{"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":14,"defined_names":58,"comments":4},"read":"whole","date_system":"1900"}

{"sheet":"Sine -- Radians","index":0,"n_rows":37,"n_cols":19,"rows_kept":37}
[null,"   Approximating Sinusoidal Functions"]
[null,"with Lagrange Interpolation Polynomials"]
[]
[null,"This DIGMath module lets you investigate how"]
[null,"Lagrange 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,"What is the first point x0 (0 to 0.35)?",null,null,{"$f":"F13/100","v":0.06}]
[{"$nulls":5},6,127]
[]
[null,"What is the third point x2 (0.40 to 0.78)?",null,null,{"$f":"F16/100","v":0.59},{"$nulls":6},"Now trace along the curve:  x = ",{"$f":"xMin+N16*dx","v":0.3832743037379548},"and  y = ",{"$f":"LOOKUP(M15+0.001,Calculations!D2:D502,Calculations!E2:E502)","v":{"v":0.3739592057378024,"d":"0.37"}}]
[{"$nulls":5},59,127,{"$nulls":6},244,{"$nulls":4}," "]
[]
[{"$nulls":11},"The corresponding error = ",{"$f":"Calculations!B13","v":{"v":1.1468126858238392E-4,"d":"0.0001"}}]
[]
[]
["Click each item below for suggestions and investigations"]
[{"$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"]
["                  Farmingdale State College      NYIT",{"$nulls":10},"The maximum error (in absolute value) =",{"$f":"Calculations!B9","v":{"v":0.008264979902445457,"d":"0.00826"}}]
["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":16,"rows_kept":38}
[null,"   Approximating Sinusoidal Functions"]
[null,"with Lagrange Interpolation Polynomials"]
[]
[null,"This DIGMath module lets you investigate how"]
[null,"Lagrange quadratic Interpolation Polynomials"]
[null,"                 are used to approximate "]
[null,"               the sine function y = sin x"]
[null,"             with x in degrees on [0°, 45°]."]
[null,"  "]
["The three interpolating points are x = x0,  22.5°,  and x2."]
[]
[null,"What is the first point x0 (0 to 20°)?",null,null,{"$f":"F13/10","v":11},"°"]
[{"$nulls":5},110,127]
[]
[null,"What is the third point x2 (25° to 45°)?",null,null,{"$f":"F16/10","v":35.2},"°",{"$nulls":6},"Now trace along the curve:  x = ",{"$f":"O16/10","v":25.3},"° and  y = ",{"$f":"LOOKUP(N15+0.001,Calculations!M2:M452,Calculations!N2:N452)","v":{"v":0.4273578633871938,"d":"0.43"}}]
[{"$nulls":5},352,127,{"$nulls":7},253]
[]
[{"$nulls":12},"The corresponding error = ",{"$f":"Calculations!K13","v":{"v":-3.1090646340053985E-4,"d":"-0.0003"}}]
[]
[]
[]
[]
["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!K9","v":{"v":0.007336988716913739,"d":"0.00734"}}]
["                  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":42,"n_cols":15,"rows_kept":42}
[null,"   Approximating Sinusoidal Functions"]
[null,"with Lagrange Interpolation Polynomials"]
[]
[null,"This DIGMath module lets you investigate how"]
[null,"     Lagrange 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,"What is the first point x0 (0 to 0.20)?",null,null,{"$f":"G13/980","v":{"v":0.2,"d":"0.20"}}]
[{"$nulls":5},29,196]
[]
[null,"What is the second point x1 (0.20 to 0.40)?",null,null,{"$f":"G16/980+0.196","v":{"v":0.196,"d":"0.20"}},"",{"$nulls":5},"Now trace along the curve:  x = ",{"$f":"O16*dxxx0","v":{"v":0.7853981633974483,"d":"0.785"}},"and  y = ",{"$f":"LOOKUP(M15+0.00001,Calculations!V2:V502,Calculations!W2:W502)","v":{"v":0.7071067811865515,"d":"0.707"}}]
