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{"sheet":"Sines","index":0,"n_rows":41,"n_cols":17,"rows_kept":41,"cut":false,"hidden":false,"merged":[]}
[null,null,null,"Approximating the Sine Function"]
[]
[null,"                This program lets you \"build\" the best"]
[null,null,null," polynomial approximation to y = sin x "]
[]
[]
[null,null,null,"based on the numbers a, b, c, …, g."]
[]
[null,null,null,"Start by setting all the coefficients to 0 "]
[null,null,null,"and then adjust each coefficient in turn "]
[null,null,null,"to get the best possible fit to the sine curve."]
[null,null,null,"We start with the constant coefficient a = 0."]
[]
[null,"First adjust b (-10 to 10) in the linear coefficient 1/b."]
[null,null,null,null,{"$f":"F15-10","v":1},11]
[null,"Next adjust c (-10 to 10) in the quadratic coefficient 1/c."]
[null,null,null,null,{"$f":"F17-10","v":0},10,null,null,null,null,null,{"$f":"\"  y = \"&Sheet3!G13 & Sheet3!G15 & Sheet3!G17 & Sheet3!G19 & Sheet3!G21 & Sheet3!G23","v":"  y = 0 + (1/1)x  + 0x^2 - (1/5)x^3  + 0x^4 +  (1/120)x^5 + 0x^6"}]
[null,"Then adjust d (-10 to 10) in the cubic coefficient 1/d."]
[null,null,null,null,{"$f":"F19-10","v":-5},5]
[null,"Then adjust e (-10 to 10) in the quartic coefficient 1/e",null,null,null,null,null,null,null,null,null,null,"Enter an interval from -c to +c on which to measure the accuracy"]
[null,null,null,null,{"$f":"F21-10","v":0},10]
[null,"Then adjust f (-200 to 200) in the fifth degree coefficient 1/f ",null,null,null,null,null,null,null,null,null,null,{"$f":"-P22","v":-3.4},"to",{"$f":"P22","v":3.4},{"$f":"0.1+$Q22/10","v":3.4},33]
[null,null,null,null,{"$f":"10*F23-200","v":120},32]
[null,"Finally adjust the 6th degree coefficient g (-500 to 500).",null,null,null,null,null,null,null,null,null,null,"The sum of the squares = ",null,null,{"$f":"Sheet3!G27","v":9.3188483244324036}]
[null,null,null,null,{"$f":"20*F25-500","v":0},25]
[]
[]
[]
[null,null,"Click each item below for suggestions and investigations"]
[null,null,null,"   Item 1"]
[null,null,null,"   Item 2"]
[null,null,null,"   Item 3"]
[null,null,null,"   Item 4"]
[null,null,null,"   Item 5"]
[]
[]
["Created by:  Sheldon P. Gordon"]
["                     Farmingdale StateCollege"]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Cosines","index":1,"n_rows":43,"n_cols":17,"rows_kept":43,"cut":false,"hidden":false,"merged":[]}
[null,null,"Approximating the Cosine Function"]
[]
[null,"              This program lets you \"build\" the best"]
[null,null,null," polynomial approximation to y = cos x "]
[]
[]
[null,null,null,"based on the numbers a, b, c, …, g."]
[]
[null,null,null,"Start by setting all the coefficients to 0 "]
[null,null,null,"and then adjust each coefficient in turn "]
[null,null,null,"to get the best possible fit to the cosine curve."]
[null,null,null,"We start with the constant coefficient a = 1."]
[]
[null,"First adjust b (-10 to 10) in the linear coefficient 1/b."]
[null,null,null,null,{"$f":"F15-10","v":-2},8]
[null,"Next adjust c (-10 to 10) in the quadratic coefficient 1/c.",null,null,null,null,null,null,null,null,null,{"$f":"\"  y = \"&Sheet3!P13 & Sheet3!P15 & Sheet3!P17 & Sheet3!P19 & Sheet3!P21 & Sheet3!P23","v":"  y = 1 - (1/2)x  - (1/4)x^2 + 0x^3+  (1/16)x^4+  (1/30)x^5 - (1/820)x^6 "}]
[null,null,null,null,{"$f":"F17-10","v":-4},6]
[null,"Then adjust d (-10 to 10) in the cubic coefficient 1/d."]
