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SECTION I.

PLANE TRIGONOMETRY.

THE object of Plane Trigonometry is, when of the sides and angles of a plane triangle we have enough given to limit it, to determine the parts which are not given.

As every oblique angled triangle may be divided into two right angled ones, it is found most expedient to commence the subject by examining the relations and properties of triangles of the latter kind. The terms of the science are therefore adapted chiefly to right angled triangles.

Definitions.

G

H K

L

I

B

M

C

N

E

F

ARTICLE 21. Definition 1. An arc of a circle is any part of the circumference, usually taken less than the whole. As AB, or BHD.

2. The chord of an arc is a right line drawn from one end of the arc to the other. Thus, BE is the chord of the arc BAE, or BDE.

3. The sine of an arc is a straight line drawn from one exP tremity of the arc, at right angles to the diameter, which passes through the other extremity. Thus, AD being a diameter to the circle, the line BF, at right angles to it, is the sine, or right sine, of the arc AB or DHB. 4. The tangent of an arc is the right line which touches

the circle at one extremity of the arc, and extends till it meets another right line, which is drawn from the centre through the other extremity. Thus AG, which touches the circle at A, is the tangent of AB,

5. The secant of an arc is the right line intercepted between the centre of the circle and the extremity of the tangent. Thus CG is the secant of the arc AB.

6. The versed sine of an arc is the part of the diameter intercepted between one end of the arc, and the sine which passes through the other end. Thus AF is the versed sine of AB, and DF is the versed sine of DHB.

7. The part by which an arc differs, in excess or defect, from a quadrant, or fourth part of the circumference, is called its complement. Thus, the arc ABH being a quadrant, HB is the complement of AB or of DHB.

8. The cosine, cotangent or cosecant of an arc, is the sine, tangent or secant of the complement of that arc. Thus BI, HK and CK, the sine, tangent and secant of HB, are termed the cosine, cotangent and cosecant of AB,

9. What an arc lacks of a semicircle, is called its supplement. Thus BHD is the supplement of AB.

10. The circumference of every circle is supposed to be divided into 360 equal parts, called degrees; each degree into 60 equal parts, called minutes; each minute into 60 cqual parts, called seconds, &c, Degrees, minutes and seconds are designated thus, °, ', ". "

11. As angles at the centre of a circle have to each other the same ratio as the arcs on which they stand (33.6); the latter are usually termed the measures of the former. Hence an angle at the centre of a circle is said to contain as many degrees, minutes and seconds, as the arc which subtends it. The sine, tangent, &c., of an arc, is also called the sine, tangent, &c. of the angle which is measured by the arc. Thus BF, the sine of AB, is called the sine of the angle ACB.

General Properties and Relations of Arcs, Sines, Tangents, &c.

ART. 22. If the arcs AH and DH are quadrants, and therefore equal, it follows (33.6) that the angles ACH and DCH are equal, and therefore are right angles. Hence the angle at the centre of a circle, subtended by a quadrant, is always a right angle.

ART. 23. The lines AG and HK, which touch the circle at A and H, are respectively at right angles to CA and CH (18.3); hence CAG and KHC are right angled triangles. Now the lines AG and CH, being at right angles to AC, are parallel to each other (28.1); consequently, the alternate angles HCK and CGA are equal; wherefore the triangles CHK and GAC are similar. The triangle CFB is also evidently similar to CAG; and CIB to CHK; therefore those four triangles are similar to each other. Also, the figure CFBI being a parallelogram, CI FB, and CF BI. From these triangles we have of course the following analogies:

=

=

1. As CF: FB:: CA: AG, or as cosine: sine :: radius: tang. 2. As CF: CB:: CA: CG, or as cosine: rad. : : radius: sec't. 3. As CI : CB::CH: CK, or as sine: radius:: radius : cosec. 4. As AG: CA::CH: HK, or as tang. : radius:: radius: cotan. 5. As CG: AG:: CB: BF, or as secant : tang. :: radius: sine.

In the algebraic formulæ used to express the relations of sines, tangents, &c., it is most convenient to assume the radius = 1. Making, therefore, this assumption, we may convert the foregoing analogies into the following equations.

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Hence, taking P and Q any arcs, tan P. cotan P = tan Q. cotan Q; consequently (16.6),

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ART. 24. It is sometimes necessary to attend to the algebraic signs of these quantities, particularly when they are reduced to general formulæ.

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An arc estimated in one direction is considered as positive, and in the opposite direction as negative. The same may be said of the sines, tangents, &c. Thus the arc AB, its sine FB, cosine CF, tangent AG, and secant CG, are considered as positive; but, when estimated in the opposite direction, they are considered as negative. Now, we readily perceive that when

the arc is less than a quadrant, as AB is, the sine, tangent, &c., are all positive. But if we take the arc more than a quadrant, but less than a semicircle, as AL, the sine LM is still positive, but the cosine CM is negative, being measured from C in a direction opposite to CF. The tangent AP and secant CP are also negative; the former being drawn in a direction opposite to AG, and the latter not produced from C through L, the extremity of the arc, but in the oppo

site direction. If we take the arc more than a semicircle, but less than three quadrants, as AHDN; the sine MN becomes negative, the cosine CM also negative, the tangent AG positive; but the secant CG, not being produced through N, but in the opposite direction, is negative. If we take the arc more than three quadrants, but less than four, as AHDE; the sine EF is still negative, but the cosine CF and the secant CP are positive; the secant being produced from the centre, through the extremity of the arc, till it meets the tangent; but the tangent AP is negative.

These signs, when prefixed to the several quantities in the preceding equations, are found to be conformable to the algebraic rules for the adaptation of signs. In the first quadrant,

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ART. 25. It is easily perceived that the sine, tangent, &c., of a given arc are limited, being dependent upon the length of the arc; but the sine, tangent, &c., of an angle, being the sine, tangent, &c., of the measuring arc, whatever may be the radius with which that arc is described, evidently admit various values. Thus EC, HI, MN, which are the sines of

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