[{"$nulls":5},50,0,{"$nulls":7},500]
[]
[null,"What is the third point x2 (0.39 to 0.59)?",null,null,{"$f":"G19/980+0.196*2","v":{"v":0.392,"d":"0.39"}},"",{"$nulls":5},"The corresponding error = ",{"$f":"Calculations!T13","v":{"v":-4.3788409126443995E-6,"d":"-0.000004"}}]
[{"$nulls":5},50,0]
[]
[null,"What is the fourth point x3 (0.59 to 0.79)?",null,null,{"$f":"G22/980+0.196*3","v":{"v":0.5880000000000001,"d":"0.59"}},""]
[{"$nulls":5},50,0]
[]
[]
[]
["Click each item below for suggestions and investigations"]
[{"$nulls":3},"   Item 1"]
[{"$nulls":3},"   Item 2"]
[{"$nulls":3},"   Item 3"]
[{"$nulls":3},"   Item 4"]
[{"$nulls":3},"   Item 5"]
[]
[]
[{"$nulls":11},"The maximum error (in absolute value) =",{"$f":"Calculations!T9","v":{"v":2.8867100836271753E-4,"d":"0.000289"}}]
["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"]
["                  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":15,"rows_kept":41}
[null,"   Approximating Sinusoidal Functions"]
[null,"with Lagrange Interpolation Polynomials"]
[]
[null,"This DIGMath module lets you investigate how"]
[null,"   Lagrange 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,"What is the first point x0 (0 to 11°)?",null,null,{"$f":"F13/10","v":7.3},"°"]
[{"$nulls":4},29,73]
[]
[null,"What is the second point x1 (11.5° to 22.5°)?",null,null,{"$f":"F16/10+11.5","v":18.4},"°",{"$nulls":5},"Now trace along the curve:  x = ",{"$f":"N16*dxxxx0","v":{"v":25.3,"d":"25.30"}},"° and  y = ",{"$f":"LOOKUP(M15+0.00001,Calculations!AE2:AE452,Calculations!AF2:AF502)","v":{"v":0.4273578633871938,"d":"0.427"}}]
[{"$nulls":4},50,69,{"$nulls":7},253]
[]
[null,"What is the third point x2 (23° to 34°)?",null,null,{"$f":"F19/10+11.5*2","v":34},"°",{"$nulls":5},"The corresponding error = ",{"$f":"Calculations!AC13","v":{"v":-5.1529636473923723E-5,"d":"-0.000052"}}]
[{"$nulls":4},50,110]
[]
[null,"What is the fourth point x3 (34.5° to 45°)?",null,null,{"$f":"F22/10+11.5*3","v":41.5},"°"]
[{"$nulls":4},50,70]
[]
[]
[]
["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!AC9","v":{"v":1.7392530246373994E-4,"d":"0.000174"}}]
["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,"dx =",{"$f":"PI()/4/500","v":1.5707963267948967E-3},"x","Sine","L2(x)","Error","ABS(Error)",null,null,"dxx =",{"$f":"45/450","v":0.1},"x","Sine","L2(x)","Error","ABS(Error)",null,null,"dxxx0 =",{"$f":"PI()/4/500","v":1.5707963267948967E-3},"x","Sine","L3(x)","Error","ABS(Error)",null,null,"dxxxx0 =",0.1,"x","Sine","L3(x)","Error","ABS(Error)"]
[{"$nulls":3},{"$f":"0","v":0},{"$f":"SIN(x)","v":0},{"$f":"y0*(x-x00)*(x-x000)/(x0-x00)/(x0-x000)+y00*(x-x0)*(x-x000)/(x00-x0)/(x00-x000)+y000*(x-x0)*(x-x00)/(x000-x0)/(x000-x00)","v":-2.2255987689207835E-3},{"$f":"F2-E2","v":-2.2255987689207835E-3},{"$f":"ABS(G2)","v":2.2255987689207835E-3},{"$nulls":4},{"$f":"0","v":0},{"$f":"SIN(xx*PI()/180)","v":0},{"$f":"yy0*(xx-xx00)*(xx-xx000)/(xx0-xx00)/(xx0-xx000)+yy00*(xx-xx0)*(xx-xx000)/(xx00-xx0)/(xx00-xx000)+yy000*(xx-xx0)*(xx-xx00)/(xx000-xx0)/(xx000-xx00)","v":-0.007336988716913739},{"$f":"O2-N2","v":-0.007336988716913739},{"$f":"ABS(P2)","v":0.007336988716913739},null,null,"dxxx =",{"$f":"T5-T4","v":-4.0000000000000036E-3},0,{"$f":"SIN(xxx)","v":0},{"$f":"yyy0*(xxx-xxx00)*(xxx-xxx000)*(xxx-xxx0000)/(xxx0-xxx00)/(xxx0-xxx000)/(xxx0-xxx0000)+yyy00*(xxx-xxx0)*(xxx-xxx000)*(xxx-xxx0000)/(xxx00-xxx0)/(xxx00-xxx000)/(xxx00-xxx0000)+yyy000*(xxx-xxx0)*(xxx-xxx00)*(xxx-xxx0000)/(xxx000-xxx0)/(xxx000-xxx00)/(xxx000-xxx0000)+yyy0000*(xxx-xxx0)*(xxx-xxx00)*(xxx-xxx000)/(xxx0000-xxx0)/(xxx0000-xxx00)/(xxx0000-xxx000)","v":-1.0195857795031182E-4},-2.8867100836271753E-4,2.8867100836271753E-4,null,null,"dxxxx =",13.5,0,{"$f":"SIN(xxxx*PI()/180)","v":0},{"$f":"yyyy0*(xxxx-xxxx00)*(xxxx-xxxx000)*(xxxx-xxxx0000)/(xxxx0-xxxx00)/(xxxx0-xxxx000)/(xxxx0-xxxx0000)+yyyy00*(xxxx-xxxx0)*(xxxx-xxxx000)*(xxxx-xxxx0000)/(xxxx00-xxxx0)/(xxxx00-xxxx000)/(xxxx00-xxxx0000)+yyyy000*(xxxx-xxxx0)*(xxxx-xxxx00)*(xxxx-xxxx0000)/(xxxx000-xxxx0)/(xxxx000-xxxx00)/(xxxx000-xxxx0000)+yyyy0000*(xxxx-xxxx0)*(xxxx-xxxx00)*(xxxx-xxxx000)/(xxxx0000-xxxx0)/(xxxx0000-xxxx00)/(xxxx0000-xxxx000)","v":-2.5192469811829987E-4},-1.3972281566543288E-4,1.3972281566543288E-4]
[null,"x","y",{"$f":"D2+dx","v":1.5707963267948967E-3},0.001570795680830879,-5.8193900518380137E-4,{"$f":"F3-E3","v":-2.1527346860146803E-3},{"$f":"ABS(G3)","v":2.1527346860146803E-3},null,null,"x","y",{"$f":"M2+dxx","v":0.1},1.7453283658983088E-3,-0.00547130132138024,{"$f":"O3-N3","v":-0.007216629687278549},{"$f":"ABS(P3)","v":0.007216629687278549},null,null,"x","y",{"$f":"V2+dxxx0","v":1.5707963267948967E-3},0.001570795680830879,1.4710054441895437E-3,-2.7798897286768209E-4,2.7798897286768209E-4,null,null,"xxxx","yyyy",0.1,1.7453283658983088E-3,1.4992881835167449E-3,-1.3110398558335845E-4,1.3110398558335845E-4]
[1,{"$f":"x0","v":0.06},{"$f":"SIN(x0)","v":0.059964006479444595},3.1415926535897933E-3,3.1415874858795635E-3,1.0608852273145813E-3,-0.002080702258564982,0.002080702258564982,null,1,{"$f":"xx0","v":11},{"$f":"SIN(K4*PI()/180)","v":0.1908089953765448},0.2,0.003490651415223732,-3.6067948481843803E-3,-7.0974462634081125E-3,7.0974462634081125E-3,null,1,{"$f":"xxx0","v":0.2},{"$f":"SIN(T4)","v":0.19866933079506122},3.1415926535897933E-3,3.1415874858795635E-3,3.0439343879946623E-3,-2.6748398509035543E-4,2.6748398509035543E-4,null,1,{"$f":"xxxx0","v":7.3},{"$f":"SIN(AC4*PI()/180)","v":0.12706460860135046},0.2,0.003490651415223732,3.2504155746252428E-3,-1.2266869253252377E-4,1.2266869253252377E-4]
[2,{"$f":"PI()/8","v":0.39269908169872414},{"$f":"SIN(B5)","v":0.3826834323650898},0.00471238898038469,0.004712371539373423,2.7028739285744896E-3,-2.009497610798933E-3,2.009497610798933E-3,null,2,{"$f":"22.5","v":22.5},{"$f":"SIN(K5*PI()/180)","v":0.3826834323650898},0.30000000000000004,5.2359638314195805E-3,-1.7434692973267718E-3,-0.006979433128746352,0.006979433128746352,null,2,{"$f":"xxx00","v":0.196},{"$f":"SIN(T5)","v":0.19474748558520416},0.00471238898038469,0.004712371539373423,4.616824614242343E-3,-2.5715448179097766E-4,2.5715448179097766E-4,null,2,{"$f":"xxxx00","v":18.4},{"$f":"SIN(AC5*PI()/180)","v":0.3156490369471024},0.3,0.00523596383141958,0.005001452694546216,-1.1441495124453471E-4,1.1441495124453471E-4]
[3,{"$f":"x000","v":0.59},{"$f":"SIN(x000)","v":0.5563610229127838},0.006283185307179587,0.006283143965558951,0.004344027098595785,-1.9391168669631665E-3,1.9391168669631665E-3,null,3,{"$f":"xx000","v":35.2},{"$f":"SIN(K6*PI()/180)","v":0.5764323161697933},0.4,6.9812602979615525E-3,1.1867533119302998E-4,-0.006862584966768523,0.006862584966768523,null,3,{"$f":"xxx000","v":0.392},{"$f":"SIN(T6)","v":0.38203747163308743},0.006283185307179587,0.006283143965558951,0.006189672483721986,-2.4699890581847406E-4,2.4699890581847406E-4,null,3,{"$f":"xxxx000","v":34},{"$f":"SIN(AC6*PI()/180)","v":0.5591929034707469},0.4,6.9812602979615525E-3,0.006752394762620462,-1.0634078573168448E-4,1.0634078573168448E-4]
[{"$nulls":3},0.007853981633974483,0.007853900888711334,0.005984344737378439,-1.8695561513328955E-3,1.8695561513328955E-3,{"$nulls":4},0.5,0.008726535498373935,1.9796390373756356E-3,-0.006746896460998299,0.006746896460998299,null,4,{"$f":"xxx0000","v":0.5880000000000001},{"$f":"SIN(T7)","v":0.5546980299408293},0.007853981633974483,0.007853900888711334,7.7624743571998445E-3,-2.3701570610967782E-4,2.3701570610967782E-4,null,4,{"$f":"xxxx0000","v":41.5},{"$f":"SIN(AC7*PI()/180)","v":0.6626200482157375},0.5,0.008726535498373935,0.008503236998186059,-9.8444229284289086E-5,9.8444229284289086E-5]
[{"$nulls":3},0.00942477796076938,0.009424638433144006,0.00762382684492266,-1.800811588221346E-3,1.800811588221346E-3,{"$nulls":4},0.6,0.010471784116245792,3.8394218212196574E-3,-0.006632362295026135,0.006632362295026135,{"$nulls":4},0.00942477796076938,0.009424638433144006,0.00933522659544711,-2.2720333768921908E-4,2.2720333768921908E-4,{"$nulls":4},0.6,0.010471784116245792,0.010253974620583417,-9.072332447179754E-5,9.072332447179754E-5]
["MaxError =",{"$f":"MAX(H2:H502)","v":0.008264979902445457},null,0.010995574287564275,0.010995352723218221,9.26247342122824E-3,-1.7328793019899813E-3,1.7328793019899813E-3,null,"MaxError =",{"$f":"MAX(Q2:Q452)","v":0.007336988716913739},null,0.7,0.012217000835247169,0.00569802368272615,-0.006518977152521019,0.006518977152521019,null,"MaxError =",{"$f":"MAX(Z2:Z502)","v":2.8867100836271753E-4},null,0.010995574287564275,0.010995352723218221,0.010907925559247938,-2.1756026166941389E-4,2.1756026166941389E-4,null,"MaxError =",{"$f":"MAX(AI2:AL452)","v":1.7392530246373994E-4},null,0.7,0.012217000835247169,0.012004602849152834,-8.3176123142680858E-5,8.3176123142680858E-5]
[{"$nulls":3},0.012566370614359171,0.012566039883352606,0.010900284466295318,-1.6657554170572877E-3,1.6657554170572877E-3,{"$nulls":4},0.7999999999999999,0.013962180339145272,0.007555444621895113,-0.006406735717250159,0.006406735717250159,{"$nulls":4},0.012566370614359171,0.012566039883352606,0.012480567609366222,-2.080849452511524E-4,2.080849452511524E-4,{"$nulls":4},0.8,0.013962180339145272,0.013755116903232112,-7.5800686424098984E-5,7.5800686424098984E-5]
[{"$nulls":3},0.014137166941154067,0.014136696038032732,0.012537259980123727,-0.001599436057909005,0.001599436057909005,{"$nulls":4},0.8999999999999999,0.015707317311820672,0.009411684638726048,-0.006295632673094624,0.006295632673094624,{"$nulls":4},0.014137166941154067,0.014136696038032732,0.01405314910658298,-1.9877586172223349E-4,1.9877586172223349E-4,{"$nulls":4},0.9,0.015707317311820675,0.015505512002161825,-6.8595084721234656E-5,6.8595084721234656E-5]
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