 70° 0'56", or 109° 59′ 4′′; C = 59° 59' 4", or 20° 0'56". 8. Given a =7, b=8, A = 27° 47′ 45′′; find B, C, c. Ans. B = 32° 12′ 15′′, or 147° 47′ 45′′; 11. Given a 35, b=21, C= 50°; find A and B. = 12. Given a = 601, b=289, C=100° 19' 6"; find A and B. Ans. A = 56° 8' 42"; B = 23° 32′ 12", |