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{"sheet":"TrigForm","index":0,"n_rows":36,"n_cols":12,"rows_kept":36,"cut":false,"hidden":false,"merged":[]}
[null,null,null,"          DeMoivre's Theorem"]
[]
[null,"                 This DIGMath module lets you investigate "]
[null,null,null,"DeMoivre's Theorem for successive powers "]
[null,null,null,"                   of a complex number "]
[null,null,null,"           z = x + yi =  z  (cos q + i sin q):"]
[null,null,null,"      zn = (x + yi)n =  zn  (cos nq + i sin nq)"]
[]
[]
[]
["First enter the real and imaginary parts of the complex number z:  "]
[null,null,null,null,"x =",0.5]
[null,null,null,null,"y =",0.3]
[]
[null,null,"so",{"$f":"\"z = \"&TEXT(x,\"0.00\")&\" + \"&TEXT(y,\"0.00\")&\" i  =  \" &TEXT(z,\"0.000\") &\" (cos (\"& TEXT(Q,\"0.0\") &\") + i sin (\" & TEXT(Q,\"0.0\") & \") )\"","v":"z = 0.50 + 0.30 i  =  0.583 (cos (31.0) + i sin (31.0) )"}]
[null,null,null,null,"q =",{"$f":"Q","v":30.963756532073521},"°"]
[null,null,null,null,null,null,null,null,null,null,{"$f":"\"z = \"&TEXT(x,\"0.00\")&\" + \"&TEXT(y,\"0.00\")&\" i  =  \" &TEXT(z,\"0.000\") &\" (cos (\"& TEXT(Q,\"0.0\") &\") + i sin (\" & TEXT(Q,\"0.0\") & \") )\"","v":"z = 0.50 + 0.30 i  =  0.583 (cos (31.0) + i sin (31.0) )"}]
["    What is the highest power n (1 to 6)?",null,null,null,{"$f":"F18+1","v":5},4]
[null,null,null,null,null,null," ",null,null,null,{"$f":"IF(n>=2,\"  z^2 = (\" &TEXT(z,\"0.000\") &\")^2  (cos (\"& TEXT(2*Q,\"0.0\") &\") + i sin (\" & TEXT(2*Q,\"0.0\") & \") )\",\"\")","v":"  z^2 = (0.583)^2  (cos (61.9) + i sin (61.9) )"}]
[null,null,null,null,null,null,null,null,null,null,null,{"$f":"IF(n>=2,\" = \"&TEXT(z^2,\"0.000\")&\" (cos (\"&TEXT(2*Q,\"0.0\")&\") + i sin (\"&TEXT(2*Q,\"0.0\")&\") ) = \"&TEXT(Sheet2!F3,\"0.000\")&\" + \"&TEXT(Sheet2!F4,\"0.000\")&\"i  \",\"\")","v":" = 0.340 (cos (61.9) + i sin (61.9) ) = 0.210 + -0.267i  "}]
[]
[null,null,null,null,null,null,null,null,null,null,{"$f":"IF(n>=3,\"  z^3 = (\" &TEXT(z,\"0.000\") &\")^3  (cos (\"& TEXT(3*Q,\"0.0\") &\") + i sin (\" & TEXT(3*Q,\"0.0\") & \") )\",\"\")","v":"  z^3 = (0.583)^3  (cos (92.9) + i sin (92.9) )"}]
[null,"Click each item below for suggestions and investigations",null,null,null,null,null,null,null,null,null,{"$f":"IF(n>=3,\" = \"&TEXT(z^3,\"0.000\")&\" (cos (\"&TEXT(3*Q,\"0.0\")&\") + i sin (\"&TEXT(3*Q,\"0.0\")&\") ) = \"&TEXT(Sheet2!H3,\"0.000\")&\" + \"&TEXT(Sheet2!H4,\"0.000\")&\"i  \",\"\")","v":" = 0.198 (cos (92.9) + i sin (92.9) ) = 0.042 + -0.194i  "}]
[null,null,null,"   Item 1"]
