{"text_format":"3","workbook":"xlsx","n_sheets":5,"sheets":["Cosine -- Radians","Cosine -- 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":"Cosine -- Radians","index":0,"n_rows":39,"n_cols":19,"rows_kept":39}
[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 cosine function y = cos 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,{"$f":"F13/100","v":0.11}]
[{"$nulls":5},11,127]
[]
[null,"What is the third point x2 (0.40 to 0.78)?",null,null,{"$f":"F16/100","v":0.74},{"$nulls":6},"Now trace along the curve:  x = ",{"$f":"xMin+N16*dx","v":0.5434955290710343},"and  y = ",{"$f":"LOOKUP(M15+0.001,Calculations!D2:D502,Calculations!E2:E502)","v":{"v":0.8559062676711312,"d":"0.86"}}]
[{"$nulls":5},74,127,{"$nulls":6},346,{"$nulls":4}," "]
[]
[{"$nulls":11},"The corresponding error = ",{"$f":"Calculations!B13","v":{"v":9.133702035041269E-4,"d":"0.0009"}}]
[]
[]
[]
[]
["Click each item below for suggestions and investigations"]
[{"$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"]
[{"$nulls":11},"The maximum error (in absolute value) =",{"$f":"Calculations!B9","v":{"v":1.6147752978665775E-3,"d":"0.00161"}}]
["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":"Cosine -- Degrees","index":1,"n_rows":37,"n_cols":16,"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 cosine function y = cos 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,{"$f":"F16/10","v":25},"°",{"$nulls":6},"Now trace along the curve:  x = ",{"$f":"O16/10","v":17.2},"° and  y = ",{"$f":"LOOKUP(N15+0.001,Calculations!M2:M452,Calculations!N2:N452)","v":{"v":0.9552783621223438,"d":"0.96"}}]
[{"$nulls":5},250,127,{"$nulls":7},172]
[]
[{"$nulls":12},"The corresponding error = ",{"$f":"Calculations!K13","v":{"v":-7.3595768360967817E-5,"d":"-0.0001"}}]
[]
[]
[]
[]
[null,"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":11},"    The maximum error (in absolute value) =",{"$f":"Calculations!K9","v":{"v":0.005888921991268337,"d":"0.00589"}}]
["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 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 cosine function y = cos 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,{"$f":"G16/980+0.196","v":{"v":0.396,"d":"0.40"}},"",{"$nulls":5},"Now trace along the curve:  x = ",{"$f":"O16*dxxx0","v":{"v":0.3440043955680824,"d":"0.344"}},"and  y = ",{"$f":"LOOKUP(M15+0.00001,Calculations!V2:V502,Calculations!W2:W502)","v":{"v":0.9414116962513962,"d":"0.941"}}]
[{"$nulls":5},50,196,{"$nulls":7},219]
[]
[null,"What is the third point x2 (0.39 to 0.59)?",null,null,{"$f":"G19/980+0.196*2","v":{"v":0.5920000000000001,"d":"0.59"}},"",{"$nulls":5},"The corresponding error = ",{"$f":"Calculations!T13","v":{"v":-4.1769885078979385E-5,"d":"-0.000042"}}]
[{"$nulls":5},50,196]
[]
[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"]
[{"$nulls":3},"   Item 5"]
[]
[]
["To access/download all the DIGMath Apps for College Algebra/Precalculus",{"$nulls":10},"The maximum error (in absolute value) =",{"$f":"Calculations!T9","v":{"v":2.8867100836271753E-4,"d":"0.000289"}}]
["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 cosine function y = cos 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,{"$f":"F13/10","v":6.2},"°"]
[{"$nulls":4},29,62]
[]
[null,"What is the second point x1 (11.5° to 22.5°)?",null,null,{"$f":"F16/10+11.5","v":19.2},"°",{"$nulls":5},"Now trace along the curve:  x = ",{"$f":"N16*dxxxx0","v":{"v":33.1,"d":"33.10"}},"°  and  y = ",{"$f":"LOOKUP(M15+0.00001,Calculations!AE2:AE452,Calculations!AF2:AF502)","v":{"v":0.8377187166204368,"d":"0.838"}}]