[null,null,null,null,{"$f":"F19-10","v":0},10]
[null,"Then adjust e (-30 to 30) in the quartic coefficient 1/e",null,null,null,null,null,null,null,null,null,null,"Enter an interval from -c to +c on which to measure the accuracy"]
[null,null,null,null,{"$f":"2*F21-30","v":16},23]
[null,"Then adjust f (-200 to 200) in the fifth degree coefficient 1/f ",null,null,null,null,null,null,null,null,null,null,{"$f":"-P22","v":-3.1},"to",{"$f":"P22","v":3.1},{"$f":"0.1+$Q22/10","v":3.1},30]
[null,null,null,null,{"$f":"10*F23-200","v":30},23]
[null,"Finally adjust the 6th degree coefficient g (-1000 to 1000).",null,null,null,null,null,null,null,null,null,null,"The sum of the squares = ",null,null,{"$f":"Sheet3!P27","v":770.60275190390075}]
[null,null,null,null,{"$f":"20*F25-1000","v":-820},9]
[]
[]
[]
[null,null,"Click each item below for suggestions and investigations"]
[null,null,null,"   Item 1"]
[null,null,null,"   Item 2"]
[null,null,null,"   Item 3"]
[null,null,null,"   Item 4"]
[null,null,null,"   Item 5"]
[]
[]
[]
[]
["Created by:  Sheldon P. Gordon"]
["                     Farmingdale StateCollege"]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Sheet3","index":2,"n_rows":203,"n_cols":17,"rows_kept":203,"cut":false,"hidden":false,"merged":[]}
[null,null,null,"Sines",null,null,null,null,null,null,null,null,"Cosine"]
[-6,{"$f":"SIN(A2)","v":0.27941549819892586},{"$f":"0+H$3*A2+H$4*A2^2+H$5*A2^3+H$6*A2^4+H$7*A2^5+H$8*A2^6","v":-27.599999999999994},{"$f":"IF(A2<Sines!M$22,0,(B2-C2)^2)","v":0},0,1,null,"coefficients",null,-6,{"$f":"COS(J2)","v":0.96017028665036597},{"$f":"1+Q$3*J2+Q$4*J2^2+Q$5*J2^3+Q$6*J2^4+Q$7*J2^5+Q$8*J2^6","v":-240.09756097560975},{"$f":"IF(J2<Cosines!M$22,0,(K2-L2)^2)","v":0},0,1,null,"coefficients"]
[{"$f":"A2+12/200","v":-5.94},0.33648835845850422,-25.647238433520002,{"$f":"IF(A3<Sines!M$22,0,(B3-C3)^2)","v":0},null,null,null,{"$f":"IF(Sines!E15=0,0,1/Sines!E15)","v":1},null,{"$f":"J2+12/200","v":-5.94},0.94168762581967758,-227.10716833194965,{"$f":"IF(J3<Cosines!M$22,0,(K3-L3)^2)","v":0},null,null,null,{"$f":"IF(Cosines!E15=0,0,1/Cosines!E15)","v":-0.5}]
[-5.8800000000000008,0.39235022399145303,-23.794573232640026,{"$f":"IF(A4<Sines!M$22,0,(B4-C4)^2)","v":0},-6,{"$f":"a","v":0},null,{"$f":"IF(Sines!E17=0,0,1/Sines!E17)","v":0},null,-5.8800000000000008,0.91981590643663946,-214.69032045054894,{"$f":"IF(J4<Cosines!M$22,0,(K4-L4)^2)","v":0},-6,{"$f":"a","v":0},null,{"$f":"IF(Cosines!E17=0,0,1/Cosines!E17)","v":-0.25}]
[-5.8200000000000012,0.44680005240542925,-22.038491265360022,{"$f":"IF(A5<Sines!M$22,0,(B5-C5)^2)","v":0},6,{"$f":"a","v":0},null,{"$f":"IF(Sines!E19=0,0,1/Sines!E19)","v":-0.2},null,-5.8200000000000012,0.8946338430724079,-202.82731729286388,{"$f":"IF(J5<Cosines!M$22,0,(K5-L5)^2)","v":0},6,{"$f":"a","v":0},null,{"$f":"IF(Cosines!E19=0,0,1/Cosines!E19)","v":0}]
