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And thus we have gone thro' all Geometry that is neceffary for our prefent Bufinefs, both as to Theory and Practice. The next thing we go on, is the Principles of Plain Trigonometry.

SE C T. II.

Of Plain TRIGONOMETRY, Right and Oblique Angled.

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LAIN TRIGONOMETRY is that Science by which we measure the Sides and Angles of plain Triangles.

2. Since Triangles are either right or oblique angled; therefore Trigonometry is commonly divided into two kinds, viz. Rectangular and Oblique-angular : and firft we fhall treat of Rectangular.

3. In any right angled Triangle as ABC, if the Hypothenufe be made the Radius, and with that a Circle be described on the one end A as a Center; then 'tis plain that BC will be the Sine of the Angle BAC (by Art. 21. of Sect. I.); and if with the fame distance,

G 2

B

distance, and on B as a Center, a Circle be defcribed, 'tis plain that AC will be the Sine of the Angle ABC; therefore, in general, if the Hypothenufe of a right angled Triangle be made the Radius, the twoLegs will be the Sines of their oppofite Angles. 4. If in a right angled Triangle DEF, one of the Legs, as DF, be made the Radius, and on the Extremity D (at one of the oblique Angles, viz. that which is form'd by the Hypothenuse and the Leg made Radius) as a Center, a Circle be defcribed; 'tis plain, that the other Leg EF will be the Tangent of the Angle at D, and the Hypothenuse D E will be the Secant of the fame Angle (by Art. 24, 25, and 67 of Sect. 1.). The fame way, making the Leg E F the Radius, and on the Center E defcribing a Circle, the other Leg GF will become the Tangent of the Angle at E, and the Hypothenufe DE the Secant of the fame.

E

F

5. It has been already fhewn, at Art. 72. of Sect. 1. that the Chord, Sine, Tangent, &c. of any Arch, or Angle, in one Circle, is proportionable to the Chord, Sine, Tangent, &c. of the fame Arch in any other Circle; from which, and the two foregoing Articles the Solutions of the feveral Cafes of rectangular Trigonometry naturally follows.

6. Since Trigonometry confifts in determining Angles and Sides from others given, there arifes various Cafes, which are feven in Rectangular and fix in Oblique-angular Trigonometry.

We fhall now proceed to the Solution of the fe ven Cafes of Rectangular Trigonometry.

CASE I.

The Angles and one of the Legs given, to find the other Leg.

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Example. In the Triangle ABC rightangled at B, fuppofe the Leg AB, 86 equal parts, (as Feet, Yards, Miles, &c.) and the Angle A 33°, 40' requir'd the other Leg BC in the fame parts with A B.

Geometrically.

Draw A B equal to 86, from any Line of equal parts, then (by Prob. 4. of Sect. 1.) upon the point

C

B⋅

and applying it to the AB was taken from.

B, erect the Perpendicular

BC; laftly, from the point

A draw the line AC, making with AB an Angle equal to 33°, 40', and that line produc'd will meet BC in C, and fo conftitute the Triangle. The length of BC may be found by taking it in your Compaffes, fame line of equal parts that

By Calculation.

Firft by making the Hypothenufe AC Radius, the other two Legs will be the Sines of their oppofite Angles (by Art. 3. of this) viz. AB the Sine of C, and CB the Sine of A; now fince (by Art. 72. of Sect. 1.) the Sine, Tangent, &c. of any Arch in

one

one Circle is proportionable to the Sine, Tangent, &c. of the fame Arch in any other Circle, 'tis plain the Sines of the Angles A and C in the Circle defcribed by the Radius A C, must be proportional to the Sine of the fame Arches or Angles, in the Circle, that the fecond Table at the end of this Book was calculated for; fo the proportion for finding BC will be

S, C: AB:: S, A: BC.

i. e. As the Sine of the Angle C in the Tables, is to the length of AB (or Sine of C in the Circle whofe Radius is AC) fo is the Sine of the Angle A in the Tables, to the length of BC (or Sine of the fame Angle in the Circle whofe Radius is A C).

Now the Angle A being 33°, 40', the Angle C must be 56°, 20' (by Art. 51. Cor. 2. Sect. 1.); therefore looking in the fecond Table at the end of this Book for the Sines of the two Angles, and in the firft for the Logarithm of 86 the given Leg, we fhall find by proceeding according to the foregoing proportion, that the required Leg BC, is 57.28; and the Operation will ftand as follows.

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2dly, Making AB the Radius, 'tis plain BC, the Leg required, will be the Tangent of the given Angle A (by the 4th of this), and fo the proportion for finding BC, when AB is made the Radius, will be,

R:T, A:AB: BC

i. e. as the Radius in the Tables, is to the Tangent of the Angle A in the fame, fo is the length of BA,

or Radius in the Scheme, to the length of BC or Tangent of A in the Scheme; therefore looking in the Tables for the parts given in the foregoing proportion, and proceeding with them according to that Rule, we shall find BC to be 57.28 as before, and the Operation will be as follows:

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Lastly, by making BC, the Leg requir'd, the Radius, 'tis plain that AB will be the Tangent of C, and the proportion for finding BC will be as follows:

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The Angles and one of the Legs given, to find the Hypotbenufe.

Example. In the Triangle ABC, suppose A B 124, and the Angle A 34, 20'; confequently the Angle C 55°, 40 requir'd the Hypothenuse AC, in the fame parts with AB,

Geometrically.

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