[null,null,null,"   Item 2",null,null,null,null,null,null,{"$f":"IF(n>=4,\"  z^4 = (\" &TEXT(z,\"0.000\") &\")^4  (cos (\"& TEXT(4*Q,\"0.0\") &\") + i sin (\" & TEXT(4*Q,\"0.0\") & \") )\",\"\")","v":"  z^4 = (0.583)^4  (cos (123.9) + i sin (123.9) )"}]
[null,null,null,"   Item 3",null,null,null,null,null,null,null,{"$f":"IF(n>=4,\" = \"&TEXT(z^4,\"0.000\")&\" (cos (\"&TEXT(4*Q,\"0.0\")&\") + i sin (\"&TEXT(4*Q,\"0.0\")&\") ) = \"&TEXT(Sheet2!J3,\"0.000\")&\" + \"&TEXT(Sheet2!J4,\"0.000\")&\"i  \",\"\")","v":" = 0.116 (cos (123.9) + i sin (123.9) ) = -0.027 + -0.112i  "}]
[null,null,null,"   Item 4"]
[null,null,null,null,null,null,null,null,null,null,{"$f":"IF(n>=5,\"  z^5 = (\" &TEXT(z,\"0.000\") &\")^5  (cos (\"& TEXT(5*Q,\"0.0\") &\") + i sin (\" & TEXT(5*Q,\"0.0\") & \") )\",\"\")","v":"  z^5 = (0.583)^5  (cos (154.8) + i sin (154.8) )"}]
[null,null,null,null,null,null,null,null,null,null,null,{"$f":"IF(n>=5,\" = \"&TEXT(z^5,\"0.000\")&\" (cos (\"&TEXT(5*Q,\"0.0\")&\") + i sin (\"&TEXT(5*Q,\"0.0\")&\") ) = \"&TEXT(Sheet2!L3,\"0.000\")&\" + \"&TEXT(Sheet2!L4,\"0.000\")&\"i  \",\"\")","v":" = 0.067 (cos (154.8) + i sin (154.8) ) = -0.043 + -0.052i  "}]
[]
[null,null,null,null,null,null,null,null,null,null,{"$f":"IF(n=6,\"  z^6 = (\" &TEXT(z,\"0.000\") &\")^6  (cos (\"& TEXT(6*Q,\"0.0\") &\") + i sin (\" & TEXT(6*Q,\"0.0\") & \") )\",\"\")","v":""}]
[null,"Created by:  Sheldon P. Gordon",null,null,null,null,null,null,null,null,null,{"$f":"IF(n=6,\" = \"&TEXT(z^6,\"0.000\")&\" (cos (\"&TEXT(6*Q,\"0.0\")&\") + i sin (\"&TEXT(6*Q,\"0.0\")&\") ) = \"&TEXT(Sheet2!N3,\"0.000\")&\" + \"&TEXT(Sheet2!N4,\"0.000\")&\"i  \",\"\")","v":""}]
[null,"                     Farmingdale StateCollege"]
[null,"Development of this module was supported by the "]
[null,"NSF's Division of Undergraduate Education  "]
[null,"under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Rectangular","index":1,"n_rows":35,"n_cols":11,"rows_kept":35,"cut":false,"hidden":false,"merged":[]}
[null,null,null,"          DeMoivre's Theorem"]
[]
[null,null,null,"     This DIGMath module lets you investigate"]
[null,null,null,"        investigate the successive powers "]
[null,null,null,"                   of a complex number "]
[null,null,null,"                            zn = (x + yi)n "]
[]
[]
[]
[]
["First enter the real and imaginary parts of the complex number z:  ",null,null,null,"x =",3]
[null,null,null,null,"y =",4]
[]
[null,null,null,"so  ",{"$f":"\"z = \"&TEXT(x,\"0.00\")&\" + \"&TEXT(y,\"0.00\")&\" i  \"","v":"z = 0.50 + 0.30 i  "}]
[]
