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(sin A — sin B), that is, to the rectangle of the sum and dif. ference of the sines; it follows, that the first of these equations converted into an analogy, becomes sin (A–B) : sin A-sin B :: sin B + sin B : sin (A+B) (XX.) That is to say, the sine of the difference of any two arcs or angles, is to the difference of their sines, as the sum of those sines is to the sine of their sum. If A and B be to each other as n + 1 to n, then the preceding proportion will be converted into sin A: sin (n + 1) Asin na :: sin (n + 1) A + sin na : sin (2n+1) A. . . . (XXI.) These two proportions are highly useful in computing a table of sines; as will be shown in the practical examples at the end of this chapter.

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factors of the first member of this equation, are tank(A'—B') r - - —and HIV.F. respectively; so that the equation

tank(A'—b') sin A'-sin B' tani (V-En') T sin Voism p' ' ' ' ' (XXII.) This equation is readily converted into a very useful proportion, viz. The sum of the sines of two arcs or angles, is to their difference, as the tangent of half the sum of those arcs or angles, is to the tangent of half their difference.

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manifestly becomes

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- cos b-cos A' , tan (A–B)' Making H = 0, in one or other of these expressions, there results,

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These theorems will find their application in some of the investigations of spherical trigonometry.

27. Once more, dividing the expression for sin(A + h) by that for cos (A + B), there results

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cos(a+b) T cos A. cos s r. sin, a sin a . then dividing both numerator and denominator of the second

- - si tan fraction, by cos A. cos B, and recollecting that * = +, we

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secon Tian n Tsec A-stan A :

and so, for many other analogies. We might investigate also some of the usual formulae of

verification in the construction of tables, such as

deduce the expression

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34. The last equation, however, only shows the sums of sines and cosines of arcs or angles in arithmetical progression, when the common difference is to the first term in the ratio of 2 to 1. To investigate a general expression for an infinite series of this kind, let

=sin A+-sin (A+B) +sin (A+2B)+sin (A+38) + &c.

Then, since this series is a recurring series, whose scale of relation is 2 cos B–1, it will arise from the developement of a fraction whose denominator is 1 2: . cos B + 2*, making z = 1. sin A+z [sin (A + h) – 2 sin a . cos n]

Now this fraction will be = 1 – 22 cos s --22

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