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In order to this we must remark that when straight lines are submitted to calculation, all those which are concerned in the same inquiry must be measured in reference to one common standard of measure, called the linear unit; the choice of which unit is, however, arbitrary. Thus if we estimate any one of the lines concerned in any inquiry in feet, all must be estimated in feet, and the linear unit adopted will be a foot, which is represented by the numeral unit 1. Also if one of the lines is measured in yards all must be measured in yards, the linear unit being then a yard, which, as before, is represented in the calculation by the numeral unit 1. As far as the accurate representation of the lines are concerned, it is obviously a matter of indifference what length be assumed for the linear unit, for the length of any line will always be expressed numerically by that number which denotes the units it contains, but, for the purpose of facilitating computation, some scales of measure are often preferable to others.

B'

(3) Let now BAC be any angle concerned in any inquiry: then having chosen the linear unit AB, describe the circumference BCD about the centre A. The arc BC may be taken for the measure or representative of the angular magnitude CAB: for let there be any other angle B'AC' about the same centre A; then we know, by Geometry, that the angle BAC is to the angle B'AC' as the intercepted arc BC to the intercepted

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arc B'C', (Geometry, p. 102); hence, as the intercepted arcs always vary as the angles, the former may, obviously, be taken to represent the latter.

It is usual to consider the circumference of every circle to consist of 360 equal parts, called degrees of that circle; an arc consisting of any number of these, 24 for instance, is called an arc of 24 degrees, and represented for brevity thus, 24°; moreover each degree is supposed to consist of 60 equal parts, called minutes, and each minute of 60 equal parts called seconds. To express any number of minutes, we mark one accent over the number, and to express seconds we mark two; thus, 24° 16′ 26", is 24 degrees 16 minutes 26 seconds. circular arcs applies equally to the angles which they

What we say of measure, so that

we call that an angle of 20° whose sides include an arc of 20° or the eighteenth part of a whole circumference.

Let us now speak of the trigonometrical lines before adverted to, and which are introduced for the purpose of reducing the entire theory of angular magnitude to the investigation of linear quantities only; we must, first, however, mention one or two further particulars respecting the arcs to which these lines refer.

(4.) The arc CD which must be added to BC to make up a quadrant, or 90° is called the complement of the arc BC; and every arc will have a complement, even those which are themselves greater than 90°, provided we consider the arcs measured in the direction BCD &c. as positive, and those measured in the opposite direction as negative; thus the complement CD of the arc BC commences at C where BC terminates, and may be considered as generated by the motion of C, the extremity of the radius AC, in the direction CD; but the complement C,D of the arc BC1, commencing in like manner at the extremity C, of the proposed arc, must be generated by the motion of C, in the opposite direction, and the angular magnitude BAC1, will here be diminished by the motion of AC,, in generating the complement; the complement of BAC1, or of the arc BC,, is, therefore, with propriety considered as negative. Calling the arc BC, or BC1, w, the complement will be 90° thus the complement of 24° 16′ 4′′′ is 65° 43′ 56′′, and the complement of 120° 36′ 10′′ is - - 30° 36′ 10′′. The arc CB1, which must be added to BC to make up a semicircle, or 180°, is called the supplement of the arc BC. If the arc is greater than 180°, as the arc BC, its supplement C, B, measured in the reverse direction is negative. The expression for the supplement of any arc or angle w is, therefore, 180°-w; thus the supplement of 110° 30' 20' is 69° 29′ 40′′, and the supplement of 200° 25′ is -20° 25'.

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In the same manner as the complementary and supplementary arcs are considered as positive or negative, according to the direction in which they are measured, so are the arcs themselves positive or negative; thus, still taking B for the commencement of the arcs, as BC is positive BC, will be negative. In the doctrine of triangles we consider only positive angles or arcs, and the magnitudes of these are comprised

between 0 and w=180°; but in the general theory of angular quantity, we consider both positive and negative angles, according as they are situated above or below the fixed line AB from which they are measured, as the angles CAB, C, AB; moreover, an angle may consist of any number of degrees whatever, thus if the revolving line AC set out from the fixed line AB and make n revolutions, and a part the angular magnitude generated is measured by n times 360°, plus the degrees in the additional part.

Of the Sine.

