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Plane Trigonometry.

I.

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DEFINITIONS.

LANE Trigonometry is the Art whereby, having given any three Parts of a plane Triangle (except the three Angles) the reft

are determined. In order to which, it is not only requifite that the Peripheries of Circles, but alfo certain Right-lines in, and about, the Circle, be fuppofed divided into fome affigned Number of equal Parts.

2. The Periphery of every Circle is fuppofed to be 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, or fecond Minutes, &c.

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G 3. Any Part AB of the Periphery of the Circle is called an Arch, and is faid to be the Measure of the Angle ACB at the Center, which it fub

A

tends.

D

C

E

Note, The Degrees, Minutes, Seconds, &c. contained in any Arch, or Angle, are wrote in this Manner, 50° 18′ 35", which fignifies that the given Arch, or Angle, contains 52 Degrees, 18 Minutes, and 35 Seconds.

4. The Difference of any Arch from 90° (or a Quadrant) is call'd its Complement; and its Difference from 180° (or a Semicircle) its Supple

ment.

5. A Chord, or Subtenfe, is a Right-line drawn. from one Extremity of an Arch to the other: Thus the Right-line BE is the Chord, or Subtense, of the Arch BAE or BDE.

6. The Sine, or Right-fine, of an Arch, is a Right-line drawn from one Extremity of the Arch, perpendicular to the Diameter paffing thro' the other Extremity. Thus BF is the Sine of the Arch

AB or DB.

7. The Verfed Sine of an Arch is the Part of the Diameter intercepted between the Sine and the Periphery. Thus AF is the Verfed Sine of AB; and DF of DB.

8. The Co-fine of an Arch is the Part of the Diameter intercepted between the Center and Sine; and is equal to the Sine of the Complement of that Arch. Thus CF is the Co-fine of the Arch AB, and is equal to BI, the Sine of its Complement HB.

9. The Tangent of an Arch is a Right-line touching the Circle in one Extremity of that Arch, produced from thence till it meets a Right-line paffing thro' the Center and the other Extremity. Thus AG is the Tangent of the Arch AB.

10. The Secant of an Arch is a Right-line reaching, without the Circle, from the Center to the Extremity of the Tangent. Thus CG is the Secant of AB.

11. The Co-tangent, and Co-fecant, of an Arch are the Tangent, and Secant, of the Complement of that Arch. Thus HK and CK are the Cotangent and Co-fecant of AB.

12. A Trigonometrical-Canon is a Table exhibiting the Length of the Sine, Tangent, and Secant, to every Degree and Minute of the Quadrant, with refpect to the Radius; which is fuppofed Unity, and conceived to be divided into 10000, or more, Decimal Parts. By the Help of this Table, and the Doctrine of fimilar Triangles, the whole Bufinefs of Trigonometry is performed; which I fhall now proceed to fhew. But, first of all, it will be proper to obferve, that the Sine of any Arch Ab greater than 90°, is equal to the Sine of another Arch AB as much below 90°; and that, its Go-fine Cf, Tangent Ag, and Secant Cg, are alfo refpectively equal to the Co-fine, Tangent, and Secant of its Supplement AB; but only are negative, or fall on contrary Sides of the Points C and A, from whence they have their Origin: All which is manifeft from the Definitions.

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THEOREM I.

In any right-angled plane Triangle ABC, it will be as the Hypothenufe is to the Perpendicular, fo is the Radius (of the Table) to the Sine of the Angle at the Bafe.

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4.) 2. E. D.

Thus, if AC=,75, and BC=,45; then it will be, 75,45 I (Radius): the Sine of A=,6; which, in the Table, answers to 36° 52′, the Meafure, or Value of A.

THEOREM II.

In any right-angled plane Triangle ABC, it will be, as the Bafe AB is to the Perpendicular BC, fo is the Radius (of the Table) to the Tangent of the Angle at the Bafe.

For, let AE or AF be the Radius of the Table, or Canon (fee the preceding Figure), and FG the Tangent of the Angle A, or Arch EF (Vid. Def. 3. and 9.); then, by reason of the Similarity of the Triangles ABC, AFG, it will be, AB: BC:: AF : FG. 2. E. D.

Note, In the Quotations where you meet with two Numbers (as 5.4.) without any mention of Prop. Theor. &c. Reference is made to the Elements of Geometry published by the fame Author ; `to which this little Tract is defigned as an Appendix.

Thus

Thus let AB,8 and BC=,5; then we fhall have,8,5: 1 (Radius): Tangent A,625; whence A itself is found, by the Canon, to be 32° 00'.

THEOREM III.

In every plane Triangle ABC, it will be, as any one Side is to the Sine of its oppofite Angle, fo is any other Side to the Sine of its oppofite Angle.

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to the equal Radii A D AB and CF. Now

F

E

the Triangles CBD, CFE being fimilar, we have CB: BD (Sin. A) :: CF (AB) : F E (Sin. C). 2, E. D.

THEOREM IV.

As the Bafe of any plane Triangle ABC, is to the Sum of the two Sides, fo is the Difference of the Sides to twice the Distance DE of the Perpendicular from the Middle of the Bafe.

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