PROP. XXI. 84. In any spherical triangle, the rectangle under the sines of any two sides is to the sine of (the third side the difference between the other two sides) x the sine of (the third side — the difference between the other two sides), as the square of radius is to the square of the sine of half the angle included by the first two sides. But cos. a cos. b + sine a sine b = r cos. (a —b) (122 Pl. Tr.). 2 sine2 C r cos. (ab)- - COS. C But Consequently sine a sine b r cos. (a - b) -r cos. c = 2 s. (c + b − a) s. 1 (c + a—b) (123 Pl. Tr.). Hence sine2 C s. § (c + b— a) s. § (c+a—b), therefore r2 = sine a sine b sine a sine b : sine (c+ b − a) sine 1⁄2 (c+a—b) :: r2 : sine2 C. 85. Cor. Hence sine Cr✔ s. 1⁄2 (c+b—a) s. 1⁄2 (c+a—b) sine a X sine b which is a convenient logarithmic formula for the value of the sine of half an angle, when the three sides of a spherical triangle are known. PROP. XXII. 85. In any spherical triangle, the rectangle under the sines of any two sides is to the sine of (the sum of the same two sides the third side) x the sine of (the sum of the same two sides-the third side), as the square of radius is to the square of the cosine of half the angle included by the first two sides. If the value of the cosine of C (83) be substituted in the formula Because r cos. (a+b) = cos. a cos, b 2 2 cos. C sine a sine b (122 Pl. -- cos. a cos. b+ sine a sine b. ↑ cos. c —r cos. (a+b). sine a sine b But r cos.crcos. (a+b)=2s. § (a + b −c) s. § (a+b+c) (123 Pl. Tr.). Consequently cos.3 { C _s. } (a+b+c) s. § (a + b —c) therefore sine a X sine b: sine (a+b+c) sine (a+b—c) ::: cos. C. 86. Cor. 1. Cos. § C=r√ s. § (a+b+c) s. § (a + b —c) sine a sine b which is a convenient log. formula for the value of the cosine of half an angle, when the three sides of a spherical triangle are known. 87. Cor. 2. Because r sine C tan. C (41 Pl. Tr.), By similar triangles ACL, BDF (fig. p. 10), CA: CL:: BD or 2 CL: DF, that is, radius: cos. any arc C :: 2 cos, C: coversed sine C, therefore 2 cos.2 C = r2 + r cos. C. N the sine of 88. Now each of these three formulæ, or values of the sine, cosine, and tan. of C, will determine an angle of any spherical triangle, when the three sides are given; and is well adapted to logarithmic calculation. By the same method of investigation logarithmic formulæ for determining each of the sides of a spherical triangle, in terms of the three angles, may be obtained. PROP. XXIII. 89. In any spherical triangle, the rectangle under the sines of any two angles is to the cosine of (the sum of the same two angles + the third angle) x cosine of (the sum of the same two angles — the third angle), as the square of radius is to the square of half the side contained between the first two angles. If the value of the cosine of a (83) be substituted in the 2 sine2 1 a дов = (46, no. 3 Pl. Tr.), cos. B cos. Cr cos. A sine B sine C Cr cos. (B+C) ―r cos. (B+C)—r cos. A But r cos. (B+C) + r cos. A=2 cos. cos. (B+C - A) (123 Pl. Tr.). Hence (A+B+C) X = — cos. & (A+B+C) cos. (B+C—A), therefore sine B sine C 1 sine B sine C-cos. (A+B+C) cos. (B+C-A) :: r2: sine2 } a. 90. Cor. Sine a=r√ -cos. (A+B+C) cos. (B+C—A) sine B sine C Remark. The second member of this equation is always positive, though under a negative form; for (A+B+C) is greater than 90° (29), therefore its cosine will be negative (35 Pl. Tr.); consequently the expression—cos. § (A+B+C) will become positive. Again, B+C-A cannot exceed 180° (25), or (B+C-A) cannot exceed 90°; therefore cos. (B+C — A) must also be positive. PROP. XXIV. 91. In any spherical triangle, the rectangle under the sines of any two angles is to the cosine of (the third angle+the difference between the other two angles) x. the cosine of (the third angle the difference between the other two angles), as the square of radius is to the square of the cosine of half the side contained between the first two angles. If the value of the cosine of a (83) be substituted in the formula sine B sine C+ cos. B cos. C+r cos. A sine B sine C But sine B sine C+ cos. B cos. C = r cos. (B—C) (122 Pl. Tr.). Consequently 2 cos.2 a r cos. (B-C) + r cos. A (123 Pl. Tr.). Hence p2 Butr cos. (B-C)+ (A + C—B) cos. (A+B-C) cos. (A+C-B), therefore sine B sine C sine B sine C: cos. (A + B - C) cos. 1⁄2 (A+ C —B) :: r2 : cos.2a. 92. Cor. 1. Cos. a=r√ cos. (A+B-C) cos. (A+C-B) r sine a 93. Cor. 2. Because cos. a and 92, that tan. a=ry sine B sine C =tan. a, it follows from art. 90 cos. (B+C+A) cos. ž(B+C−A) cos. (A+B-C) cos. (A+C-B)' Each of these three formula will determine a side of a spherical triangle, when the three angles are given, and is well. adapted to logarithmic calculation. 257827B 94. Remark. Prop. 21 is not proper to be used when the angle sought is near 180°, because the difference of the logarithmic sines of C for 1" is then very small; and very near 180° the log. sines of C (in the tables) are the same for arcs which differ by many seconds. In this case therefore it will be proper to use Prop. 22. Prop. 21 is most proper to be used when C is less than 90°, for then there is a considerable difference between the log. sines of C for 1". The reason of the very small variation of the sines of arcs near 90° is, that the logarithms are not continued to more than seven decimals. Prop. 22 is not proper to be used when the angle sought is very small, because the difference between the log. cosines of C for 1" is then very small; and very near 0° the log. cosines of C are the same for arcs which differ by many seconds. Therefore when an arc is extremely small, Prop. 21 must be used. Prop. 22 is most proper to be used when C is between 90° and 180°, for then there is a considerable difference between the log. cosines of C for 1" (86 Pl. Tr.). Prop. 23 is not proper to be used, when the arc a is near 180°, but is most proper to be used when a is less than 90°. See observation on Prop. 21. Prop. 24 is not proper to be used when the arc a is near 0o, but is most proper to be used when a is between 90° and 180°. See observation on Prop. 22. Formula for the Solution of Spherical Triangles. 95. The following formulæ are sufficient for the logarithmic solution of all the cases of a right-angled spherical triangle. sine c sine A= tan. b cot. B r sine a r cos. B = cos. b sine A = tan. a cot. c r cos. c = cos. a cos. b = cot. A cot. B These forms will also hold in any case, by taking such other sides and angles as are similarly situated with respect to one another. 96. The following formulæ are sufficient for the logarithmic solution of all the cases of any spherical triangle. But there are other simple formula which are better adapted to the solution of right-angled triangles. See articles 95 and 101. |