[null,null,null,null,null,null,null,null,null,{"$f":"\"z = \"&TEXT(x,\"0.00\")&\" + \"&TEXT(y,\"0.00\")&\" i  \"","v":"z = 0.50 + 0.30 i  "}]
[null,null,null,null,{"$f":"F17+1","v":3},2]
["    What is the highest power n (1 to 6)?",null,null,null,null,null," ",null,null,{"$f":"IF(nn>=2,\"z^2 = (\" &TEXT(xx,\"0.000\") &\")^2  + 2(\"& TEXT(xx,\"0.0\") &\")(\" & TEXT(yy,\"0.0\") &\") i - (\" & TEXT(yy,\"0.0\") & \") ^2\", \"\")","v":"z^2 = (3.000)^2  + 2(3.0)(4.0) i - (4.0) ^2"}]
[null,null,null,null,null,null,null,null,null,null,{"$f":"IF(nn>=2,\"= \"&TEXT(xx^2-yy^2,\"0.000\")&\"   +  \"&TEXT(2*xx*yy,\"0.000\")&\" i \", \"\")","v":"= -7.000   +  24.000 i "}]
[]
[null,"Click each item below for suggestions and investigations",null,null,null,null,null,null,null,{"$f":"IF(nn>=3,\"z^3 = (\" &TEXT(xx,\"0.000\") &\")^3  + 3(\"& TEXT(xx,\"0.000\") &\")^2 (\" & TEXT(yy,\"0.000\") &\") i + 3(\" & TEXT(xx,\"0.000\") &\")(\" & TEXT(yy,\"0.000\") & \") i^2  + (\" & TEXT(yy,\"0.000\") & \")^3 i^3\", \"\")","v":"z^3 = (3.000)^3  + 3(3.000)^2 (4.000) i + 3(3.000)(4.000) i^2  + (4.000)^3 i^3"}]
[null,null,null,"   Item 1",null,null,null,null,null,null,{"$f":"IF(nn>=3,\"= \"&TEXT(xx^3-3*xx*yy^2,\"0.000\") &\"   +  \"&TEXT(-3*xx^2*yy+yy^3,\"0.000\")&\" i \", \"\")","v":"= -117.000   +  -44.000 i "}]
[null,null,null,"   Item 2"]
[null,null,null,"   Item 3",null,null,null,null,null,{"$f":"IF(nn>=4,\"z^4 = (\" &TEXT(xx,\"0.000\") &\")^4  + 4(\"& TEXT(xx,\"0.000\") &\")^3 (\" & TEXT(yy,\"0.000\") &\") i + 6(\" & TEXT(xx,\"0.000\") &\")^2 (\" & TEXT(yy,\"0.000\") & \")^2 i^2  + 4(\" & TEXT(xx,\"0.000\") &\") (\" & TEXT(yy,\"0.000\") & \")^3 i^3 + (\" & TEXT(yy,\"0.000\") & \")^4 i^4\", \"\")","v":""}]
[null,null,null,"   Item 4",null,null,null,null,null,null,{"$f":"IF(nn>=4,\"= \"&TEXT(xx^4-6*xx^2*yy^2+yy^4,\"0.000\") &\"   +  \"&TEXT(4*xx^3*yy- 4*xx*yy^3,\"0.000\")&\" i \", \"\")","v":""}]
[]
[null,null,null,null,null,null,null,null,null,{"$f":"IF(nn>=5,\"z^5 = (\"&TEXT(xx,\"0.000\")&\")^5  + 5(\"&TEXT(xx,\"0.000\")&\")^4 (\"&TEXT(yy,\"0.000\")&\") i + 10(\"&TEXT(xx,\"0.000\")&\")^3 (\"&TEXT(yy,\"0.000\")&\")^2 i^2  + 10(\"&TEXT(xx,\"0.000\")&\")^2 (\"&TEXT(yy,\"0.000\")&\")^3 i^3 + (\"&TEXT(xx,\"0.000\")&\") (\"&TEXT(yy,\"0.000\")&\")^4 i^4 + (\"   &TEXT(yy,\"0.000\")    &   \")^5 i^5\", \"\")","v":""}]
[null,null,null,null,null,null,null,null,null,null,{"$f":"IF(nn>=5,\"= \"&TEXT(xx^5-10*xx^3*yy^2+xx*yy^4,\"0.000\") &\"   +  \"&TEXT(5*xx^4*yy-10*xx^2*yy^3+ yy^5,\"0.000\")&\" i \", \"\")","v":""}]
[]
