By Rule II. (without haversines.) (3) Given a 512, b = 627, c=430: required C. In the above examples the angles A B and C have been taken out of the tables by inspection, namely, to the nearest 15". This degree of accuracy will be found in most cases sufficient. Rules for proportioning for seconds are given in the explanation of the tables. EXAMPLES. Required the angles of the plane triangle ABC whose three sides a, b, c, are Of two sides and two opposite angles, any three being given to find the fourth. Write down a proportion, having for the two first terms the two sides concerned, and for the third term the sine of the angle opposite to the first side put down, and for the fourth term the sine of the angle opposite the side in the second term of the proportion: mark the term required with a stroke of the pen underneath. E If the term marked be a middle term; add the logarithms of the two extreme terms and subtract the logarithm of the middle term not marked. If the term marked be an extreme term; add the logarithms of the two middle terms, and subtract the logarithm of the extreme term. The result will be the logarithm of the required term. NOTE 1. Since the sine of an angle has the same numerical value as the sine of its supplement; if one of the angles be greater than 90° subtract it from 180° and look out the log. sine of the remainder in the table of log. sines, as the log. sine of the angle in question. 2. When two angles are known, the third can be found by adding together the two known angles and subtracting the result from 180°: the remainder will be the third angle. 3. Therefore if two angles and the adjacent sides be given, the third angle is also known, and thence by the above rule the other two sides. 4. This rule depends on the property of triangles that the sides are to one another in the same proportion as the sines of the angles opposite to them; and it will be, in general the easiest way of solving the example to write down such proportion, and find the unknown term by Art. (24, p. 20). 504, 6=25, and A= 68° 48': to Ex. Given a find the other parts. (fig. 1). 35. When two sides and the angle opposite the less side are given, each of the quantities sought will have two distinct values. For suppose that in the triangle ABC, (fig. 2) the sides BC and AC and the angle A opposite to the less side are given, then if BC be not perpendicular to AB, and from C as a center with radius CB an arc be described, it will cut AB in another point B': then if CB' be joined, it is manifest that there will be two triangles ACB and ACB' having the given parts, namely the sides AC CB or CB' and the angle A the same in both while the remaining parts are different. The angle ABC or AB,C is found by the formula a: b :: sin. A sin. B or a b:: sin. A: sin. A B,C: B and AB,C have the same b sin. A : hence the sines of the angles numerical value (each being = a they are there |
