Given two Sides and an Angle opposite one of them, to find the other Angles and the third Side Example I.-Given the side A B = 336, the side B C = 355, and the angle A 49° 26'; required the angles B and C, and the side A C. BY CONSTRUCTION Draw the line A B, which make equal to 336; draw the line A C so as to make an angle of 49° with A B; take the length of B C in the compasses, and setting one foot in B, let the other cut the line A C in C, and draw the line BC; then the angle B will measure 84°, the angle C 46°, and the side A C 465.3. BY CALCULATION There is no ambiguity here, inasmuch as the given angle is opposite to the greater given side. Example II.-Given the side A B = 355, the side B C = 336, and the angle A= 49° 26'; required the angles B and C to the nearest second, and the side A C. NOTE. The given angle being opposite to the less side, there are two different triangles which may possess the same data, because a circle described with B as centre and radius B C will cut C A in C', hence B C' has the same value as B C, so that both triangles C B A and CBA will satisfy the conditions of the question; it is therefore doubtful, or ambiguous, whether we must make C less than 90°, or greater than 90°. This example constructed with the side A B 355, and side BC = 336, the projection makes A C 30.4 or 4313, since the angle C may be either 531° or 1261°, i.e., either acute or obtuse. = To find the other two angles, the two sides and the included angle being given First Method Tan. (AB) = a b a+b × tan. (A + B) L. tan. (A~ B) = log. (a - b) + L. tan. L. (A + B) — log. (a + b) The value of (A + B) being known, and that of † (A ~ B) being found, their sum and difference give respectively the values of the angles A and B. The third side may be found by the Rule of Sines. GENERAL RULE for the half difference of the Angles.-Find the sum and difference of the given sides, subtract the given angle from 180°, and take half the remainder, for half the sum of the unknown angles: to the log. of the diff. of the sides add the tan. of the half-sum of the unknown angles: from the sum of these logs. take log. of the sum of the sides; the remainder is the tan. of the half-diff. of unknown angles, which take out add the half-diff. to the half-sum of unknown angles for the angle opposite the greater side, and subtract it to get the less angle. A Given two Sides and the included Angle, to find the other Angles and Example. Given the side A B = 85, the side A C= 47, and the angle 52° 40'; required the angles C and B, and the side B C. L. tan. (CB) = log. (cb) + L. tan. (C+B)-log. (c + b) : sin. B sin. B a sin. A b (c + b) = 132 The greater angle is opposite the greater side. A C, hence the angle C is greater than the angle B. the angle C, and the difference is the angle B. A B is greater than NOTE. This triangle may be solved by letting fall a perpendicular from the angle C on the side A B, which will divide it into two right-angled triangles; then with the hypotenuse A C and angle A find the perpendicular and the base, which base subtracted from the side A B will leave the base of the other triangle; then, with the perpendicular and base find the angle B, which added to angle A, and their sum subtracted from 180°, will give the angle C; and, with one of the angles and its opposite side find the side B C. The third side can be found direct by the following formula To find the Angles, the Sides being given A triangle may be determined from its three sides, by the following formulæ where s sum of the sides— = The usual method of finding the angles is through the cosine (1), except when the angle is very small, for then the change in log. cosines is very small, and consequently not conducive to accuracy. In this case the angle should be found through the tangent. From (I) Cos. A= {log. s + log. (sa) +20 (log. blog. c)} where s = a + b + c 2 RULE I. Cosine Method.-Find half the sum of the sides, and from it subtract the side opposite the angle sought, which call the remainder. Add together the logarithms of the half sum of the sides, and the remainder, and increase the index by 20; also add together the logarithms of the two sides containing the angle sought; subtract the latter sum from the former and divide by 2; this will give the cos. of half the angle sought. But the method by the tangent is the readiest, when all the three angles are sought, as only four logarithms are required to be taken out of the Tables; and the rule may be simplified as follows From (2)- - 1⁄2 {log. (s—6) + log. (s — c) —— log. s — log. (s—a)} + 10 RULE II. Tangent Method. Add together the three sides a, b, and c; take half their sum, which calls; from this half sum subtract each side in succession, and thus you have the value of s, sa, s — b, and s — c; subtract the logarithm of s from 20.000000; to the remainder add the logarithms of s — a, s — b, and s c; divide the sum by 2 for the half-sum, which becomes a constant; from this constant subtract in order the logarithm (sa), for tan. A; the logarithm (s—b) for tan. B; and the log. three angles of the (s c) for tan. C; multiply each angle by 2, and the triangle are obtained, the sum of which should be 180°. Given the three Sides, to find the Angles = Example. Given the side A B = 157, the side B C 110, and the side AC 88, to find the angles A, B, and C. = |