[null,null,null,null,null,null,null,null,null,{"$f":"IF(nn>=6,\"z^6 = (\"&TEXT(xx,\"0.000\")&\")^6  + 6(\"&TEXT(xx,\"0.000\")&\")^5 (\"&TEXT(yy,\"0.000\")&\") i + 15(\"&TEXT(xx,\"0.000\")&\")^4 (\"&TEXT(yy,\"0.000\")&\")^2 i^2  + 20(\"&TEXT(xx,\"0.000\")&\")^3 (\"&TEXT(yy,\"0.000\")&\")^3 i^3 + (\"&TEXT(xx,\"0.000\")&\")^2 (\"&TEXT(yy,\"0.000\")&\")^4 i^4 + (\"&TEXT(xx,\"0.000\")&\") (\"&TEXT(yy,\"0.000\")&\")^5 i^5 + (\"   &TEXT(yy,\"0.000\")    &   \")^6 i^6\", \"\")","v":""}]
[null,"Created by:  Sheldon P. Gordon",null,null,null,null,null,null,null,null,{"$f":"IF(nn>=6,\"= \"&TEXT(xx^6-15*xx^4*yy^2+15*xx^2*yy^4-yy^6,\"0.000\") &\"   +  \"&TEXT(6*xx^5*yy-20*xx^3*yy^3+ 6*xx*yy^5,\"0.000\")&\" i \", \"\")","v":""}]
[null,"                     Farmingdale StateCollege"]
[null,"Development of this module was supported by the "]
[null,"NSF's Division of Undergraduate Education  "]
[null,"under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Non-IntegerPowers","index":2,"n_rows":36,"n_cols":11,"rows_kept":36,"cut":false,"hidden":false,"merged":[]}
[null,null,null,"          DeMoivre's Theorem"]
[null,null,"            This DIGMath module lets you investigate "]
[null,null,null,"DeMoivre's Theorem for non-integer powers "]
[null,null,null,"                   of a complex number "]
[null,null,null,"           z = x + yi =  z  (cos q + i sin q):"]
[null,null,null,"      zn = (x + yi)n =  zn  (cos nq + i sin nq)"]
[]
[]
[]
["First enter the real and imaginary parts of the complex number z:  "]
[null,null,null,null,"x =",1]
[null,null,null,null,"y =",2]
[]
[null,null,"so",{"$f":"\"z = \"&TEXT(x,\"0.00\")&\" + \"&TEXT(y,\"0.00\")&\" i  =  \" &TEXT(z,\"0.000\") &\" (cos (\"& TEXT(Q,\"0.0\") &\") + i sin (\" & TEXT(Q,\"0.0\") & \") )\"","v":"z = 0.50 + 0.30 i  =  0.583 (cos (31.0) + i sin (31.0) )"}]
[null,null,null,null,"q =",{"$f":"Q","v":30.963756532073521},"°"]
[null,null,null,null," z =",{"$f":"TEXT((xxx^2+yyy^2)^0.5,\"0.000\")","v":"2.236"}]
[null,null,null,null,null,null,null,null,null,{"$f":"\"z = \"&TEXT(x,\"0.00\")&\" + \"&TEXT(y,\"0.00\")&\" i  =  \" &TEXT(z,\"0.000\") &\" (cos (\"& TEXT(Q,\"0.0\") &\") + i sin (\" & TEXT(Q,\"0.0\") & \") )\"","v":"z = 0.50 + 0.30 i  =  0.583 (cos (31.0) + i sin (31.0) )"}]
["    What is the power n (0.1 to 2.5)?",null,null,null,{"$f":"F18/10","v":1.3},13]
[null,null,null,null,null,null," ",null,null,{"$f":"\"  z^\"&TEXT(nnn,\"0.0\")&\" = \"&TEXT(F16,\"0.000\")&\"^\" & TEXT(nnn,\"0.0\") & \"  [ cos (\" & TEXT(nnn,\"0.0\") & \"(\" & TEXT(F15,\"0.0\") & \") ) + i sin (\" & TEXT(nnn,\"0.0\") & \"(\" & TEXT(F15,\"0.0\")   & \") ) ]\"","v":"  z^1.3 = 2.236^1.3  [ cos (1.3(31.0) ) + i sin (1.3(31.0) ) ]"}]
