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5. Given the side b 58, the side c 67, and the included angle
A=36°; to find the angles B and C, and the side a. 6. Given the three sides, 631,
268, and 546; to find the angles.
155. The three theorems demonstrated in this section, have been here applied to oblique angled triangles only. But they are equally applicable to right angled triangles.
Thus, in the triangle ABC, (Fig. 17.) according to theorem I, (Art. 143.)
Sin B : AC::sin A : BC
This is the same proportion as one stated in art. 134, except that, in the first term here, the sine of B is substituted for radius. But, as B is a right angle, its sine is equal to radirs. (Art. 95.) Again, in the triangle ABC, (Fig. 21.) by the same theorem ;
Sin A: BC::sin C : AB This is also one of the proportions in rectangular trigonometry, when the hypothenuse is made radius.
The other two theorems might be applied to the solution of right angled triangles. But, when one of the angles is known to be a right angle, the methods explained in the preceding section, are much more simple in practice.*
* For the application of Trigonometry to the Mensuration of Heights and Distances, see Navigation and Surveying:
GEOMETRICAL CONSTRUCTION OF TRIANGLES, BY THE PLANE
Art. 156. To facilitate the construction of geometrical figures, a number of graduated lines are put upon the common two feet scale; one side of which is called the Plane Scale, and the other side, Gunter's Scale. The most important of these are the scales of equal parts, and the line of chords. In forming a given triangle, or any other right lined figure, the parts which must be made to agree with the conditions proposed, are the lines, and the angles. For the former, a scale of equal parts is used; for the latter, a line of chords.
157. The line on the upper side of the plane scale, is divided into inches and tenths of an inch. Beneath this, on the left hand, are two diagonal scales of equal parts,* divided into inches and half inches, by perpendicular lines. On the larger scale, one of the inches is divided into tenths, by lines which pass obliquely across, so as to intersect the parallel lines which run from right to left. The use of the oblique lines is 10 measure hundredths of an inch, by inclining more and more to the right, as they cross each of the parallels.
To take off, for instance, an extent of 3 inches, 4 tenths, and 6 hundredths ;
Place one foot of the compasses at the intersection of the perpendicular line marked 3 with the parallel line marked 6, and the other foot at the intersection of the latter with the oblique line marked 4.
The other diagonal scale is of the same nature. The divisions are smaller, and are numbered from left to right.
153. In geometrical constructions, what is often required, is to make a figure, not equal to a given one, but only similar. Now figures are similar which have cqual angles, and the
* These lines are not represented in the plate, as the learner is supposed to have the scale before him.
sides about the equal angles proportional. (Euc. Def. 1. 6.) Thus a land surveyor, in plotting a field, makes the several lines in his plan to have the same proportion to each other, as the sides of the field. For this purpose, a scale of equal parts may be used, of any dimensions whatever. If the sides of the field are 2, 5, 7, and 10 rods, and the lines in the plan are 2, 5, 7, and 10 inches, and if the angles are the same in each, the figures are similar. One is a copy of the other, upon a smaller scale.
So any two right lined figures are similar, if the angles are the same in both, and if the number of smaller parts in each side of one, is equal to the number of larger parts in the corresponding sides of the other. The several divisions on the scale of equal parts may, therefore, be considered as representing any measures of length, as feet, rods, miles, &c. All that is necessary is, that the scale be not changed, in the construction of the same figure ; and that the several divisions and subdivisions be properly proportioned to each other. If the larger divisions, on the diagonal scale, are units, the smaller ones are tenths and hundredths. If the larger are tens, the smaller are units and tenths.
159. In laying down an angle, of a given number of degrees, it is necessary to measure it. Now the proper measure of an angle is an arc of a circle. (Art. 74.) And the measure of an arc, where the radius is given, is its chord. For the chord is the distance, in a straight line, from one end of the arc to the other. Thus the chord AB (Fig. 33.) is a measure of the arc ADB, and of the angle ACB.
To form the line of chords, a circle is described, and the lengths of its chords determined for every degree of the quadrant. These measures are put on the plane scale, on the line marked CHO.
