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PLANE Trigonometry is that part of Geometry, which teaches how to measure the fides and angles of plane triangles. It is divided into right-angled and oblique-angled trigonometry.
The circumference of any circle, is divided into 360 equal parts, called degrees, and each degree into 60 equal parts, called minutes, and each minute into 60 equal parts, called seconds, and so on.
Note, Degrees are frequently marked °, and minutes'. Thus,
30 degrees, 14 minutes, are marked 30°, 14.
A femi-circle contains 180°, and a quarter of a circle or quadrant, 90°. Thus the arch ABD, is 185 °, and BD is 90°,
1. THE complement of an arch, is what it wants of 90°, or of a quadrant. Thus, the complement of the arch ED, is ED. See fig. 50. Plate 3.
2. The supplement of an arch, is what it wants of a semi-circle, Thus the supplement of the arch ED, is EBA.
Note, An arch and an angle measure each other.
3. A line drawn through one extremity of an arch perpendicular upon the diameter passing through the other extremity, is called the fine of that arch. Thus, EH is the fine of the arch ED, or of the angle ECD.
4. The fegment of the diameter intercepted between the fino and extremity of an arch, is called the versed fine of that arch. Thus, HD is the versed fine of the arch ED, or of the angle ECD.
5. A straight line passing through D, one extremity of an arch, and meeting the diameter produced through E, the other extremity, is called the tangent of the arch. Thus, GD is the angent of the arch ED, or of the angle ECD.
6. A straight line drawn from the centre, through one extromity of an arch, meeting the tangent drawn through the other extremity, is called the secant of that arch. Thus, CG is the secant of the arch ED, or of the angle GCD.
Corollary 1. The fine, tangent and fecant of any arch, is the fine, tangent, or fecant of its supplement.
BK is the tangent, CK thé secant, and EL the fine of the arch BE, according to definitions 3, 5, and 6, but BE is the complement of the arch ED; therefore LE, BK and CK, are the fine complement, tangent complement, and secant com* plement of the arch ED. But for brevity's fake, they are cal
led the co-fine, co-tangent, and co-fecant of the arch ED, or of th: angle E CD.
Corol. 2. Since the triangle CEH and GCD are similar, CH:CD (=CE):: CE: CG. Hence,
In words, The radius is a mean proportional between the co-line and fecant of any arch.
Corol. 3. Because the triangles BKC, GCD are similar, GD:DC(=CB):: CB: BK. Hence,
In words, The radius is a mean proportional between the tangent and co-tangent of
Note, The least possible secant, the tangent of 45°, and the fine of 90°, are each of them equal to the radius
In every triangle, there are six things to be considered, viz. three fides and three angles.
All the angles in a triangle, are together equal to two right angles, or 180°. If, therefore, two angles of a triangle are given, the third is also given,' for it is found, by subtracting the sum of the other two from 180°.
When one angle of a triangle is given, the sum of the other two may be found, by subtracting the given angle from 180°.
When one angle of a triangle is a right-angle, the other iwo are acute, and are together equal to one right-angle, and consequently are complements of each other.
IN any right argled plane triangle, if the hypothenuse be
mede radi!ls, the legs become the fines of the opposite angles : but if either of the legs be made radius, the other leg becomes the tangent of the opposite angle, and the hypothenuse becomes the secant of the fame angle. Fig. 51. plate 3.
LET ABC be a right angled triangle, if the hypothenuse BC be made radius, the fide AC will be the fine of the opposite angle ABC ; and if either side, BA be made radius, the other leg AC will be the tangent of the opposite angle ABC, and the hypothenuse BC, the fecant of the fame angle.
With the centre E, and radii BC, BA, describe two arches CD, LA, meeting BC, BA in E and D. Since CAB is a right angie, BC being radius, AC is the sme of the angle ABC, by definition 3, and BA being radius, AC is the tangent, and BC the fecant of the angle ABC, by def. 5, 6.
Since circles are to one another as their radii, similar arches of the fime circles will be in the same proportion; therefore, the lines, tangents, and fecants of similar arches, that is, of equal angles, are as their radii; consequently, the tabular radius is to the tabular sine, tangent or secant of either of the acute angles of a right angled triangle, as the radius of the given triangle, is to the fine, tangent or fecant, in the same triangle.
And, because any one of the three fides may be called the radius, any of the fides required, may be obtained by three enalogies or varictics.
X. B. All the varieties which can occur in the folution of
right angled triangles, may be comprehended under two prollems.
First, When all the angles and one side are given, to find the other two sides.
21., When two sides and the right angle are given, to find two acute angles and the third side.
We come now more fully to shew how each of these problems are solved by logarithms.
CASE I. The angles and one of the legs given, to find the hypothes
nuse and the other leg. Plate 3. fig. 43.
Ex. 1. In the triangle ABC, right-angled at B, fuppofe AB 300 equal parts, as feet, yards, miles, &c., and the angle at A 30° 40°, (and consequently the angle at C 59° 20') Required the fides BC, AC.
Variety 1. making AB rad. BC becomes the tangent, and AC the fecant of angle A. Whence arise the following proportions:
To find BC.
To find AC. ralius 90, 10.00000
As rad. 90°
10.00000 is to AB 300
2.47712 Sotang. ang. A 30°40' 9.77303 So fec. A 30° 40' - 10.06543 2.25015 | To AC 348.8
To BC 177.9
Variety 2. making BC rad. BA becomes the tangent, and AC the fecant of the angle at C. Hence the following pro- portions: