Tanar√ COS (B+C+A) COS (B+C-A) Either of which formulæ will determine a side when the three angles of the triangle are given. 112. It may be still further observed, that by taking all the varieties of which these last formula for the tangents are susceptible, we shall have Tan Asin (a+b—c) sin 1 (a+c—b) sin(b+c-a) sin (a+b+c) Tan B =sin(b+c—a) sin † (a+b−c) Tanc = √ sin (a+c-b) sin 1 (b+c—a)· TanacOS (B+C—A) COS ↓ (A+B+C) Tanb Tanc COS (A+B-C) cos (A+C-B) ∙COS (A+B-C) COS 1 (A+B+C) 113. In like manner, by taking all the varieties of which the preceding formulæ for the tangents are susceptible, we shall have -a sin (B-A) = Tan tanc 114. And from these last 12 formulæ may be de. duced the following, which serve to find the third side or the third angle, when two sides and the angle opposite to one of them are known. Which formulæ, joined to those of art. 102, for determining a side or angle, when two sides and an angle opposite to one of them, or two angles and a side opposite to one of them, are given, are sufficient for resolving, logarithmically, every case of spherical triangles. It may here also be observed, that these equations furnish the means of determining the affections of the sides and angles of spherical triangles, in all the cases which are not necessarily ambiguous, by barely attending to the signs of the quantities of which they are composed. Thus, in the equation r cos a = cos b cos c, for right-angled spherical triangles, the 3 sides must be all equal to 90°, or all less, or two of them greater and the third less, as no other combination can render the sign of cos b cos c like that of cos a, as the equation requires. Also, in the last analogy for oblique-angled spherical triangles, art. 113, as cot B and cos (ac) are both positive, tan (A+c) and cos (a+c) must have the same sign; hence, half the sum of any two sides is of the same kind as half the sum of their opposite angles: which consideration will sometimes take away an ambiguity that might otherwise arise, in cases where the quantity sought is to be determined by means of a sine. SPHERICAL THEOREMS. THEOREM I. 115. If two arcs of circles meet each other, they make two angles, which are, together, equal to two right angles, or 180°, لا Let the arc AB meet the arc CD in the point в; then will the two / ABC, ABD be equal to two right angles, For, suppose the arc E B to be perpendicular to CD, then the EBC, EBD are right angles. S And since the EBD is equal to the S EBA ABD, the three s EBC, EBA, ABD are equal to two right 2s. ABC, But the two s EBC, EBA are equal to the whence the two Z ABC, ABD are also equal to two right angles. THEOREM II. Q. E. D. 116. If two arcs of circles intersect each other, the vertical or opposite angles will be equal. D E B Let the two arcs AB, CD intersect each other in E, then will the AEC be equal to DEB, and AED to СЕВ, For since the arc A E meets the arc CD, the 4s AEC, AED are, together, equal to two right ≤s (theo. 1). S And because the arc DE meets the arc AB, the s DEB, DE A are also equal to two right s. Whence the sum of the LS AEC, AED is equal to the sum of the 4S DEB, DEA. And if the AED, which is common, be taken away, the remaining AEC will be equal to the reAnd in the same manner it may be maining DEB. shown, that the AED is equal to CEB. Q. E. D. COR. If two arcs of circles intersect each other, the ▲s about the point of intersection are, together, equal to two right 2s. THEOREM III. (Þ) 117. An angle made by any two great circles of the sphere is equal to the angle of inclination of the planes of those circles. B Let BAE be a spherical angle, made by the two great circles CBA, CEA; then will this angle be equal to the angle of inclination of the planes of those circles. For take the arcs AB, AE each equal to 90°, and through the points B, E draw the arc of a great circle BE, and from D, the centre of the sphere, draw DB, de. (p) Any two great circles of the sphere A B C, AFC, which pass through the poles A, C of another great circle BE, cut all the parallels HG, BE into similar arcs, or such as contain the same number of degrees. |