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Sine B : sine A :: AC: BC,
Sine 27° 11' : sine 39° :: 8.00 : 11:02 = BC,
4568 : 6293 :: 8.00: 11.02 = BC.
The above solution of that trigonometrical proposition, "Where two sides are given, and the included angle,” is not the most mathematical, but it has been given to show the advantage of acquiring a certain readiness in applying the different signs, and of assuming one line and then another as radius. The same proposition will now be solved in a different manner.
If the sum of any two unequal quantities be given, and their difference, half the sum, plus half the difference, will be equal to the greater quantity; and balf the sum, minus half the difference, will be equal to the lesser quantity. Example.—Let the sum of 2 quantities = 14 and their diffe2.
7+1 = 8 = the greater quantity ; and
7-1 = 6 = the lesser quantity. Again, let the sum of 2 quantities = 28, and their difference = 2.
12 14, and
14+6 = 20 = the greater quantity; and
1446 = 8 = the lesser quantity. The two following propositions are taken from “Young's Plane Trigonometry.
Let ABC, Fig. 103, be any plane triangle; the sides AC, BC, and the included angle C, are supposed to be given ; to find the angles A and B.
From the longer of the two given sides, CA, cut off a part, CD, equal to the shorter CB; draw BD, and perpendicular to it draw C E, which produce to F, and draw E G parallel to A B.
Then, because the right-angled triangles C E B, C E D have the angles at B and D equal, and the side CE common, CE bisects the angle at C, and also the line B D. And because DE=E B, therefore D G=GA; therefore CG= } (BC+CA);
CG= half the sum of the given sides ; and since A D is the difference of those sides, therefore
AG= half the difference of the given sides. Again, since the three angles of every triangle make two right angles, the sum of the angles CBD, CDB must be equal to the sum of the angles A' B of the proposed triangle, so that CDB is half that sum ; therefore
CBD = half the sum of the unknown angles; also
ABD = half their difference, because the half sum added to the half difference must make up the greater of the two.
These preliminaries being settled, we have, by right-angled triangles
CE=B Ex tangent CBD;
EF=BER tangent ABD, therefore
CE : EF:: tangent CBD : tangent A BD. But
CE: EF::CG : GA, therefore
2 CG: 2GA :: tangent CBD : tangent A BD, and this proportion expressed in words is the following rule.
As the sum of the two given sides is to their difference, so is the tangent of half the sum of the opposite angles to the tangent of half the difference.
When the given parts are the three sides of a plane triangle. Figs. 104 and 105. When B is acute,
A C = B C? +BA?_2 BAX BD. When B is obtuse,
AC=BCP+BA” +2 BARBD. But B D=B C x cosine CBD; therefore by substitution,
putting a, b, c, for the three sides of the triangle, we have, when B is acute,
6° = a + C2 — 2 a cx cosine B; When B is obtuse,
72 - a" + C + 2 ac x cosine CBD. But C B D being the supplement of B, in the latter case, its cosine is-cosine B; hence, whatever be the character of the angle B,
6°= a + CP-2 a cx cosine B. Here one side of a triangle is expressed in terms of the other two and the included angle, and when the two sides are small numbers this formula inay be conveniently used, the natural cosine of the included angle being employed. As b is any side of the triangle, we of course have similar expressions for a’ and 62. Hence, for the cosines of the three anyles in terms of the sides, we have
60+ c -a Cosine A =
2 b c
a +-62 Cosine B
a + 12 Cosine C
2 a b To resume then,
I. In any triungle, having a side given and two angles, one of which is opposite to the given side.-As the sine of the angle opposite to the given side is to the sine of the other angle, so is the given side to the side opposite the other angle.
In the triangle A B C, given the side A B, the angle A, and the angle C opposite the side A B; required the side opposite the angle A.
Sine C: sine A:: AB:BC.
29654 : 40274 :: 600 : 814 8= BC.
sine 139° = sine 41', then
sine C: sine B :: AB: AC, 29654 : 65606 :: 600 : 1327.4
II. When the three given parts are two sides and the included angle. As the sum of the two given sides is to their difference, so is the tangent of half the sum of the opposite angles to the tangent of half their difference.
In the triangle ABC, A B = 345, and B C = 174, and their included angle B = 37° 20'; required the other parts. A B=345
A + B + C = 180°
A+C=142° — 40'
519 : 171 :: tangent 71° 40' = 2.9600+ : -97527 = tangent 44° 17'; then 71° 20' being half the sum, and 44° 17' half the difference of the opposite angles, it follows from what has been said before, that 71° 20' + 44° 17' = 115° 37' = the greater angle,
and, 71° 20'— 44° 17' = 27° 3' = the lesser angle. The third side may now be found by the former Rule I.
Sine 115° 37' = sine 64° 23'; then, sine 64° 23' : sine 37° 20' :: 345 : 232 = C A. III. When the three sides are given, to find the angles.Calling the three sides by the letters a, b, and c, accordingly as they are opposite the angles A, B, and C, we shall find the angles by calculating the cosines, thus :
1° + C + Cos. A
2 b c
a' + '-12 Cos: B
a +1-C Cos. C=
2 a b
These formulæ the reader may work out by applying them to any of the foregoing triangles. It will, however, be as well to observe, as regards the last proposition of " Three sides given to find the angles,” that whichever angle is sought, the sum of the two sides squared (b2 + c^), is made up of the squares of the two vides including thut angle; and that the square of the side, (a”), subtracted from the sum, is the square of the third side opposite the angle sought; and that the divisor, (2 b c) is made up of the two sides including or containing the angle sought, the one multiplied by the other and taken twice.
The working out of these few rules is far more simple than many of the operations constantly performed, either before or after the construction of works; there is scarcely more than halfa-dozen to remember, and they can therefore be easily remembered; moreover, an abstract of them may be easily enclosed in the field-book or note-book which the engineer or surveyor has always to carry with him, with the Portable Trigonometry.
In the preceding pages of this chapter, and in that on “Setting out Curves,” the reader will find all that he requires for engineering field-work generally, and if it be once impressed on the memory in a rational manner, he will never be at a loss in any matter of setting out skew lines or curves, wherever or however they may be, or in any survey, even of the highest order, presupposing of course that he has attended to former advice, and put it in practice.
A few observations must now be made as to the application of triangulation to different cases, for there is considerable difference as to the manner of applying the system to large areas of land, as for instance, hilly or flat, wooded or clear; again, to the different natural conditions of a navigable river, or an estuary; and again to the survey of a harbour, or lands to be reclaimed always supposing of course that ordinary triangulation and traversing are not applicable.
It is to be understood that in this system of surveying as few as possible of the principal lines are to be actually chained, and that at the angular points triangles are fixed by observations made from one or two base lines, the remaining angles and sides being calculated by the rules given in the preceding pages; in many of the above cases it will be readily seen that to a great extent this must be unavoidably so, as for instance, in a large navigable river which cannot be chained across, but of which the deflections should be reciprocally fixed from both banks, also very likely sandbanks, shoals, or other features in the river itself; again, such a tract of land as a marsh, or any other area more or less permanently covered with water, or so intersected by dykes and cuts that an ordinary system of surveying applied alone would be insufficient; again, an extensive range of downs or steppes, without a vestige of cultivation, where the labour of chaining would be thrown away, or where so hilly or broken that good chaining is all but impracticable, and useless unless contour lines have to be surveyed.
if not all the principal angles have to be taken