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is the absolute length of a degree. The circumference of a carriage wheel, the circumference of the earth, or the still greater and indefinite circumference of the heavens, has the same number of degrees; yet the same number of degrees in each and every circumference is the measure of precisely the same angle.

DEFINITIONS.

1. The Complement of an arc is 90° minus the arc. 2. The Supplement of an arc is 180° minus the arc. 3. The Sine of an angle, or of an arc, is a line drawn from one end of an arc, perpendicular to a diameter drawn through the other end. Thus, BF is the sine of the arc AB, and also of the arc BDE. BK is the sine

of the arc BD.

4. The Cosine of an arc is the perpendicular distance from the center of the circle to the sine of the arc; or, it is the same in magnitude as the sine of the complement of the arc. Thus, CF is the cosine of the arc AB; but CF= KB, which is the sine of BD.

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5. The Tangent of an arc is a line touching the circle in one extremity of the arc, and continued from thence, to meet a line drawn through the center and the other extremity. Thus, AH is the tangent to the arc AB, and DL is the tangent of the arc DB.

6. The Cotangent of an arc is the tangent of the complement of the arc. Thus, DL, which is the tangent of the arc DB, is the cotangent of the arc AB.

REMARK.-The co is but a contraction of the word complement.

7. The Secant of an arc is a line drawn from the center of the circle to the extremity of the tangent. Thus, CH is the secant of the arc AB, or of its supplement BDE.

8. The Cosecant of an arc is the secant of the complement. Thus, CL, the secant of BD, is the cosecant of AB.

9. The Versed Sine of an arc is the distance from the extremity of the arc to the foot of the sine. Thus, AF is the versed sine of the arc AB, and DK is the versed sine of the arc DB.

For the sake of brevity, these technical terms are contracted thus: for sine AB, we write sin. AB; for cosine AB, we write cos. AB; for tangent AB, we write tan. AB, etc.

From the preceding definitions we deduce the following obvious consequences:

1st. That when the arc AB becomes insensibly small, or zero, its sine, tangent, and versed sine are also nothing, and its secant and cosine are each equal to radius.

2d. The sine and versed sine of a quadrant are each equal to the radius; its cosine is zero, and its secant and tangent are infinite.

3d. The chord of an arc is twice the sine of one half the arc. Thus, the chord, BG, is double the sine, BF. 4th. The versed sine is equal to the difference between the radius and the cosine.

5th. The sine and cosine of any arc form the two sides of a right-angled triangle, which has a radius for its hypotenuse. Thus, CF and FB are the two sides of the right-angled triangle, CFB.

Also, the radius and tangent always form the two sides of a right-angled triangle, which has the secant of the arc for its hypotenuse. This we observe from the right-angled triangle, CAH.

To express these relations analytically, we write

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From the two equiangular triangles CFB, CAH, we

have

CF: FB CA AH.
= •

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cos. : R

=

CF CB CA: CH.

R: sec.; whence, cos. sec. = R. (4)

The two equiangular triangles, CAH and CDL, give

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cos. : sin. = cot.: R; whence, cos. R From equations (4) and (5), we have

cos. sec. = tan. cot.

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Or,

cos. tan. = cot. : sec.

R-cos.

(8)

We also have ver. sin. =

The ratios between the various trigonometrical lines are always the same for arcs of the same number of degrees, whatever be the length of the radius; and we may, therefore, assume radius of any length to suit our convenience. The preceding equations will be more concise, and more readily applied, by making the radius equal unity. This supposition being made, we have, for equations 1 to 6, inclusive,

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equal parts by the diameters, AD and EH, the one h

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The center of the circle is taken as the origin of distances, or the zero point, and the different directions in which distances are estimated from this point are indicated by the signs + and If those from C to the right be marked +, those from C to the left must be marked ; and if distances from Cupwards be considered plus, those from С downwards must be considered minus.

-.

If one extremity of a varying are be constantly at A, and the other extremity fall successively in each of the several quadrants, we may readily determine, by the above rule, the algebraic signs of the sines and cosines of all ares from 0° to 360°. Now, since all other trigonometrical lines can be expressed in terms of the sine and cosine, it follows that the algebraic signs of all the circular functions result from those of the sine and cosine.

We shall thus find for arcs terminating in the

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The chord of 60° and the tangent of 45° are each equal to radius; the sine of 30°, the versed sine of 60°, and the conine of 60° are each equal to one half the radius.

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the circle, (Prob. 28, B. IV), and as the arc subtended by each side of the hexagon contains 60°, we have the chord of 60° equal to the radius.

2d. The triangle CAH is right-angled at A, and the angle Cis equal to 45°, being measured by the arc AD; hence the angle at H is also equal to 45°, and the triangle is isosceles. Therefore AH = CA = radius of the circle.

=

3d. The triangle ABC is isosceles, and Bn is a perpendicular from the vertex upon the base; hence An nC= Bm. But Bm is the sine of the arc BE, Cn is the cosine of the arc AB, and An is the versed sine of the same arc, and each is equal to one half the radius. Hence the proposition; the chord of 60°, etc.

PROPOSITION II.

Given, the sine and the cosine of two arcs, to find the sine and the cosine of the sum and of the difference of the same ares expressed by the sines and cosines of the separate arcs. Let G be the center of the

circle, CD the greater arc, and DF the less, and denote these arcs by a and b respectively.

Draw the radius GD; make the arc DE equal to the arc DF, and draw the chord EF. From F and E, the extremities, and I, the middle point

M

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