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= + (± a) × (±c) + (±a) × (±d) + (±6) × (±c) + (± b) × (±d).

(ii). For division:

(±a+b)/(±c) =+ (±a) / (±c) + (±b) / (±c). A divisor consisting of two or more terms cannot be distributed over the dividend.

35. Laws of Exponents:

(i).

Involution: (am)×n=(a>n)×m ̧

(ii). Index law: Xam yan =α
= a±m±n

Also, as a consequence of these two:

(iii). Corollary: (ab)m = x/ am xy bm.

36. Laws of Logarithmic Operation:

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Zero may be regarded as the origin of additions, unity as the origin of multiplications.

38. Agenda: Involution and Logarithmic Operation in Arithmetic.

(1). Show that, if n be an integer, the index law (Art. 35) leads to the result:

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(2). Show, by the law of involution and the index law (Art. 35), that 8/3: = 2, 811/4= 3, etc.

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(3). Show, by the law of metathesis (Art. 36), that *log 325, 3log 7296, etc.

(4). Show, by the law of metathesis and the addition theorem (Art. 36), that

4log 8+log 22, log 21,

'log (1/8) + log (1/27)=-3.

(5). Find the logarithms: of 16 to base 22, of 125 to base 5 X 52, of 128 to modulus 1 /ln 8, of 1/81 to modulus 1 /ln 27.

CHAPTER II.

GONIOMETRIC AND HYPERBOLIC RATIOS.

IX. GONIOMETRIC RATIOS.

39. Definition of Arc-Ratio. In the accompanying figures N'N is a straight line fixed in position and direction, OP is supposed to have reached its position by turning about the fixed point O in the positive sense of rotation from the initial position ON. Any point

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on OP at a constant distance from O describes an arc AVQ, a linear magnitude. Let the ratio of this arc to the radius OQ, both taken positively, be denoted by 0, that is,

0=(length of arc AVQ)! (line-segment OQ).

The amount of turning of OQ, that is, the angle AOQ, fixes the value of this ratio; and since the arcs of concentric circles intercepted by common radii are proportional to those radii (Prop. 18), the ratio may be replaced by an arc CD provided only OC be taken equal to the linear unit. In

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 T 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 O 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:

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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 v and is (Prop. 17.)

πυ= 180 θ.

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

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(6). sin (0+ [2 n + }] π) = ± cos 0.

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(10). sin, or cos of (0 ± 2 n π) = sin, or cos of 0. (11). sin, or cos of (0 ± [2 n+1] π) == — (sin, or cos of 0).

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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