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the geometrical figures a description of the angle will be sufficient to identify the ratio itself. This magnitude ✪ will be called the arc-ratio of the angle AOQ. The letter stands for ratio of a semi-circumference to its radius, that is, the arc-ratio of 180°.

Lines drawn parallel or perpendicular to N'N, shall be regarded as positive when laid off from 0 to the right or upwards, negative when extending to the left or downwards. OP drawn outwards from O is to be considered positive in all cases.

40. Definitions of the Goniometric Ratios. LQ in the above figures being drawn perpendicular to NʼN, upwards or downwards according as Q is above or below N'N and correspondingly positive or negative, the goniometric ratios, called sine, cosine, tangent, cotangent, secant, cosecant, are defined as functions of the arc-ratio by the following identities:

sin 0=LQ/OQ, cos 0=OLOQ,
tan 0LQ/OL, cot OL LQ,
sec=0Q/OL, csc 0=0Q/LQ.

It must be borne in mind that is here not an angle expressed in degrees, but a ratio, which can therefore be represented by a linear magnitude. In elementary trigonometry sin usually means "sine of angle AOQ in degrees"; here it may be read "sine of magnitude 0," where 0 =arc AVQ/ radius OA.* If v be the number of degrees in the angle AOQ, the relation between ʊ and ✪ is

πυ= 180 θ,

*See Lock's Elementary Trigonometry, p. 31.

(3)

(Prop. 17.)

41. Agenda. Properties of Goniometric Ratios. Prove the following:

(i). sin

= = 0, COS O=1.

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:

1

(6). sin ([2 n + 1] π) = ± cos 0. (7). cos ([2n+])=sin 0. (8). sin (0+ [2 n + }] π) = cos 0. (9). cos (@[2 n + 1] π)=sin 0. (10). sin, or cos of (± 2 n ) = sin, or cos of 0. (II). sin, or cos of (0 ± [2 n +1]π) == — (sin, or cos of ◊).

42. Line-Representatives of Goniometric Ratios. If in the foregoing definitions the denominators OL, LQ be replaced by the radius OQ; the numerators of the six goniometric ratios will be six straight lines drawn, either from the centre, or from Q, or from one of the fixed points A, B on the circumference a quadrant's distance apart.

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If a be the radius of the circle, they may be indicated

as follows:

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a sin

a cos 0

a tan 6

OL,

AT,

LQ, perpendicular distance of Q from A’A. distance from centre to foot of LQ. distance along a tangent from A to OP. = BS, distance along a tangent from B to OP. a sec = OM, intercept of tangent at Q upon OA. 0

a cot

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a csc = ON, intercept of tangent at Q upon OB.

These constructions are evidently only variations in the statement of the definitions of the goniometric ratios. When = = 1, the six ratios have as their geometric representatives these lines themselves.

Formerly they were defined as such for all values of the radius and were therefore not ratios, but straight lines dependent for their lengths upon the arc AQ, that is upon both the angle AOQ and the radius of the circle. The older form of definition is now rare.*

43. To Prove Limit [(sin 0)/0]= 1, when 00. Let the arc-ratio of the angle POQ in Fig. 23, draw

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* See Todhunter's Plane Trigonometry, p. 49, and the reference there given: Peacock's Algebra, Vol. II, p. 157. See also Buckingham's Differential and Integral Calculus, 3d ed., p. 139, where the older definitions are still retained.

When = o, cos 0:

= 1; therefore by making smaller than any previously assigned arbitrarily small magnitude, (sin 0) / 0 is made to differ from unity by a like arbitrarily small magnitude. Under these conditions (sin 0) / is said to have I as its limit, and the fact is expressed by the formula

limit (sine}

0=0

=

in which stands for 'approaches.'

44.

I,

Area of Circular Sector. Let the sector OAQ be divided into n equal smaller sectors by radii to the points P1, P2, P3, etc., which set off the arc AQ into the same number of equal parts AP,, PIP21

P P

M

P

2

P1P3, etc., and draw PM perpendicular to OA. The area of each of the triangles OAP1, OP,P2, OP¿P3, A... is a MP1,

or if (arc AQ) | OA=0, it is

a asin (AP/OA) = † a2 sin(✪ | n), and hence the area of the entire polygon OAP, P... Q is

Fig. 24.

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2 0 / n

Now when the number of points of division P,,P1⁄2‚P3, . . . is indefinitely increased, the polygon OAP,P,... Q approaches coincidence with the circular sector OAP,P2. . . Q, that is,

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area of sector OAP,P2. Q, when n∞;

but at the same time

0 no and

sin (0/n)
Ꮎ ! 22

= 1,

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and from the area of the circular sector, by quantities that are less than any previously assigned arbitrarily small magnitude. Under these circumstances it is assumed as axiomatic that the two limits which the varying quantity approaches cannot differ, and that therefore

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The limits in fact could not be different unless the area of the sector were susceptible of two distinct values, which is manifestly impossible.

45. Agenda. The Addition Theorem for Goniometric Ratios.

From the foregoing definitions of the goniometric ratios prove for all real arc-ratios the following formula:

(1). sin (a± ẞ) = sin a cos ẞcos a sin ẞ.
(2). cos (aẞ) =cosa cos ẞ sin a sin ß.
tan atan ß
I tan a tan ẞ

(3). tan (a+B)=

(4). sin 2 a = 2 sin a cos a.
COS 2 α== Cos2 a sin2 a.

(5).

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(6). I + cos 2 α = 2 COS2 a.
(7). I - COS 2 α = 2 sin2 c.

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