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Containing Plane Trigonometry, right-angled and
oblique, with its Application in determining the Measures of inaccessible Heights and Distances.
S the science of measuring the sides and an
gles of plane triangles. It is divided into two paris, viz. into rectangular and oblique angular trigonometry, because every triangle is either rightangled or oblique ; therefore we shall begin with
Plate V. fig. 1.
1. In every right-angled plane triangle, ABC, if the hypothenuse AC be made the radius, and with it a circle, or an arc of one, be described from each end; it is plain (from def. 22.) that BC is the sine of the angle A, and AB is the sine of the angle C; that is, the legs are the sines of their opposite angles.
2. If one leg AB be made the radius, and with it, on the point A, an arc be described; then BC is the tangent, and AC is the secant of the angle A, by def. 24 and 25. fig. 2.
3. If BC be made the radius, and an arc be described with it on the point C; then is AB the tangent, and AC is the secant of the angle C, as before. Fig. 3.
Because the sine, tangent, or secant of any given arc in one circle, is to the sine, tangent, or secant of a like arc (or to one of the like number of degrees) in another circle; as the radius of the one is to the radius of the other therefore the sine, tangent, or secant of any arc is proportional to the sine, tangent, or secant of a like arc, as the radius of the given arc is to 10.00000, the radius from whence the logarithmic sines, tangents, and secants, in most tables, are calculated, i. e.
If AC be made the radius, the sines of the angle A and C, described by the radius AC, will be proportional to the sines of the like arcs, or angles in the circle, that the tables now mentioned were calculated for. So if BC was required, having the angles and AB given, it will be fig. 1.
As S.C: AB::SA: BC.
i. e. As the sine of the angle C in the tables, is to the length of AB; (or sine of the angle C, in a circle whose radius is AC;) so is the sine of
the angle A in the tables, to the length of BC, (or sine of the same angle, in the circle, whose radius is AC.)
In like manner, the tangents and secants represented by making either leg the radius, will be proportional to the tangents and secants of a like arc, as the radius of the given arc is to 10.00000, the radius of the tables aforesaid.
Hence it is plain, that if the name of each side of the triangle be placed thereon, a proportion will arise to answer the same end as before : thus if AC be made the radius, let the word radius be written thereon; and as BC and AB, are the sines of their opposite angles ; upon the first let S.A, or sine of the angle A, and on the other let S.C, or sine of the angle C, be wrote: then,
When a side is required, it may be obtained by this proportion, viz. As the name of the side given
is to the side given,
to the side required. Thus, if the angles A and C, and the hypothenuse AC were given, to find the legs; the proportions will be
1. R: AC :: S.A: BC. fig. 1.
That is, as radius is to AC, so is the sine of the angle A, to BC. And,
2. R: AC:: S.C : AB.
That is, as radius is to AC, so is the sine of the angle C, to AB.
When an angle is required we use this proportion, viz.
As the side that is made the radius,
is to radius, So is the other given side,
to its name.
Thus if the legs were given to find the angle A, and if AB be made the radius, it will be
AB: R:: BC: T.A. Fig. 2.
That is, as AB, is to radius, so is BC, to the tangent of the angle A.
After the same manner, the sides or angles of all right-angled plane triangles may be found, from their proper data.
We here, in plate 4, give all the proportions requisite for the solution of the six cases in rectangular trigonometry; making every side possible the radius.
In the following triangles this mark --in an angle, denotes it to be known, or the quantity of degrees it contains to be given ; and this mark'on a side, denotes its length to be given in feet, yards, perches, or miles, &c. and this marko, either in an angle or on a side, denotes the angle or side required.
From these proportions it may be observed ; that, to find a side, when the angles and one side are given, any side may be made the radius: and to find an angle, one of the given sides must be made the radius. So that in the 1st, 2d, and 3d cases, any side, as well required as given, may be made the radius, and in the first statings of the 4th, 5th, and 6th cases, a given side only is made the radius.
HE angles and hypothenuse given, to find the
base and perpendicular. Fig. 4. In the right-angled triangle ABC, suppose the angle A 46o. 30', and consequently the angle C 43o. 30', (by cor. theo. 5;) and AC 250 parts, (as feet, yards, miles, &c.) required the legs AB and BC.
Make an angle of 46°. 30' in blank lines, (by prob. 16. sect. 1.) as CAB; lay 250, which is the given hypothenuse, from a scale of equal parts, from A to C; from C, let fall the perpendicular BC; (by prob. 7. sect, 1.) and that will constitute the triangle ABC. Measure the lines BC, and AB, from the same scale of equal parts that AC was taken from ; 'and you have the answer..