Or, Radxc=cos Bxa......(1) Radxc=cot Bxb......(4) And by plane geometry, a2-b2+c2. In the above equations, any three quantities being given, the fourth may be found; and as the radius is always known, any other two quantities besides are sufficient to discover the fourth. The equation a2-b2+c2 is not adapted to logarithms. The above equations are investigated in the beginning of Section 3. The Solution of Oblique-angled Triangles. 1. In any triangle let there be given two sides and the angle opposite to one of them, to find the remaining side and angles. Let the three angles be represented as before, by A, B, C; and their opposite sides by a, b, c, respec tively; and let c=345 feet a=232 feet Required the other parts. A=37°20′ It was observed before that this case in oblique-angled triangles is ambiguous, as the geometrical construction, from the given data, would answer the triangle ABC or ABC'; and as there is no restraint or limitation included in the question, either of the angles C, C', may be taken as the result of the analogy. Now as CB=C'B, the angle BCC=BC'C; but BCC' is the supplement of ACB; therefore CC'B and ACB are the supplements of each other. The degrees in the table, answering to the sine, is acute; therefore if it were required to find the obtuse angle ACB, we must deduct C' from 180°, and the remainder, viz. 180° - C', will be the angle ACB. Then to find the angle B, 115°36′ C or 37 20-A 64° 24'-C' 101 44 =C+A 152 56=C+A Hence 180° 152 56' or 180° 101 44' 27 4-B 78 16-B By the Scale and Compasses. In the first analogy-Extend the compasses from 232 to 345 on the line of numbers; then that extent will reach from 37° to 64° on the line of sines. In the second analogy-Extend the compasses from 37° to 27° or 78°1, on the line of sines; then that extent will reach, on the line of numbers, from 232 to 174 or 3742, the two values nearly of the sides b. 2. Given the two sides and the contained angle, to find the other parts of the triangle. In the triangle ABC, let c=345 feet b=174.07 feet A=37°20′. Required the other parts. By Prop. 5, sec. 3, we have A C a B Asc+b:c-b::tan (C+B): tan (C-B); that is, 2 2 180°-37°20′=142°40′=B+C; then By the Scale and Compasses. In the first analogy-Extend the compasses from 519 to 171, on the line of numbers; and that extent will reach from 71°3, on the line of tangents, the contrary way, (because the tangents are set back again from 45°,) a little beyond 45, which being set so far back from 45, falls on 44°, the fourth term. In the second analogy-Extend the compasses from 64° to 37°, on the line of sines; and that extent will reach, on the line of numbers, from 345 to 232, the fourth term sought. 3. Given the three sides of a triangle, to find the angles. |