[]
[null,null,null,null,null,null,null,null,null,null,{"$f":" \"= \" &TEXT(Sheet2!F34,\"0.000\") & \" + \" & TEXT(Sheet2!F35,\"0.000\") & \" i\"","v":"= 2.172 + 1.839 i"}]
[]
[null,"Click each item below for suggestions and investigations"]
[null,null,null,"   Item 1"]
[null,null,null,"   Item 2"]
[]
[]
[]
[]
[]
[]
[null,"Created by:  Sheldon P. Gordon"]
[null,"                     Farmingdale StateCollege"]
[null,"Development of this module was supported by the "]
[null,"NSF's Division of Undergraduate Education  "]
[null,"under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Sheet2","index":3,"n_rows":39,"n_cols":14,"rows_kept":39,"cut":false,"hidden":false,"merged":[]}
["Rectangular Form"]
[]
["x =",{"$f":"x","v":0.5},null,{"$f":"x","v":0.5},null,{"$f":"z^2*COS(2*Q)","v":0.21018952318434872},null,{"$f":"z^3*COS(3*Q)","v":4.2158389272062752E-2},null,{"$f":"z^4*COS(4*Q)","v":-2.724072868707221E-2},null,{"$f":"z^5*COS(5*Q)","v":-4.2909091694764166E-2},null,{"$f":"z^6*SIN(6*Q)","v":-1.6333679161936464E-2}]
["y =",{"$f":"y","v":0.3},null,{"$f":"y","v":0.3},null,{"$f":"z^2*SIN(2*Q)","v":-0.26724588742118388},null,{"$f":"z^3*SIN(3*Q)","v":-0.19371801726732907},null,{"$f":"z^4*SIN(4*Q)","v":-0.11234457130007354},null,{"$f":"z^5*SIN(5*Q)","v":-5.1984153834897753E-2},null,{"$f":"z^6*COS(6*Q)","v":-3.574934042797042E-2}]
["z =",{"$f":"(x^2+y^2)^0.5","v":0.58309518948452999},null,"v0",null,"v1",null,"v2",null,"v3",null,"v4",null,"v5"]
["Q =",{"$f":"IF(x<>0,180/PI()*ATAN(y/x),IF(y>0,90,-90))","v":30.963756532073521},null,1,null,2,null,3,null,4,null,5,null,6]
["plotting points:",null,0,{"$f":"D3","v":0.5},0,{"$f":"F3","v":0.21018952318434872},0,{"$f":"IF(H6<=n,H3,0)","v":4.2158389272062752E-2},0,{"$f":"IF(J6<=n,J3,0)","v":-2.724072868707221E-2},0,{"$f":"IF(L6<=n,L3,0)","v":-4.2909091694764166E-2},0,{"$f":"IF(N6<=n,N3,0)","v":0}]
[null,null,0,{"$f":"D4","v":0.3},0,{"$f":"F4","v":-0.26724588742118388},0,{"$f":"IF(H6<=n,H4,0)","v":-0.19371801726732907},0,{"$f":"IF(J6<=n,J4,0)","v":-0.11234457130007354},0,{"$f":"IF(L6<=n,L4,0)","v":-5.1984153834897753E-2},0,{"$f":"IF(N6<=n,N4,0)","v":0}]
[]
["vector points",null,0,{"$f":"D3","v":0.5},0,{"$f":"F3","v":0.21018952318434872},{"$f":"IF(H$6<=n,G7,\"\")","v":0},{"$f":"IF(H$6<=n,H7,\"\")","v":4.2158389272062752E-2},{"$f":"IF(J$6<=n,I7,\"\")","v":0},{"$f":"IF(J$6<=n,J7,\"\")","v":-2.724072868707221E-2},{"$f":"IF(L$6<=n,K7,\"\")","v":0},{"$f":"IF(L$6<=n,L7,\"\")","v":-4.2909091694764166E-2},{"$f":"IF(N$6<=n,M7,\"\")","v":""},{"$f":"IF(N$6<=n,N7,\"\")","v":""}]