[-5.7600000000000016,0.49964188311690094,-20.375555604480041,{"$f":"IF(A6<Sines!M$22,0,(B6-C6)^2)","v":0},null,null,null,{"$f":"IF(Sines!E21=0,0,1/Sines!E21)","v":0},null,-5.7600000000000016,0.86623206396172914,-191.49894092628196,{"$f":"IF(J6<Cosines!M$22,0,(K6-L6)^2)","v":0},null,null,null,{"$f":"IF(Cosines!E21=0,0,1/Cosines!E21)","v":6.25E-2}]
[-5.700000000000002,0.55068554259763614,-18.802404750000051,{"$f":"IF(A7<Sines!M$22,0,(B7-C7)^2)","v":0},null,null,null,{"$f":"IF(Sines!E23=0,0,1/Sines!E23)","v":8.3333333333333332E-3},null,-5.700000000000002,0.83471278483916078,-180.68644841951254,{"$f":"IF(J7<Cosines!M$22,0,(K7-L7)^2)","v":0},null,null,null,{"$f":"IF(Cosines!E23=0,0,1/Cosines!E23)","v":3.3333333333333333E-2}]
[-5.6400000000000023,0.59974732879404158,-17.315751851520055,{"$f":"IF(A8<Sines!M$22,0,(B8-C8)^2)","v":0},null,null,null,{"$f":"IF(Sines!E25=0,0,1/Sines!E25)","v":0},null,-5.6400000000000023,0.80018944107280732,-170.37156477894473,{"$f":"IF(J8<Cosines!M$22,0,(K8-L8)^2)","v":0},null,null,null,{"$f":"IF(Cosines!E25=0,0,1/Cosines!E25)","v":-1.2195121951219512E-3}]
[-5.5800000000000027,0.64665067225618145,-15.912383930640058,{"$f":"IF(A9<Sines!M$22,0,(B9-C9)^2)","v":0},null,null,null,null,null,-5.5800000000000027,0.76278627941949018,-160.53647592597085,{"$f":"IF(J9<Cosines!M$22,0,(K9-L9)^2)","v":0}]
[-5.5200000000000031,0.69122677159712453,-14.589161103360063,{"$f":"IF(A10<Sines!M$22,0,(B10-C10)^2)","v":0},null,null,null,null,null,-5.5200000000000031,0.72263791087059404,-151.16382171527664,{"$f":"IF(J10<Cosines!M$22,0,(K10-L10)^2)","v":0}]
[-5.4600000000000035,0.73331520099565406,-13.343015802480071,{"$f":"IF(A11<Sines!M$22,0,(B11-C11)^2)","v":0},null,null,null,null,null,-5.4600000000000035,0.67988882619785962,-142.23668899409719,{"$f":"IF(J11<Cosines!M$22,0,(K11-L11)^2)","v":0}]
[-5.4000000000000039,0.77276448755598492,-12.170952000000078,{"$f":"IF(A12<Sines!M$22,0,(B12-C12)^2)","v":0},null,null,"Polynomial:",null,null,-5.4000000000000039,0.63469287594263735,-133.7386047024396,{"$f":"IF(J12<Cosines!M$22,0,(K12-L12)^2)","v":0},null,null,"Polynomial:"]
[-5.3400000000000043,0.80943265644661688,-11.07004442952007,{"$f":"IF(A13<Sines!M$22,0,(B13-C13)^2)","v":0},null,"linear",{"$f":"IF(Sines!E15>0,\"0 + (1/\" &TEXT(Sines!E15,\"#\") &\")x \",IF(Sines!E15<0,\"0 - (1/\" &TEXT(-1*Sines!E15,\"#\") & \")x \",\"0 - 0x \"))","v":"0 + (1/1)x "},null,null,-5.3400000000000043,0.58721271672007669,-125.65352901427129,{"$f":"IF(J13<Cosines!M$22,0,(K13-L13)^2)","v":0},null,"linear",{"$f":"IF(Cosines!E15>0,\"1 + (1/\" &TEXT(Cosines!E15,\"#\") &\")x \",IF(Cosines!E15<0,\"1 - (1/\" &TEXT(-1*Cosines!E15,\"#\") & \")x \",\"1 + 0x \"))","v":"1 - (1/2)x "}]
[null,null,0,{"$f":"IF(A14<Sines!M$22,0,(B14-C14)^2)","v":0},null,null,null,null,null,-5.2800000000000047,0.53761922583096,-117.9658485196756,{"$f":"IF(J14<Cosines!M$22,0,(K14-L14)^2)","v":0}]
