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1.3.5...(2r-l)arar+1 + 2.4.6... 2r(2r+l) + ( 2) cos-1 tan-1 x, log (1 4- J?), are other functions which may be conveniently expanded by this method; and which the student is recommended to apply the process to.

60.] As many properties of some series which have been expanded in the last article will be required in the sequel of the Treatise, it is most convenient to introduce them here, though they may more properly be considered to belong to analytical trigonometry.

By an imaginary or impossible quantity is meant, one of the form' a + b*f-i; (23)

a and b being symbols of positive or negative possible quantities, and the symbol J 1 being that, which, when squared, is equal to —1. Two such expressions, which differ only in the sign of <f—1, are said to be conjugate to each other; thus

a + bV—1 and a—bj 1 are called conjugate imaginary expressions; and it is to be observed, that the product of two such conjugate expressions, viz.

(a+4V^T) {a-b V^T) = o»+b\ (24) Now such an expression as (23) may always be put under the form f (cQ8 Q + y^i Bin e). (25)

in which case r is called the modulus of the expression a + b V—l. For let

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and as a and b are possible quantities, a2 + b2 is a positive quantity, and therefore r is possible; and as tan 0 passes through all

values, from — oo through 0 to + co, as 6 increases from — ^

to + ~, whatever are the relative signs and magnitudes of a


and there is always some angle between — - and - which will satisfy the equation, tan 6 = -; therefore the substitutions

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the series being expanded by equation (18), Art. 59; whence, equating impossible parts, and dividing both sides by 2 A/— 1, we have

(tana?)3 (tana?)8 ,„., a? a= tana?- 3 + 5 ~ >■ (340

which is a series useful for the calculation of it. Again, by equation (31),

e^f-i cos a? + v—T sin x;

. eyS^\ = C08 y _|_ J _ 1 sin y .

therefore by multiplication,

e;x+y) -/^T _ cos cos y _ sin a? siu y

+ -J— 1 (cos a? sin y 4- cos y sin a?), (35) but e(*+" v'-r = cos (.r + y) + sin (a? + y); (36)

wherefore, equating the possible and impossible parts of the equal quantities (35) and (36), we have the fundamental trigonometrical formulae,

cos (a? + y) = cos x cos y sin x sin y, (37) sin (x -f y) = sin a? cos y + cos a? sin y. (38)


(cos x+ V—1 sin a?) (cos y 4- \/ — 1 sin y) (cosz + \/—1 sin 2)...

= cos (a?4 y + 2+ ...) + A/ 1 sin (a? + y+ z +...)...; (39) whence, if a? = y = z = ... torn quantities,

(cos a? + sin a?)"1 = cos ma? 4- sin ma?:

in a similar way it may be shewn that

(cos x -J — 1 sin x)m = cos ma? — */ 1 sin ma?; "and therefore generally,

(cos x ± »J—1 sin a?)m = cos ma? + v—1 sin ma?; (40) which is De Moivre's Theorem.

By these processes therefore the multiplication of a series of factors of the form cos a? 4- v —1 sin a?, and therefore of all imaginary expressions, is reduced to the addition of the arcs under the circular functions; and the involution of such quantities to the multiplication of the arcs.

62.] Equivalent expression of (cos x)n, in terms of the cosines of the multiple arcs.

To abbreviate the notation, let us substitute as follows:

gxJ—l Z, ~\ gmx-J—1 _ gin


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