{"text_format":"3","workbook":"xlsx","n_sheets":5,"sheets":["Sine -- Radians","Sine -- Degrees","4 Points - Radians","4 Points - Degrees","Calculations"],"not_rendered":{"charts":8,"images":10,"defined_names":68,"comments":4},"read":"whole","date_system":"1900"}

{"sheet":"Sine -- Radians","index":0,"n_rows":38,"n_cols":20,"rows_kept":38}
[null,"    Approximating the Sine Function"]
[null,"with Newton Interpolation Polynomials"]
[]
[null,"This DIGMath module lets you investigate how"]
[null,"  Newton quadratic Interpolation Polynomials"]
[null,"                    are used to approximate "]
[null,"                 the sine function y = sin x"]
[null,"                with x in radians on [0, p/4]."]
[]
["The three interpolating points are x = x0,  p/8,  and x2."]
[null,"Since Newton's formula requires that the points be uniformly spaced,"]
[null,"the third point x2 is determined by the first point x0 and the center point."]
[]
[null,"What is the first point x0 (0 to 0.35)?",null,null,{"$f":"F15/100","v":0}]
[{"$nulls":5},0,127,{"$nulls":5},"Now trace along the curve:  x = ",{"$f":"O16*dx0","v":0.24818581963359368},"  and  y = ",{"$f":"LOOKUP(N15,Calculations!D2:D502,Calculations!E2:E502)","v":{"v":0.24412280216829127,"d":"0.24"}}]
[{"$nulls":14},158,{"$nulls":4}," "]
[null,"Then the third interpolating point x2 is",null,null,{"$f":"PI()/4-x0","v":0.7853981633974483}]
[{"$nulls":12},"The corresponding error = ",{"$f":"Calculations!B13","v":{"v":3.0076371955506664E-3,"d":"0.0030"}}]
[]
[]
[]
[]
["Click each item below for suggestions and investigations"]
[{"$nulls":3},"   Item 1"]
[{"$nulls":3},"   Item 2"]
[{"$nulls":3},"   Item 3"]
[{"$nulls":3},"   Item 4"]
[{"$nulls":3},"   Item 5"]
[]
[]
["To access/download all the DIGMath Apps for College Algebra/Precalculus"]
["Click this link:",null,"DIGMath Modules for College Algebra and Precalculus"]
[]
["Created by:  Sheldon P. Gordon   &   Florence S. Gordon",{"$nulls":11},"The maximum error (in absolute value) =",{"$f":"Calculations!B9","v":{"v":0.003636273200089135,"d":"0.00364"}}]
["                  Farmingdale State College      NYIT"]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Sine -- Degrees","index":1,"n_rows":38,"n_cols":17,"rows_kept":38}
[null,"    Approximating the Sine Function"]
[null,"with Newton Interpolation Polynomials"]
[]
[null,"This DIGMath module lets you investigate how"]
[null,"  Newton quadratic Interpolation Polynomials"]
[null,"                    are used to approximate "]
[null,"                 the sine function y = sin x"]
[null,"              with x in degrees on [0°, 45°]."]
[]
["The three interpolating points are x = x0,  22.5°,  and x2."]
[null,"Since Newton's formula requires that the points be uniformly spaced,"]
[null,"the third point x2 is determined by the first point x0 and the center point."]
[]
[null,"What is the first point x0 (0 to 20°)?",null,null,{"$f":"F15/10","v":4.3},"°"]
[{"$nulls":5},43,127,{"$nulls":6},"Now trace along the curve:  x = ",{"$f":"P16*dxx0","v":39.5},"and  y = ",{"$f":"LOOKUP(O15,Calculations!M2:M452,Calculations!N2:N452)","v":{"v":0.6347305132022715,"d":"0.63"}}]
[{"$nulls":15},395]
[null,"Then the third interpolating point x2 is",null,null,{"$f":"45-xx0","v":40.7},"°"]
[{"$nulls":13},"The corresponding error = ",{"$f":"Calculations!K13","v":{"v":-6.0622329553217114E-4,"d":"-0.001"}}]
[]
[]
[]
[]
["Click each item below for suggestions and investigations"]
[{"$nulls":3},"   Item 1"]
[{"$nulls":3},"   Item 2"]
[{"$nulls":3},"   Item 3"]
[{"$nulls":3},"   Item 4"]
[{"$nulls":3},"   Item 5"]
[]
[]
["To access/download all the DIGMath Apps for College Algebra/Precalculus"]
["Click this link:",null,"DIGMath Modules for College Algebra and Precalculus"]
[]
["Created by:  Sheldon P. Gordon   &   Florence S. Gordon",{"$nulls":12},"The maximum error (in absolute value) =",{"$f":"Calculations!K9","v":{"v":3.3126514530683075E-3,"d":"0.00331"}}]
["                  Farmingdale State College      NYIT"]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"4 Points - Radians","index":2,"n_rows":41,"n_cols":15,"rows_kept":41}
[null,"    Approximating the Sine Function"]
[null,"with Newton Interpolation Polynomials"]
[]
[null,"This DIGMath module lets you investigate how"]
[null,"    Newton cubic Interpolation Polynomials"]
[null,"                    are used to approximate "]
[null,"                 the sine function y = sin x"]
[null,"                with x in radians on [0, p/4]."]
[]
["The four interpolating points are x = x0,  x1,  x2,  and x3."]
[null,"Since Newton's formula requires that the points be uniformly spaced,"]
[null,"the 3rd and 4th points are determined by the first two points x0 and x1."]
[]
[null," What is the first point x0 (0 to 0.20)?",null,null,{"$f":"F15/500","v":{"v":0.058,"d":"0.06"}}]
[{"$nulls":5},29,127,{"$nulls":4},"Now trace along the curve:  x = ",{"$f":"N16*dxxx0","v":0.3157300616857742},"and  y = ",{"$f":"LOOKUP(M15,Calculations!V2:V502,Calculations!W2:W502)","v":{"v":0.3090169943749489,"d":"0.31"}}]
[{"$nulls":13},201]
[null,"What is the second point x1 (0.20 to 0.40)?",null,null,{"$f":"0.2+F18/500","v":0.30000000000000004},{"$f":"IF(xxx0000>PI()/4,\"  That's too big!\",\"\")","v":""}]
[{"$nulls":5},50,127,{"$nulls":4},"The corresponding error = ",{"$f":"Calculations!T13","v":{"v":-6.2624473801120573E-6,"d":"-0.000006"}}]
[]
[null,"The third and fourth interpolating points are:"]
[{"$nulls":3},"x2  =",{"$f":"E$17+(E$17-E$14)","v":{"v":0.542,"d":"0.54"}}]
[{"$nulls":3},"x3  =",{"$f":"E$17+2*(E$17-E$14)","v":{"v":0.7840000000000001,"d":"0.78"}}]
[]
[]
[]
["Click each item below for suggestions and investigations"]
[{"$nulls":3},"   Item 1"]
[{"$nulls":3},"   Item 2"]
[{"$nulls":3},"   Item 3"]
[{"$nulls":3},"   Item 4"]
[{"$nulls":3},"   Item 5"]
[]
[]
["To access/download all the DIGMath Apps for College Algebra/Precalculus",{"$nulls":10},"       The maximum error (in absolute value) =",{"$f":"Calculations!T9","v":{"v":1.0105576380175824E-4,"d":"0.000101"}}]
["Click this link:",null,"DIGMath Modules for College Algebra and Precalculus"]
[]
["Created by:  Sheldon P. Gordon   &   Florence S. Gordon"]
["                  Farmingdale State College      NYIT"]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"4 Points - Degrees","index":3,"n_rows":41,"n_cols":16,"rows_kept":41}
[null,"    Approximating the Sine Function"]
[null,"with Newton Interpolation Polynomials"]
[]
[null,"   This DIGMath module lets you investigate how"]
[null,"       Newton cubic Interpolation Polynomials"]
[null,"                    are used to approximate "]
[null,"                 the sine function y = sin x"]
[null,"               with x in degrees on [0°, 45°]."]
[]
["The four interpolating points are x = x0,  x1,  x2,  and x3."]
[null,"Since Newton's formula requires that the points be uniformly spaced,"]
[null,"the 3rd and 4th points are determined by the first two points x0 and x1."]
[]
[null,"What is the first point x0 (0 to 11.25°)?",null,null,{"$f":"G15/4","v":{"v":2,"d":"2.00"}},"°"]
[{"$nulls":6},8,127,{"$nulls":4},"Now trace along the curve:  x = ",{"$f":"O16*dxxxx0","v":28.3},"° and  y = ",{"$f":"LOOKUP(N15,Calculations!AE2:AE452,Calculations!AF2:AF452)","v":{"v":0.472550764869056,"d":"0.47"}}]
[{"$nulls":14},283]
[null,"What is the second point x1 (11.5° to 22.5°)?",null,null,{"$f":"G18/10","v":15.5},"°",{"$f":"IF(xxxx0000>45,\"  That's too big!\",\"\")","v":""}]
[{"$nulls":6},155,127,{"$nulls":4},"The corresponding error = ",{"$f":"Calculations!AC13","v":{"v":-5.826883185899856E-6,"d":"-0.000006"}}]
[]
[null,"The third and fourth interpolating points are:"]
[{"$nulls":3},"x2  =",{"$f":"E$17+(E$17-E$14)","v":{"v":29,"d":"29.00"}},"°"]
[{"$nulls":3},"x3  =",{"$f":"E$17+2*(E$17-E$14)","v":{"v":42.5,"d":"42.50"}},"°"]
[]
[]
[]
["Click each item below for suggestions and investigations"]
[{"$nulls":3},"   Item 1"]
[{"$nulls":3},"   Item 2"]
[{"$nulls":3},"   Item 3"]
[{"$nulls":3},"   Item 4"]
[{"$nulls":3},"   Item 5"]
[]
[]
["To access/download all the DIGMath Apps for College Algebra/Precalculus",{"$nulls":11},"The maximum error (in absolute value) =",{"$f":"Calculations!AC9","v":{"v":8.7868513770295031E-5,"d":"0.000088"}}]
["Click this link:",null,"DIGMath Modules for College Algebra and Precalculus"]
[]
["Created by:  Sheldon P. Gordon   &   Florence S. Gordon"]
["                  Farmingdale State College      NYIT"]
["Development of this module was supported by the "]
["NSF's Division of Undergraduate Education  "]
["under grants DUE-0310123 and DUE-0442160."]

{"sheet":"Calculations","index":4,"n_rows":502,"n_cols":35,"rows_kept":502}
[null,"dx0 =",{"$f":"PI()/4/500","v":1.5707963267948967E-3},"x","Sine","L(x)","Error","ABS(Error)",null,null,"dxx0 =",{"$f":"45/450","v":0.1},"x","Sine","L(x)","Error","ABS(Error)",null,null,"dxxx0 =",{"$f":"PI()/4/500","v":1.5707963267948967E-3},"x","Sine","L3(x)","Error","ABS(Error)",null,null,"dxxxx0 =",{"$f":"45/450","v":0.1},"x","Sine","L3(x)","Error","ABS(Error)"]
[null,"dx =",{"$f":"x00-B4","v":0.39269908169872414},{"$f":"0","v":0},{"$f":"SIN(x)","v":0},{"$f":"y0+dy*(x-x0)/dx+d2y*(x-x0)*(x-x0-dx)/2/dx^2","v":0},{"$f":"F2-E2","v":0},{"$f":"ABS(G2)","v":0},null,null,"dxx =",{"$f":"K5-K4","v":18.2},0,{"$f":"SIN(xx*PI()/180)","v":0},{"$f":"yy0+dyy*(xx-xx0)/dxx+d2yy*(xx-xx0)*(xx-xx0-dxx)/2/dxx^2","v":-3.3126514530683075E-3},{"$f":"O2-N2","v":-3.3126514530683075E-3},{"$f":"ABS(P2)","v":3.3126514530683075E-3},null,null,"dxxx =",{"$f":"T5-T4","v":0.24200000000000005},{"$f":"0","v":0},{"$f":"SIN(xxx)","v":0},{"$f":"yyy0+dyyy0*(xxx-xxx0)/dxxx+d2yyy0*(xxx-xxx0)*(xxx-xxx0-dxxx)/2/dxxx^2+d3yyy0*(xxx-xxx0)*(xxx-xxx0-dxxx)*(xxx-xxx0-2*dxxx)/6/dxxx^3","v":-1.0105576380175824E-4},{"$f":"X2-W2","v":-1.0105576380175824E-4},{"$f":"ABS(Y2)","v":1.0105576380175824E-4},null,null,"dxxxx =",{"$f":"AC5-AC4","v":13.5},0,{"$f":"SIN(xxxx*PI()/180)","v":0},{"$f":"yyyy0+dyyyy0*(xxxx-xxxx0)/dxxxx+d2yyyy0*(xxxx-xxxx0)*(xxxx-xxxx0-dxxxx)/2/dxxxx^2+d3yyyy0*(xxxx-xxxx0)*(xxxx-xxxx0-dxxxx)*(xxxx-xxxx0-2*dxxxx)/6/dxxxx^3","v":-4.4851129816778253E-5},{"$f":"AG2-AF2","v":-4.4851129816778253E-5},{"$f":"ABS(AH2)","v":4.4851129816778253E-5}]
[null,"x","y",{"$f":"D2+dx0","v":1.5707963267948967E-3},0.001570795680830879,1.6467878158792743E-3,{"$f":"F3-E3","v":7.5992135048395413E-5},{"$f":"ABS(G3)","v":7.5992135048395413E-5},null,null,"xx","yy",{"$f":"M2+dxx0","v":0.1},1.7453283658983088E-3,-1.46764674630398E-3,{"$f":"O3-N3","v":-3.212975112202289E-3},{"$f":"ABS(P3)","v":3.212975112202289E-3},null,null,"x","y",{"$f":"V2+dx0","v":1.5707963267948967E-3},0.001570795680830879,1.4733814181448622E-3,{"$f":"X3-W3","v":-9.74142626860167e-05},{"$f":"ABS(Y3)","v":9.74142626860167e-05},null,null,"xx","yy",{"$f":"AE2+dxxxx0","v":0.1},1.7453283658983088E-3,0.001703193421803897,{"$f":"AG3-AF3","v":-4.21349440944119e-05},{"$f":"ABS(AH3)","v":4.21349440944119e-05}]
[1,{"$f":"x0","v":0},{"$f":"SIN(x0)","v":0},3.1415926535897933E-3,3.1415874858795635E-3,3.2926434704218504E-3,1.5105598454228687E-4,1.5105598454228687E-4,null,1,{"$f":"xx0","v":4.3},{"$f":"SIN(xx0*PI()/180)","v":0.0749787268263277},0.2,0.003490651415223732,3.7620200906338629E-4,-0.003114449406160346,0.003114449406160346,null,1,{"$f":"xxx0","v":0.058},{"$f":"SIN(xxx0)","v":0.057967486802534995},3.1415926535897933E-3,3.1415874858795635E-3,3.0477601281730946E-3,-9.3827357706468932E-5,9.3827357706468932E-5,null,1,{"$f":"xxxx0","v":2},{"$f":"SIN(xxxx0*PI()/180)","v":0.03489949670250097},0.2,0.003490651415223732,0.003451180223923781,-3.9471191299951264E-5,3.9471191299951264E-5]
[2,{"$f":"PI()/8","v":0.39269908169872414},{"$f":"SIN(B5)","v":0.3826834323650898},0.00471238898038469,0.004712371539373423,0.004937566963627729,2.2519542425430625E-4,2.2519542425430625E-4,null,2,22.5,{"$f":"SIN(K5*PI()/180)","v":0.3826834323650898},0.30000000000000004,5.2359638314195805E-3,2.2188948130337444E-3,-0.003017069018385836,0.003017069018385836,null,2,{"$f":"xxx00","v":0.30000000000000004},{"$f":"SIN(xxx00)","v":0.2955202066613396},0.00471238898038469,0.004712371539373423,0.004622076854742253,-9.0294684631169776E-5,9.0294684631169776E-5,null,2,{"$f":"xxxx00","v":15.5},{"$f":"SIN(AC5*PI()/180)","v":0.26723837607825685},0.30000000000000004,5.2359638314195805E-3,0.005199104389880645,-3.6859441538935819E-5,3.6859441538935819E-5]
[3,{"$f":"x000","v":0.7853981633974483},{"$f":"SIN(x000)","v":0.7071067811865475},0.006283185307179587,0.006283143965558951,6.5815582954969086E-3,2.9841432993795748E-4,2.9841432993795748E-4,null,3,{"$f":"xx000","v":40.7},{"$f":"SIN(K6*PI()/180)","v":0.6520984038303922},0.4,6.9812602979615525E-3,0.004060431665607143,-2.9208286323544096E-3,2.9208286323544096E-3,null,3,{"$f":"xxx000","v":0.542},{"$f":"SIN(xxx000)","v":0.5158503796005889},0.006283185307179587,0.006283143965558951,0.006196328086311682,-8.6815879247269245E-5,8.6815879247269245E-5,null,3,{"$f":"xxxx000","v":29},{"$f":"SIN(AC6*PI()/180)","v":0.48480962024633706},0.4,6.9812602979615525E-3,0.006946961033012246,-3.4299264949306206E-5,3.4299264949306206E-5]
[{"$nulls":3},0.007853981633974483,0.007853900888711334,0.008224617466029392,3.7071657731805804E-4,3.7071657731805804E-4,{"$nulls":4},0.5,0.008726535498373935,0.005900812566783563,-0.002825722931590372,0.002825722931590372,null,4,{"$f":"xxx0000","v":0.7840000000000001},{"$f":"SIN(xxx0000)","v":0.7061174395415357},0.007853981633974483,0.007853900888711334,0.007770510311340686,-8.3390577370647846E-5,8.3390577370647846E-5,null,4,{"$f":"xxxx0000","v":42.5},{"$f":"SIN(AC7*PI()/180)","v":0.6755902076156602},0.5,0.008726535498373935,0.008694745266656364,-3.1790231717570769E-5,3.1790231717570769E-5]
[{"$nulls":3},0.00942477796076938,0.009424638433144006,0.009866744475225175,4.4210604208116955E-4,4.4210604208116955E-4,{"$nulls":4},0.6,0.010471784116245792,0.007740037516563006,-2.7317465996827862E-3,2.7317465996827862E-3,{"$nulls":4},0.00942477796076938,0.009424638433144006,0.00934462001828859,-8.0018414855415054E-5,8.0018414855415054E-5,{"$nulls":4},0.6,0.010471784116245792,0.010442452204150757,-2.9331912095035412E-5,2.9331912095035412E-5]
["MaxError =",{"$f":"MAX(H2:H502)","v":0.003636273200089135},null,0.010995574287564275,0.010995352723218221,0.011507939323084259,5.1258659986603769E-4,5.1258659986603769E-4,null,"MaxError =",{"$f":"MAX(Q2:Q452)","v":3.3126514530683075E-3},null,0.7,0.012217000835247169,0.009578106514945475,-0.002638894320301694,0.002638894320301694,null,"MaxError =",{"$f":"MAX(Z2:Z502)","v":1.0105576380175824E-4},null,0.010995574287564275,0.010995352723218221,0.010918653695614736,-7.6699027603484987E-5,7.6699027603484987E-5,null,"MaxError =",{"$f":"MAX(AI2:AI452)","v":8.7868513770295031E-5},null,0.7,0.012217000835247169,0.012190076958833191,-2.6923876413977726E-5,2.6923876413977726E-5]
[{"$nulls":3},0.012566370614359171,0.012566039883352606,0.013148202009606645,5.8216212625403913E-4,5.8216212625403913E-4,{"$nulls":4},0.7999999999999999,0.013962180339145272,0.011415019561930955,-0.002547160777214317,0.002547160777214317,{"$nulls":4},0.012566370614359171,0.012566039883352606,0.012492607831778434,-7.3432051574172028E-5,7.3432051574172028E-5,{"$nulls":4},0.7999999999999999,0.013962180339145272,0.013937614644041434,-2.4565695103838034E-5,2.4565695103838034E-5]
[{"$nulls":3},0.014137166941154067,0.014136696038032732,0.014787532534792334,6.5083649675960233E-4,6.5083649675960233E-4,{"$nulls":4},0.8999999999999999,0.015707317311820672,0.013250776657519468,-2.4565406543012036E-3,2.4565406543012036E-3,{"$nulls":4},0.014137166941154067,0.014136696038032732,0.014066478915239024,-7.0217122793708384E-5,7.0217122793708384E-5,{"$nulls":4},0.8999999999999999,0.015707317311820672,0.015685060373113267,-2.2256938707405222E-5,2.2256938707405222E-5]
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