Routes the mathematical language Steinmetz used to train engineers in calculation itself.
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#1Engineering Mathematics: A Series of Lectures Delivered at Union CollegeChapter 1: The General Number- line 3090Priority OCR formula candidateLet A = a(cos a+j sin a) be divided by J5 = 6(cos ,5+y sin /5),
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Priority OCR formula candidateIf, A=ai +ja2 = a (cos a+j sin a), then
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Priority OCR formula candidateand ai + ja2 = a (cos 6 + j sin d) ;
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Priority OCR formula candidateor ai -\-ja2 = A(cos 0+j sin 6).
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Priority OCR formula candidateand A ==5 (cos 37 deg. H-j sin 37 deg).
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Priority OCR formula candidateC = AB = ah (cos a+j sin a) (cos /?+ / sin /5)
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Priority OCR formula candidateA = ai +ja2 = a(cos a+j sin a),
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Priority OCR formula candidate20. If ai+y6i=6+2.5 J is represented by the point Pi;
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Priority OCR formula candidateand the vertical distance 6i=2.5. If a2+jb2 = S+4:j is repre-
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Priority OCR formula candidatePi: a:=+3, y= +2 P2: x= +S y= -2,
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Priority OCR formula candidate• Pi=3+2/ P2 = 3-2/ P3=-3+2/ P4=-3-2y
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Priority OCR formula candidatePi = - -^a, P2 = - 7— a,
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Priority OCR formula candidateV +l=cos qX + 7 sm qX - ,
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Priority OCR formula candidatecos x+y sin X = ~1,
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Priority OCR formula candidatea+y6 = 3 4-2?s, the point P2, 90 deg. away from Pi, would
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Priority OCR formula candidateP2 = jPi-Ka+jb)=j(3-V2j)= -2+3/,
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Priority OCR formula candidateizontal distances of Pi and P2: ao = ai+a2 = 6+3 = 9, and a
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Priority OCR formula candidateao +jho = {ai +a2) +j(bi +62)
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Priority OCR formula candidatea2 +jb2 = (ao -di) +j(bo -61)
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Priority OCR formula candidate^ = a(cos 0 +j sin d),
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Priority OCR formula candidatebut since ai=a cos 6 is negative, -4, cos 6 must be negative,
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Priority OCR formula candidate4 = 5 (cos 217 deg. +j sin 217 deg.)
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Priority OCR formula candidate= 5( - cos 37 deg. -j sin 37 deg.)
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Priority OCR formula candidate^0 = So (cos do +j sin ^0)
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Priority OCR formula candidateisi,n 6i=j^ = 0A50, as ^i = 24.3deg.
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Priority OCR formula candidate♦^2 = S2(cos d2 +j sin dz)
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Priority OCR formula candidate/S4 = 1105 (cos 30 deg. +y sin 30 deg.)
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Priority OCR formula candidateS6 = sq{cos Oq +/ sin Oq)
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Priority OCR formula candidateAB = {aihi -a2?>2) +j{aih2 +^2*^1),
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Priority OCR formula candidateFor instance, A=2+j multiplied by B=^l+1.5 j gives
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Priority OCR formula candidate= a6 {cos (a +/?) +j sin (a +/5)} :
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Priority OCR formula candidateThus, to multiply the vector quantity, A = ai+ja2 = a (cos
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Priority OCR formula candidate« + y sin^ by ^ = 61 +J62 = & (cos /? +f sin ^) the vector OA in Fig.
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Priority OCR formula candidateA = 3 +4/ = 5(cos 53 deg. +/ sin 53 deg.);
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Priority OCR formula candidate(7 = 44 = 54(cos 4 X53 deg. +j sin 4 X53 deg.) ■
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Priority OCR formula candidate= 625(cos 212 deg. +j sin 212 deg.)
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Priority OCR formula candidate(7 = A/A = A”=a”(cos-+7sin-)
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Priority OCR formula candidateA = a\(ios{a-\-2q7:)-\-j ^m.{a+2qn)\,
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Priority OCR formula candidateC= ■<lA= i625(cos ^ - —+jsm 1 ^)
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Priority OCR formula candidate= 5(cos 53 + j sin 53) = 3 + ij
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Priority OCR formula candidate= 5(cos 143 +j sin 143) =5( -cos 37 +f sin 37) = -4+3/
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Priority OCR formula candidate= 5(cos 233 + J sin 233) = 5( -cos 53 + / sin 53) = -3 + 4/
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Priority OCR formula candidate= 5(cos 323 +y sin 323) =5(cos 37 -j sin 37) =4-3/
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Priority OCR formula candidatein opposite direction from A, Inversely, if we take AC= -2,
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Priority OCR formula candidatevertical distance, jBP= +2f, and therefore is given by a
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Priority OCR formula candidateIf jx = \og, (-1), then £^’^= -1,
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Priority OCR formula candidatePi and P2: 6o = &i+&2 = 2.5+4 = 6.5, hence, is given by the
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Priority OCR formula candidateper sec, making an angle ^o = 20 deg. with the x-axis; hence,
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Priority OCR formula candidate24. If A = ai+ja2 and B = hi-\-jb2, are two general, or
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Priority OCR formula candidateIf now we have two impedances, OZi and OZ2, Zi =ri -jxi
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Priority OCR formula candidateIf we have a current and a voltage, I = ii + ji2 and E = ei -\-je2,
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Priority OCR formula candidateFor instance, let 4= -529-33iy = 625 (cos 212 deg.+
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Priority OCR formula candidatebut since ai = a cos 6 is negative, ~4, cos 6 must be negative,
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Priority OCR formula candidatea2 = a sin a
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Priority OCR formula candidatetance AB, and multiply by (-1)^ we get the distance AC= -2
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Priority OCR formula candidateand multiply by (-1), we get AB=+2; that is, multiplica-
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Priority OCR formula candidatemore by V-l, we’ get +2 X v^^X V~l= -2; that is,
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Priority OCR formula candidatecombination of the distances, 05= +3 and BP=- +2/. For
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Priority OCR formula candidatePs’ x=-S, y=+2 P4: x=-S y=-2]
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Priority OCR formula candidatex= - = and 2/ = - =/; or, is represented by the general nuniber,
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Priority OCR formula candidatelog£(-l)=/7r(2n + l),
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Priority OCR formula candidatejhk= +1.
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Priority OCR formula candidatekhj= -1,
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Priority OCR formula candidateThe only feature which must be kept in mind is that f = - 1, and
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Priority OCR formula candidatef =/, f= -1, f = -], f- +1;
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Priority OCR formula candidatef = +y, f= -1, f= -/,, f = +i;
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Priority OCR formula candidatex+jy = 5-Sj,
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Priority OCR formula candidatex = 5 and y= -3.
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Priority OCR formula candidatethe general number a+jh = 6+2.5j may be considered as
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Priority OCR formula candidatedistance from the y axis, 0A = BP = a^6, and the vertical
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Priority OCR formula candidatetan <? = :== =-77- fl/)^‘
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Priority OCR formula candidatetan 6=-.
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