(5.) The sine of an arc or of the angle which it measures, is the perpendicular, from one extremity of the arc, upon the diameter passing through the other extremity: thus CS is the sine

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B S

of the arc BC; C1 S1 is the sine of the arc BC1; C2 S, is the sine of the arc BC2; C, S, the sine of the arc BC, &c. If the proposed arc were a quadrant, or 90°, the sine DA would be equal to the radius, and, therefore, its numerical value would be 1; the same would be the case if the arc consisted of 3 quadrants, or 270°, or indeed of any odd number of quadrants; for all other arcs the numerical value of the sine will be a proper fraction or decimal. These, it must be observed, are the trigonometrical values of the sines, which are estimated according to the scale AB=1; but it should be remarked that when we know the value of the sine of an arc agreeably to this scale, its value agreeably to any other scale is at once obtained by proportion; thus let R be any value assumed for the radius, and let us write the sine corresponding in capitals, SINE; then 1 sine:: R: SINE=RX sine, so that the sine of an arc, corresponding to any assumed radius, is found by multiplying its trigonometrical sine by that radius; and, on the contrary, the sine according to any value of the radius being known, the trigonometrical sine is found by dividing it by that radius; the number, in fact, which expresses the trigonometrical sine being the ratio of the geometrical line itself to the

-radius, whatever this may be. What we have said of the sine will be easily seen to apply to the other trigonometrical lines. As with the arcs so with the sines, those which lie in opposite directions take opposite signs, those above the fixed line B, B being regarded as positive, and those below as negative, so that the sines in the first and second quadrants are positive, those in the third and fourth negative, while in the fifth and sixth they are again positive, and so on.

Every arc or angle has the same sine as its supplement; thus if B, C, is equal to BC it is obvious that BC, will be the supplement of BC, and the sine CS of the latter must be equal to the sine C, S, of the former.

Of the Cosine.

B

C

S

B

(6.) The cosine of an arc or angle is the sine of its complement: thus the cosine of the arc BC is the line Cs, which is, obviously, the sine of the arc DC, the complement of BC. As the several sines are arranged on opposite sides of the diameter B, B, so the cosines are arranged on opposite sides of the diameter DD1; those on the right of DD, being regarded as positive, and those opposite as negative; hence in the first quadrant, the cosines are positive, in the second negative, in the third negative, in the fourth positive, and so on; the cosine of an arc is equal to the cosine of its supplement, but has a different sign.

When the arc is 0 the sine is 0, but the cosine BA is 1; when the arc is 90, the sine DA is 1, but the cosine is 0; when the arc is 180° the sine is 0, but the cosine is B, A = 1; when the arc is 270° the sine D, A is 1, but the cosine is 0; and when the arc is 360° the sine is 0, and the cosine 1, as at first, and so on.

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It is plain that the cosine of an arc is always equal to that part of the radius which is intercepted between the sine of that arc and the centre. Thus referring to the figure in (5) AS is equal to the cosine of BC, and AS, to the cosine of BC,C,, or of BC,C,.

Of the Tangent, Cotangent, Secant, and Cosecant.

(7.) The tangent of an arc, and, therefore, of the angle which it measures, is a line drawn from one extremity of the arc, touching it at that extremity, and terminating in the diameter produced, drawn through the other extremity: thus BT is the tangent of the arc BC.

B

The cotangent is the tangent of the complement: thus Dt is the cotangent of the arc BC. It is easy to trace the changes which these two to lines undergo as the arc BC increases from 0, for which value the tangent is obviously O, and the cotangent infinite. Observing the same rules here as for the sine and cosine, we see that in the first quadrant the tangent and cotangent are both positive, in the second the tangent BT, and cotangent Dt, re both negative; in the third the tangent BT, and cotangent Dt, are both positive; and in the fourth the tangent BT, and cotangent Dts are both negative, and so on; but as we shall soon see, the signs of the tangent and cotangent may always be at once inferred from those of the sine and cosine.

The secant of an arc is that portion of the prolonged diameter, limitting the tangent, which is included between the centre and tangent; and the cosecant is the secant of the complement. Thus in the last

figure AT is the secant of the arc BC, and At the cosecant.

In the four trigonometrical lines, sine, cosine, tangent, and cotangent, we have seen that each is posited in one or other of two directly opposite directions, and that, therefore, one or other of the opposite signs + and, prefixed to the numerical value of any such line, served to point out the proper direction for any particular value of the arc or angle. But as the secant and cosecant continually vary in direction, as well as in magnitude with the arc or angle, the geometrical position of either of these lines does not so clearly indicate to us the sign with which it should be represented. The proper sign, however, is always readily ascertained from knowing the signs of the sine and cosine, for upon these two lines all the others depend, as we shall shortly show.

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