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a sin 0LQ, perpendicular distance of Q from A'A. OL, distance from centre to foot of LQ.

a cos e

a tan 6

a cot

a sec

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=

=

AT, distance along a tangent from A to OP. = BS, distance along a tangent from B to OP.

= OM, intercept of tangent at Q upon OA. 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 a 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.*

P

43. To Prove Limit [(sin )/0]=1, when 0. Let the arc-ratio of the angle POQ in Fig. 23, draw PQP', an arc with radius OP, draw PT and P'T' tangents to the arc at P and P', join P, P', and O, T. Then assuming that an arc is greater than the subtending chord and less than the enveloping tangents at its extremities, we have

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S

Fig. 23.

T

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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 0 =o, cos 0 = 1; therefore by making 0 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

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44. 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,, P1P21 P,P,, etc., and draw P,M perpendicular to OA. The area of each of the triangles OAP, OP,P2, OP.P3, . . . is a MP1,

Pt P

P

A

M

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a asin (AP,|OA) = ↓ a2 sin(0 /n), and hence the area of the entire polygon OAP, P2... Q is

Fig. 24.

na2

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2

n

2

Ө a2 0 sin (0/n)

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Now when the number of points of division P1,P2,P3, .. is indefinitely increased, the polygon OAP,P,... Q approaches coincidence with the circular sector OAP1P2. . . Q, that is,

a2 0 sin (0/n)

2

area of sector OAP,P2....... Q, when n∞;

0 / n but at the same time

sin (0/n)

0/no and

= 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:

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X. HYPERBOLIC RATIOS.

46. Definitions of the Hyperbolic Ratios. An important class of exponentials, which because of their relation to the equilateral hyperbola are called the hyperbolic sine, cosine, tangent, cotangent, secant and cosecant, and are symbolized by the abbreviations sinh, cosh, tanh, coth, sech, csch, are defined by the following identities:

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47. Agenda. Properties of Hyperbolic Ratios. Prove the following:

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48.

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Geometrical Construction for Hyperbolic Ratios. For the representation of the hyperbolic ratios. the equilateral hyperbola is employed. Its equation in Cartesian co-ordinates is

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Let OX, OY be its axes, OJ an asymptote, P any point on the curve, x and y its co-ordinates, AQB the quadrant of

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a circle with centre at the origin and radius a, NQ a tangent to the circle from the foot of the ordinate y, RS a tangent to the hyperbola parallel to the chord PA, a, ß the co-ordinates of Q, the arc-ratio of the angle AOQ.

It is obvious from its definition that cosh u has I for its smallest and ∞ for its largest value corresponding to u=o and ∞ respectively, and if the variations of x / a be confined to the right hand branch of the hyperbola its range of values is likewise between 1 and +∞; hence we may

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