160. The chord of 60° is equal to radius. (Art. 95.) In laying down or measuring an angle, therefore, an arc must be drawn, with a radius which is equal to the extent from O to 60 on the line of chords. There are generally on the scale, two lines of chords. Either of these may be used; but the angle must be measured by the same line from which the radius is taken.
161. To make an angle, then, of a given number of degrees ; From one end of a straight line as a center, and with a radius equal to the chord of 60° on the line of chords, describe an arc of a circle cutting the straight line. From the
point of intersection, extend the chord of the given number of degrees, applying the other extremity to the arc; and through the place of meeting, draw the other line from the angular point.
If the given angle is obtuse, take from the scale the chord of half the number of degrees, and apply it twice to the arc. Or make use of the chords of any two arcs whose sum is equal to the given number of degrees.
A right angle may be constructed, by drawing a perpendicular without using the line of chords.
Ex. 1. To make an angle of 32 degrees. (Fig. 33.) With the point C, in the line CH, for a center, and with the chord of 60° for radius, describe the arc ADF. Extend the chord of 32o from A to B ; and through B, draw the line BC. Then is ACB an angle of 32 degrees.
2. To make an angle of 140 degrees. (Fig. 34.) On the line CH, with the chord of 60°, describe the arc ADF; and extend the chord of 70° from A to D, and from D to B. The arc ADB=70° X 2=140.°
On the other hand;
162. To measure an angle; On the angular point as a center, and with the chord of 60° for radius, describe an arc to cut the two lines which include the angle. The distance between the points of intersection, applied to the line of chords, will give the measure of the angle in degrees. If the angle be obtuse, divide the arc into two parts.
Ex. 1. To measure the angle ACB. (Fig. 33.) Describe the arc ADF cutting the lines CH and CB. The distance AB will extend 32° on the line of chords.
2. To measure the angle ACB. (Fig. 34.) Divide the arc ADB into two parts, either equal or unequal, and measure each part, by applying its chord to the scale. The sum of the two will be 140°.
163. Besides the lines of chords, and of equal parts, on the plane scale; there are also lines of natural sines, tangents, and secants, marked Sin. Tan. and Sec.; of semitangents, marked S. T.; of longitude, marked Lon. or M. L.; of rhumbs, marked Rhu, or Rum. &c. These are not necessary in trigonometrical constructions. Some of them are used in Navigation; and some of them, in the projections of the Sphere.
164. In Navigation, the quadrant, instead of being graduated in the usual manner, is divided into cight portions, called
Rhumbs. The Rhumb line, on the scale, is a line of chords, divided into rhumbs and quarter-rhumbs, instead of degrees.
165. The line of Longitude is intended to show the number of geographical miles in a degree of longitude, at different distances from the equator. It is placed over the line of chords, with the numbers in an inverted order: so that the fig. ure above shows the length of a degree of longitude, in any latitude denoted by the figure below.* Thus at the equator, where the latitude is 0, a degree of longitude is 60 geographical miles. In latitude 40, it is 46 miles; in latitude 60, 30 miles, &c.
166. The graduation on the line of secants begins where the line of sines ends. For the greatest sine is only equal to radius ;
but the secant of the least arc is greater than radius. 167. The semitangents are the tangents of half the given arcs. Thus the semitangent of 20° is the tangent of 10°. The line of semitangents is used in one of the projections of the sphere.
168. In the construction of triangles, the sides and angles which are given, are laid down according to the directions in Arts. 158, 161. The parts required are then measured, according to Arts. 158, 162. The following problems correspond with the four cases of oblique angled triangles ; (Art. 148.) but are equally adapted to right angled triangles.
169. Prob. I. The angles and one side of a triangle being given; to find, by construction, the other two sides.
Draw the given side. From the ends of it, lay off two of the given angles. Extend the other sides till they intersect; and then measure their lengths on a scale of equal parts.
Ex. 1. Given the side b 32 rods, (Fig. 27.) ihe angle A 56° 20', and the angle C 49° 10'; to construct the triangle, and find the lengths of the sides a and c.
Their lengths will be 25 and 27.
2. In a right angled triangle, (Fig. 17.) given the hypothenuse 90, and the angle A 32° 20', to find the base and perpendicular.
The length of AB will be 76, and of BC 48.
* Soinetimes the line of longitude is placed under the line of chords.