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The co-tangent and co-sine of any arc, may be had by the same method ; the complement of any degree, being only its residue from 90, or a quadrant, as before observed (by theo. 24. part 3 and 4.)

3. To find a tangent by the help of a co-tangent only.

From twice the radius, which is 20.00000, take the co-tangent, the remainder is the tangent (by theo. 24. part 5.)

EXAMPLE,

Required the tangent of 29o. 50' being defaced, as also the sine and co-sine defaced, by the co-tangent only.

From twice the radius,
Take the co-tangent of 29o. 50'

20.00000 10.24148

The rem. is the tang. of 29o. 50' req.

9.75852

4. To find the secant by the help of a co-sine; which may be found of great use when a table of sines and tangents can only be had.

From twice the radius, which is 20.00000, take the Co-sine, and the remainder will be the secant, (by theo. 24. part 6.)

EXAMPLE.

Required, the secant of 570. 20' by the help of

the co-sine only

From the double radius,
Take the co-sine of 57° 20'

20.00000 9.73219

The rem. is the secant of 57°. 20' req. 10.26781

5. To find a secant by the help of the sine and tangent.

From the tangent added to radius, take the sine, the remainder will be the secant (by theo. 24. part 7.)

EXAMPLE.

Required, the secant of 57°. 20' by help of the sine and tangent.

From the tan. of 570, 20'+10.00000 the radius,

Take the sine of 57° 20'

20.19303 9.92522

The rem. is the secant of 57o. 20' req. 10.26781.

The secants in these tables might have been onitted, because all proportions in which they are concerned, may be wrought by sines and tangents only, as shall be shewn in the several cases of plane trigonometry; and are here only inserted that all the various

SECTII.

PLANE TRIGONOMETRY.

DEFINITIONS.

Plane TRIGONOMETRY is the art, whereby, having given any three parts of a plane triangle, except the three angles, the rest are determined. For this purpose it is requisite that not only the periphery of circles, but also certain right lines, in, and about circles, be supposed to be divided into some assigned number of equal parts.

2. The periphery of every circle is supposed 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 ; &c.

3. Any part of the periphery, is calledlan arch; and is the measure of the angle at the centre, which it subtends.

4. The difference of any arch from 90°, or a quadrant, is called the complement of that arch, and its difference from 180°, or a semi-circle, is called its supplement.

5. A Chord, or Subtense of an arch, is a right line joining its extremities.

6. The Sine of an arch, is a right line drawn from one extremity of that arch, perpendicularly to a diameter passing through the other extremity.

7. The Versed-Sine of an arch is that part of the diameter, intercepted between the sine and the periphery.

8. The Co-Sine of an arch is that part of the diameter, between the sine and the centre : and is equal to the sine of the complement.

9. The Tangent of an arch is a right line, touching the arch in one extremity, and produced from thence till it meets a right line drawn from the centre, through the other extremity.

10. The Secant of an arch is that line drawn from the centre to the extremity of the tangent.

11. The Co-Tangent, and Co-Secant, are the tangent, and secant of the complement of the arch.

12. Every sine, co-sine, tangent, co-tangent, &c, corresponds to two arches, which are supplements to each other : the one being as much greater than 90°, as the other is less. Thus the sine, &c. of 1°, is also the sine of 179°; the sine of 10°, is the sine, &c. of 170°; and the sine, &c. of 60° is also that of 120°,

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Thus, let BH, or AH, be any arch; then is

HL, its sine,
LB, or AL, its versed sine,
CL, =EH, its co-sine,
BK, its tangent,
CK, its secant,
DI, its co-tangent, &c.
CI, its co-secant.

A Table of Logarithm sines, tangents, secants, &c. is the Logarithms of the lengths of those lines, calculated to a given radius, for every degree, and minute of the quadrant; by the help of which, and the following Theorems, any three parts of a triangle, except the three angles, being given, the rest may be found.

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