2. Given b, c, and s, to find в and c. We employ GAUSS S equations as follows: The first two by eliminating cosa determine (B+c) when the sign of cos a is known, and the second two determine (B—c) when the sign of sina is known. Hence, these equations present no ambiguity when the sign of sin a is given ; for sina is always positive, and cosa has the same sign as sin a according to the formlua, sin a2 sina cosa The equations (6) taken with the positive sign only may give values of в and c exceeding 360°, in which case the required solution will be found by diminishing such values by 360°. 3. Given B, C, and a to find a and b. The general relations between the given and required parts are cos Acos c cos B + sin c sin в cos a (7)* which determine a and b without ambiguity, when the sign of either sin A, sin b, or cos b is given. In like manner the ambiguity is removed when the sign of either sin c or cos c is given. Adapted for logarithms by the method already used, these equations becomet tan tan B cos a (p < 1800 always positive; or less than 90° with the sign of its tangent,) When the quadrant in which ▲ is to be taken is known, (9)+ Check-With (10) or (11) we may employ sin a sin B sin b sin a, and with any of the solutions (8), (9), (10), (11), the check which present no ambiguity when the sign of cosa is given; that is, when the sign of sin A is given, observing that sina is always positive, and cos▲ has the same sign as sin a. As before, when these equations lead to values of b or c greater than 360°, the true values are to be found by subtracting 360°. 5. Given a, b, and A, to find B, C, and c. given and required parts, are The general relations between the -cos c cos A+ sin c sin A cos b = sin a sin B = sin b sin A sin B cos a (13) = sin c cos A+ cosc sin a cos b sin c cos b cos c sin b cos A= sin a cos B The first equation determines в when the sign of cos B is given; and в being known, the remaining equations will fully determine c and c. Thus we find first In these solutions it may happen that 4 +☀', or 0 + 0' exceeds 360°, in which ease c= = 4 + p' — 360°, or c = 0 + 0' - 360°. Checks. One of the following* may be employed when either c or c has alone been computed : When both c and c have been computed, the obvious check is 6. Given a, b, and ▲, to find c and c without finding B. Observing that k is positive in the preceding article we deduce the following forms and conditions, by eliminating B ;t (k positive) cos ' = cos o cot a tan b (p' less than 1800, with the same sign as cos B.) (17) (1) The propriety of employing the same factor k in both (15) and (16) will be We find in both seen by comparing the values of k deduced from the two groups. + By substituting sin a sin b for sin B in the 4th of (15), and for sin ▲ its value sin a from 2nd of 15. See 3d of (15.) § The value of cos e' is obtained by taking the value of k in the 2d of (16), and substituting it in the 4th. In these solutions, when +', and +0 +0' exceed 360°, we must take c = +'-360°, c = 0 + 0'-360°; and when they are negative we must take c='+360°, c = 0+0'+360°. 7. Given A, B, and a, to find b, c, and c. We find b by the formula which determines b when the sign of cos b* is given. The remainder of the solution is by (15) and (16). 8. Given A, B, and a, to find c and c without finding b. We may eliminate b from (15) and (16) in their present form, but the conditions for determining the auxiliary angles will not be so simple as in the following method. Let ø and 'in (15) be exchanged for '900, and +90° respectively; then after eliminating b, we find In these formulas, as before, when ☀ + ø', and 0 +0' exceed 360°, we take c = +'-360°, c = 0+0'360°; and when they are negative we take c=+ '+360°, c = 0 + 0' + 360°. 9. Given a, b, and c, to find A, B, or c. The formula (see Art. 82) determines a when the sign of sin A is given; or when the sign of either sin A, sin B, or sin c is given; when the sign of any one of these functions is known, those of the other two may be discovered by an inspection of the equation * Which determines the quadrant in which b is, the sign of the sin b only determining whether it is in the first two or last two. + Compare the 3d of (18). * Compare the 3d of (17). The usual formulas for sin A, cos A, tanA (see Art. 84), derived from (21), may be employed, and the ambiguity removed, by the same conditions. Check.-Compute two of the functions sina, cosa, tana; or one of them in connexion with (21). 10. Given A, B, and c, to find a, b, or e. The formula (see Art. 88) determines a when the sign of sin a is given; or when the sign of either sin a, sin b, or sin c is given, since when the sign of any one of these functions is known, those of the other two may be discovered by an inspection of the equation the usual formulas for sina, cosa, tana, may be employed, and the ambiguity removed, by the same conditions. Check. Compute two of the functions sina, cosa, tana; or one of them in connexion with (22). 10. From the preceding sketch it appears that for the determinate solution of a spherical triangle generally considered, there are required four data; namely, the numerical values of three of the six parts composing the triangle, and the algebraic sign of one of the functions of a required part. To recapitulate, the triangle is fully determined by the following data: 1. b, c, A; and the sign of either sin a, sin B, cos B, sin c, or cos c. 11. Since is the symbol which represents the circumference of a circle whose diameter is unity, or the semicircumference whose radius is unity, will represent a quadrant of the latter or 90°. We may, therefore, for convenience, represent the and its complement by a. supplement of an arc a by π- a, 12. A triangle, one side of which is 7 or 90°, is called a quadrantal triangle; such triangles may be resolved by Napier's rules for the circular parts, if the quadrantal side be neglected, and b, π-c, B, C, and a — be taken for the circular parts. - π For let A'B'C' be the polar triangle. It will be right angled because ▲=” — ɑ . Applying Napier's rule to this triangle we obtain |