TRIGONOMETRY.

27

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 feconds, and fo 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 18°, and BD is 90°,

DEFINITIONS.

1. THE complement of an arch, is what it wants of 90°, or of a quadrant. Thus, the complement of the arch ED, is EB. See fig. 50. Plate 3.

D 2

2, The

2. The fupplement of an arch, is what it wants of a semi-cir cle, Thus the fupplement 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 paffing 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 fine and extremity of an arch, is called the verfed fine of that arch. Thus, HD is the verfed fine of the arch ED, or of the angle ECD.

5. A ftraight line paffing 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 extremity of an arch, meeting the tangent drawn through the other extremity, is called the fecant of that arch. Thus, CG is the fecant 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 fupplement.

BK is the tangent, CK the fecant, 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 fecant com'plement of the arch ED. But for brevity's fake, they are called the co-fine, co-tangent, and co-fecant of the arch ED, or o f the angle E CD.

Cǝrǝl.

Corol. 2. Since the triangle CEH and GCD are fimilar, CH: CD (=CE):: CE: CG. Hence,

In words, The radius is a mean proportional between the co-fine and fecant of any arch.

Corol. 3. Because the triangles BKC, GCD are fimilar, GD: DC (=CB):: CB: BK. Hence,

In words, The radius is a mean proportional between the tangent and co-tangent of any arch.

Note, The leaft poffible fecant, the tangent of 45°, and the fine of 90°, are each of them equal to the radius.

In every triangle, there are fix things to be confidered, 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 alfo given, for it is found, by fubtracting 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 fubtracting the given angle from

180°.

When one angle of a triangle is a right-angle, the other two are acute, and are together equal to one right-angle, and confequently are complements of each other.

PROPO

IN any right angled plane triangle, if the hypothenfe be made radius, the legs become the fines of the oppofite angles : but if either of the legs be made radius, the other leg becomes the tangent of the oppofite angle, and the hypothenufe becomes the fecant of the fame angle. Fig. 51. plate 3.

LET ABC be a right angled triangle, if the hypothenufe BC be made radius, the fide AC will be the fine of the oppofite angle ABC; and if either fide, BA be made radius, the other leg AC will be the tangent of the oppofite angle ABC, and the hypothenufe BC, the fecant of the fame angle.

With the centre B, and radii BC, BA, deferibe two arches CD, EA, meeting BC, BA in E and D. Since CAB is a right angle, BC being radius, AC is the fine 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, fimilar arches of the fame circles will be in the fame proportion; therefore, the fines, tangents, and fecants of fimilar arches, that is, of equal angles, are as their radii; confequently, the tabular radius. is to the tabular fine, tangent or fecant 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 fame triangle.

And, becaufe any one of the three fides may be called the radius, any of the fides required, may be obtained by three analogies or varieties.

N. B. All the varieties which can occur in the folution of right angled triangles, may be comprehended under two problems.

Firft, When all the angles and one fide are given, to find the other two fides.

2d, When two fides and the right angle are given, to find two acute angles and the third fide.

We come now more fully to fhew how each of thefe problems are folved by logarithms.

PROBLEM I.

CASE I. The angles and one of the

legs given, to find the hypothe

nufe 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 confequently 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 arife the following proportions:

To find BC.

radius 90

is to AB 300

is to AB 300

2.47712

Sotang. ang. A 30°40′ 9.77303 So fec. A 30° 40′ 10.06543

Variety 2. making BC rad. BA becomes the tangent, and AC the fecant of the angle at C. Hence the following proportions:

Variety