[null,null,0,{"$f":"D4","v":0.3},0,{"$f":"F4","v":-0.26724588742118388},{"$f":"IF(H$6<=n,G8,\"\")","v":0},{"$f":"IF(H$6<=n,H8,\"\")","v":-0.19371801726732907},{"$f":"IF(J$6<=n,I8,\"\")","v":0},{"$f":"IF(J$6<=n,J8,\"\")","v":-0.11234457130007354},{"$f":"IF(L$6<=n,K8,\"\")","v":0},{"$f":"IF(L$6<=n,L8,\"\")","v":-5.1984153834897753E-2},{"$f":"IF(N$6<=n,M8,\"\")","v":""},{"$f":"IF(N$6<=n,N8,\"\")","v":""}]
[]
[]
[]
[]
["Trig Form"]
[]
["xx =",{"$f":"xx","v":3},null,{"$f":"xx","v":3},null,{"$f":"xx^2-yy^2","v":-7},null,{"$f":"xx^3-3*xx*yy^2","v":-117},null,{"$f":"xx^4-6*xx^2*yy^2+yy^4","v":-527},null,{"$f":"xx^5-10*xx^3*yy^2+xx*yy^4","v":-3309},null,{"$f":"xx^6-15*xx^4*yy^2+15*xx^2*yy^4-yy^6","v":11753}]
["yy =",{"$f":"yy","v":4},null,{"$f":"yy","v":4},null,{"$f":"2*xx*yy","v":24},null,{"$f":"-3*xx^2*yy+yy^3","v":-44},null,{"$f":"4*xx^3*yy-4*xx*yy^3","v":-336},null,{"$f":"5*xx^4*yy-10*xx^2*yy^3+yy^5","v":-3116},null,{"$f":"6*xx^5*yy-20*xx^3*yy^3+6*xx*yy^5","v":-10296}]
["z =",{"$f":"(x^2+y^2)^0.5","v":0.58309518948452999},null,"v0",null,"v1",null,"v2",null,"v3",null,"v4",null,"v5"]
["Q =",{"$f":"IF(x<>0,180/PI()*ATAN(y/x),IF(y>0,90,-90))","v":30.963756532073521},null,1,null,2,null,3,null,4,null,5,null,6]
["plotting points:",null,0,{"$f":"D18","v":3},0,{"$f":"F18","v":-7},0,{"$f":"IF(H21<=nn,H18,0)","v":-117},0,{"$f":"IF(J21<=nn,J18,0)","v":0},0,{"$f":"IF(L21<=nn,L18,0)","v":0},0,{"$f":"IF(N21<=nn,N18,0)","v":0}]
[null,null,0,{"$f":"D19","v":4},0,{"$f":"F19","v":24},0,{"$f":"IF(H21<=nn,H19,0)","v":-44},0,{"$f":"IF(J21<=nn,J19,0)","v":0},0,{"$f":"IF(L21<=nn,L19,0)","v":0},0,{"$f":"IF(N21<=nn,N19,0)","v":0}]
[]
["vector points",null,0,{"$f":"D18","v":3},0,{"$f":"F18","v":-7},{"$f":"IF(H$6<=n,G22,\"\")","v":0},{"$f":"IF(H$6<=nn,H22,\"\")","v":-117},{"$f":"IF(J$6<=n,I22,\"\")","v":0},{"$f":"IF(J$6<=nn,J22,\"\")","v":""},{"$f":"IF(L$6<=n,K22,\"\")","v":0},{"$f":"IF(L$6<=nn,L22,\"\")","v":""},{"$f":"IF(N$6<=n,M22,\"\")","v":""},{"$f":"IF(N$6<=nn,N22,\"\")","v":""}]
[null,null,0,{"$f":"D19","v":4},0,{"$f":"F19","v":24},{"$f":"IF(H$6<=n,G23,\"\")","v":0},{"$f":"IF(H$6<=nn,H23,\"\")","v":-44},{"$f":"IF(J$6<=n,I23,\"\")","v":0},{"$f":"IF(J$6<=nn,J23,\"\")","v":""},{"$f":"IF(L$6<=n,K23,\"\")","v":0},{"$f":"IF(L$6<=nn,L23,\"\")","v":""},{"$f":"IF(N$6<=n,M23,\"\")","v":""},{"$f":"IF(N$6<=nn,N23,\"\")","v":""}]
[]
[]
[]
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["Non-Integer Powers",null,null,null,null,null,null,null,null,null,null,null,null," z"]
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