[{"$f":"A13+12/200","v":-5.2800000000000047},{"$f":"SIN(A15)","v":0.84318774185641432},-10.037437808640082,{"$f":"IF(A15<Sines!M$22,0,(B15-C15)^2)","v":0},null,"quad",{"$f":"IF(Sines!E17>0,\"+  (1/\"&TEXT(Sines!E17,\"#\")&\")x^2\",IF(Sines!E17<0,\" - (1/\"&TEXT(-1*Sines!E17,\"#\")&\")x^2 \",\" + 0x^2\"))","v":" + 0x^2"},null,null,-5.2200000000000051,0.48609088628794483,-110.6603694479722,{"$f":"IF(J15<Cosines!M$22,0,(K15-L15)^2)","v":0},null,"quad",{"$f":"IF(Cosines!E17>0,\"+  (1/\"&TEXT(Cosines!E17,\"#\")&\")x^2\",IF(Cosines!E17<0,\" - (1/\"&TEXT(-1*Cosines!E17,\"#\")&\")x^2 \",\" + 0x^2\"))","v":" - (1/4)x^2 "}]
[{"$f":"A15+12/200","v":-5.2200000000000051},0.87390826192901983,-9.0703460613600768,{"$f":"IF(A16<Sines!M$22,0,(B16-C16)^2)","v":0},null,null,null,null,null,-5.1600000000000055,0.4328131444694569,-103.72231093180501,{"$f":"IF(J16<Cosines!M$22,0,(K16-L16)^2)","v":0}]
[-5.1600000000000055,0.90148365596635249,-8.1660515404800691,{"$f":"IF(A17<Sines!M$22,0,(B17-C17)^2)","v":0},null,"cubic",{"$f":"IF(Sines!E19>0,\" +  (1/\"&TEXT(Sines!E19,\"#\") &\")x^3\", IF(Sines!E19<0,\" - (1/\"&TEXT(-1*Sines!E19,\"#\")&\")x^3 \",\" - 0x^3\"))","v":" - (1/5)x^3 "},null,null,-5.1000000000000059,0.37797774271298601,-97.137298312195725,{"$f":"IF(J17<Cosines!M$22,0,(K17-L17)^2)","v":0},null,"cubic",{"$f":"IF(Cosines!E19>0,\" +  (1/\"&TEXT(Cosines!E19,\"#\") &\")x^3\", IF(Cosines!E19<0,\" - (1/\"&TEXT(-1*Cosines!E19,\"#\")&\")x^3 \",\"+ 0x^3\"))","v":"+ 0x^3"}]
[-5.1000000000000059,0.92581468232773012,-7.3219042500000775,{"$f":"IF(A18<Sines!M$22,0,(B18-C18)^2)","v":0},null,null,null,null,null,-5.0400000000000063,0.32178202924972771,-90.891356484563673,{"$f":"IF(J18<Cosines!M$22,0,(K18-L18)^2)","v":0}]
[-5.0400000000000063,0.94681377559260693,-6.5353210675200799,{"$f":"IF(A19<Sines!M$22,0,(B19-C19)^2)","v":0},null,"quartic",{"$f":"IF(Sines!E21>0,\"+  (1/\"&TEXT(Sines!E21,\"#\")&\")x^4\",IF(Sines!E21<0,\" - (1/\"&TEXT(-1*Sines!E21,\"#\")&\")x^4 \", \" + 0x^4 \"))","v":" + 0x^4 "},null,null,-4.9800000000000066,0.26442824796406178,-84.970903285711941,{"$f":"IF(J19<Cosines!M$22,0,(K19-L19)^2)","v":0},null,"quartic",{"$f":"IF(Cosines!E21>0,\"+  (1/\"&TEXT(Cosines!E21,\"#\")&\")x^4\",IF(Cosines!E21<0,\" - (1/\"&TEXT(-1*Cosines!E21,\"#\")&\")x^4 \", \" + 0x^4 \"))","v":"+  (1/16)x^4"}]
[-4.9800000000000066,0.9644053617015288,-5.8037849666400767,{"$f":"IF(A20<Sines!M$22,0,(B20-C20)^2)","v":0},null,null,null,null,null,-4.920000000000007,0.20612281053396531,-79.362742921779841,{"$f":"IF(J20<Cosines!M$22,0,(K20-L20)^2)","v":0}]
[-4.920000000000007,0.97852612994113708,-5.1248442393600762,{"$f":"IF(A21<Sines!M$22,0,(B21-C21)^2)","v":0},null,"quintic",{"$f":"IF(Sines!E23>0,\"+  (1/\"&TEXT(Sines!E23,\"#\") &\")x^5\",IF(Sines!E23<0,\" - (1/\"&TEXT(-1*Sines!E23,\"#\")&\")x^5 \",\" + 0x^5\"))","v":"+  (1/120)x^5"},null,null,-4.8600000000000074,0.14707555357187013,-74.054059437161527,{"$f":"IF(J21<Cosines!M$22,0,(K21-L21)^2)","v":0},null,"quintic",{"$f":"IF(Cosines!E23>0,\"+  (1/\"&TEXT(Cosines!E23,\"#\") &\")x^5\",IF(Cosines!E23<0,\" - (1/\"&TEXT(-1*Cosines!E23,\"#\")&\")x^5 \",\" + 0x^5\"))","v":"+  (1/30)x^5"}]
