[TRIGONOMETRY.] tan (6- A + B) ,tan C = sin I c cos (a + b) since 2s - a - b = c. Hence the first of (5) easily follows, and the second in a similar manner.
The formula (6) is not easily remembered, except by the following : -Write the sides in any successive pairs, as ab, be, Ca, or ac, cb, ba.: change the last three into the corresponding angles giving ab, bc, C A, or ac, CB, BA; remembering the formula cos sin = cot make the middle terms cosines, those on the right and left sines, and those on the extreme right and left cotangents. We have then cot a sin b = cos b cos c + sin C cot A, which is a case of the formula in question.
We now proceed to the different cases of triangles, observing that these may be taken in pairs, owing to the property of the supple mental triangle. Thus, suppose it granted that we can solve the case of finding the three angles when the sides are given, it follows that we can solve that of finding the three sides from the three angles. For, if A, B, and c be given, find the angles of the triangle whose sides are A', B', and c'. If b', and c' be these angles, then a, b, c are the sides of the original triangle. Nor is it worth while to separate the several cases, since it generally happens that out of each pair one is of much more frequent occurrence than the other.
Case 1.-Given the three sides, to find the three angles. If one angle only be wanted, one of the fornaulre (3) answers as well as any thing. If all three angles be wanted, the shortest way is to calculate M from Dt = V {sin (s - a) sin (s - b) sin (s - c)-:- sin s} and then the angles from if if if tan 6 = 0- by tan 6 c = sin tan A sin (s Supplement.-Given the three angles, to find the three sides. Make the supplements of the given angles sides, calculate the three angles, and the supplements of the last three angles will be the sides required.
Case 2.-Given two sides (a and b), and the included angle c, to find the remaining parts. If all the parts be wanted, calculate 8 (A + n) and 4 (A - B) from (5), and then find A and n by addition and sub traction : lastly, find c from one of sin c sin sin C = Sin b - sin c = sin a = ; sin B' ein A or from both, which will be a verification. But if the remaining side
only be wanted, use the formula (7) or (8), which gives this third side at once, by means of the subsidiary angle 0. Or from the extremity of the shorter side given (say a), let fall a perpendicular arc z on b, dividing b into x and y. Then tan x = tan a cos C, sin z = sin a sin a = b - x, cos c= cos z COS y.
Supplentont.-Given a side (c) and the two adjacent angles (A and n); required the remaining parts. Make A' and 33' the sides of a triangle, and c' the included angle ; find C' the remaining side, and a' and b' the remaining angles. Then c is the remaining angle of the original triangle, and a and b are the remaining sides. To find the remaining sides alone, the following formula may be used : a + b C COS (A - B) = tan 2 • cos 6 + B) a - b C sin 6 (A - 33) tan 2 = tan • sin 6 (A. + B) Case 3.-Given two sides (a and b) first both less than a right angle, and an angle opposite to one of them (A); required the remaining parts. This case may afford no solution at all, or may give two solutions ; it is therefore sometimes called the ambiguous case. The formula (6) may be used by the usual introduction of a subsidiary angle ; but we should recommend a person who is not well practised in the subject to prefer the following simple method :-From the extremity of b which is not adjacent to the given angle, drop a perpendicular z on the side c, and let x be the segment adjacent to A. Let a"z and b'z be the angles made by a and b with z. And first calculate sin b sin ; if this be greater than unity, the triangle does not exist ; if it be equal to unity, the triangle is right angled at a, and may be treated as a right-angled triangle. But if sin 6 pin a be loss than unity, find : from sin : = sin it sin A, and x and a from • sin b tan x = tan 6 cos A, 8111 = 1111I A There are two values of a, supplements of each other, both of which are possible answers. Let s be the one which is less than a right angle, and a' that which is greater.