[-4.8600000000000074,0.98912526079436869,-4.4961117184800727,{"$f":"IF(A22<Sines!M$22,0,(B22-C22)^2)","v":0},null,null,null,null,null,-4.8000000000000078,8.749898343945435E-2,-69.032410224390858,{"$f":"IF(J22<Cosines!M$22,0,(K22-L22)^2)","v":0}]
[-4.8000000000000078,0.99616460883584002,-3.915264000000068,{"$f":"IF(A23<Sines!M$22,0,(B23-C23)^2)","v":0},null,"6th degree",{"$f":"IF(Sines!E25>0,\"+  (1/\"&TEXT(Sines!E25,\"#\") &\")x^6\",IF(Sines!E25<0,\" - (1/\"&TEXT(-1*Sines!E25,\"#\")&\")x^6 \",\" + 0x^6\"))","v":" + 0x^6"},null,null,-4.7400000000000082,2.760751145421951E-2,-64.285719574992569,{"$f":"IF(J23<Cosines!M$22,0,(K23-L23)^2)","v":0},null,"6th degree",{"$f":"IF(Cosines!E25>0,\"+  (1/\"&TEXT(Cosines!E25,\"#\") &\")x^6\",IF(Cosines!E25<0,\" - (1/\"&TEXT(-1*Cosines!E25,\"#\")&\")x^6 \",\" + 0x^6\"))","v":" - (1/820)x^6 "}]
[-4.7400000000000082,0.99961884001418522,-3.3800406655200739,{"$f":"IF(A24<Sines!M$22,0,(B24-C24)^2)","v":0},null,null,null,null,null,-4.6800000000000086,-3.2383317759715854E-2,-59.802272271299444,{"$f":"IF(J24<Cosines!M$22,0,(K24-L24)^2)","v":0}]
[-4.6800000000000086,0.99947552282728425,-2.8882435046400676,{"$f":"IF(A25<Sines!M$22,0,(B25-C25)^2)","v":0},null,null,null,null,null,-4.620000000000009,-9.2257601999502808E-2,-55.570707219236198,{"$f":"IF(J25<Cosines!M$22,0,(K25-L25)^2)","v":0}]
[-4.620000000000009,0.99573517306224613,-2.437735737360061,{"$f":"IF(A26<Sines!M$22,0,(B26-C26)^2)","v":0},null,"sum of squares =",null,null,null,-4.5600000000000094,-0.1517998584983459,-51.580011122069216,{"$f":"IF(J26<Cosines!M$22,0,(K26-L26)^2)","v":0},null,"sum of squares ="]
[-4.5600000000000094,0.98841125193913193,-2.0264412364800606,{"$f":"IF(A27<Sines!M$22,0,(B27-C27)^2)","v":0},null,null,{"$f":"SUM(D2:D203)","v":9.3188483244324036},null,null,-4.5000000000000098,-0.21079579943077015,-47.819512195122542,{"$f":"IF(J27<Cosines!M$22,0,(K27-L27)^2)","v":0},null,null,{"$f":"SUM(M2:M203)","v":770.60275190390075}]
[-4.5000000000000098,0.97753011766509912,-1.6523437500000586,{"$f":"IF(A28<Sines!M$22,0,(B28-C28)^2)","v":0},null,null,null,null,null,-4.4400000000000102,-0.26903310311738965,-44.278873921460416,{"$f":"IF(J28<Cosines!M$22,0,(K28-L28)^2)","v":0}]
[-4.4400000000000102,0.96313093057331933,-1.3134861235200539,{"$f":"IF(A29<Sines!M$22,0,(B29-C29)^2)","v":0},null,null,null,null,null,-4.3800000000000106,-0.32630217815367346,-40.948088848535676,{"$f":"IF(J29<Cosines!M$22,0,(K29-L29)^2)","v":0}]
[-4.3800000000000106,0.94526551218806687,-1.0079695226400478,{"$f":"IF(A30<Sines!M$22,0,(B30-C30)^2)","v":0},null,null,null,null,null,-4.3200000000000109,-0.38239691771267043,-37.817472425804731,{"$f":"IF(J30<Cosines!M$22,0,(K30-L30)^2)","v":0}]
[-4.3200000000000109,0.92399815872319202,-0.7339526553600475,{"$f":"IF(A31<Sines!M$22,0,(B31-C31)^2)","v":0},null,null,null,null,null,-4.2600000000000113,-0.43711544130701746,-34.877656883308546,{"$f":"IF(J31<Cosines!M$22,0,(K31-L31)^2)","v":0}]
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