[{"$nulls":4},50,77,{"$nulls":7},331]
[]
[null,"What is the third point x2 (23° to 34°)?",null,null,{"$f":"F19/10+11.5*2","v":32.8},"°",{"$nulls":5},"The corresponding error = ",{"$f":"Calculations!AC13","v":{"v":7.221491053477358E-5,"d":"0.000072"}}]
[{"$nulls":4},50,98]
[]
[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 =",{"$f":"PI()/4/500","v":1.5707963267948967E-3},"x","Cosine","L(x)","Error","ABS(Error)",null,null,"dxx =",{"$f":"45/450","v":0.1},"x","Cosine","L(x)","Error","ABS(Error)",null,null,"dxxx0 =",{"$f":"PI()/4/500","v":1.5707963267948967E-3},"x","Cosine","L3(x)","Error","ABS(Error)",null,null,"dxxxx0 =",0.1,"x","Cosine","L3(x)","Error","ABS(Error)"]
[{"$nulls":3},{"$f":"0","v":0},{"$f":"COS(x)","v":1},{"$f":"y0*(x-x00)*(x-x000)/(x0-x00)/(x0-x000)+y00*(x-x0)*(x-x000)/(x00-x0)/(x00-x000)+y000*(x-x0)*(x-x00)/(x000-x0)/(x000-x00)","v":1.0016147752978666},{"$f":"F2-E2","v":1.6147752978665775E-3},{"$f":"ABS(G2)","v":1.6147752978665775E-3},{"$nulls":4},{"$f":"0","v":0},{"$f":"COS(xx*PI()/180)","v":1},{"$f":"yy0*(xx-xx00)*(xx-xx000)/(xx0-xx00)/(xx0-xx000)+yy00*(xx-xx0)*(xx-xx000)/(xx00-xx0)/(xx00-xx000)+yy000*(xx-xx0)*(xx-xx00)/(xx000-xx0)/(xx000-xx00)","v":1.001380286727672},{"$f":"O2-N2","v":1.3802867276719866E-3},{"$f":"ABS(P2)","v":1.3802867276719866E-3},null,null,"dxxx =",{"$f":"T5-T4","v":0.196},0,{"$f":"COS(xxx)","v":1},{"$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":0.9989277010971875},-2.8867100836271753E-4,2.8867100836271753E-4,null,null,"dxxxx =",13.5,0,{"$f":"COS(xxxx*PI()/180)","v":1},{"$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":0.9994146961122199},-1.3972281566543288E-4,1.3972281566543288E-4]
[null,"x","y",{"$f":"D2+dx","v":1.5707963267948967E-3},0.9999987662997035,1.0015827238322816,{"$f":"F3-E3","v":1.5839575325781174E-3},{"$f":"ABS(G3)","v":1.5839575325781174E-3},null,null,"x","y",{"$f":"M2+dxx","v":0.1},0.9999984769132877,1.0013569850632882,{"$f":"O3-N3","v":1.3585081500004925E-3},{"$f":"ABS(P3)","v":1.3585081500004925E-3},null,null,"x","y",{"$f":"V2+dxxx0","v":1.5707963267948967E-3},0.9999987662997035,0.9989448473049833,-2.7798897286768209E-4,2.7798897286768209E-4,null,null,"xxxx","yyyy",0.1,0.9999984769132877,0.9994287960071527,-1.3110398558335845E-4,1.3110398558335845E-4]
[1,{"$f":"x0","v":0.11},{"$f":"COS(x0)","v":0.9939560979566968},3.1415926535897933E-3,0.9999950652018582,1.0015484322903465,1.5533670884883E-3,1.5533670884883E-3,null,1,{"$f":"xx0","v":11},{"$f":"COS(K4*PI()/180)","v":0.981627183447664},0.2,0.9999939076577904,1.0013308160227226,1.3369083649321656E-3,1.3369083649321656E-3,null,1,{"$f":"xxx0","v":0.2},{"$f":"COS(T4)","v":0.9800665778412416},3.1415926535897933E-3,0.9999950652018582,0.9989593073598257,-2.6748398509035543E-4,2.6748398509035543E-4,null,1,{"$f":"xxxx0","v":6.2},{"$f":"COS(AC4*PI()/180)","v":0.9941509639723154},0.2,0.9999939076577904,0.9994396054710699,-1.2266869253252377E-4,1.2266869253252377E-4]
[2,{"$f":"PI()/8","v":0.39269908169872414},{"$f":"COS(B5)","v":0.9238795325112867},0.00471238898038469,0.999988896715596,1.001511900672062,1.5230039564660958E-3,1.5230039564660958E-3,null,2,{"$f":"22.5","v":22.5},{"$f":"COS(K5*PI()/180)","v":0.9238795325112867},0.30000000000000004,0.9999862922474267,1.001301779605976,0.00131548735854925,0.00131548735854925,null,2,{"$f":"xxx00","v":0.396},{"$f":"COS(T5)","v":0.9226112947402},0.00471238898038469,0.999988896715596,0.9989710829221963,-2.5715448179097766E-4,2.5715448179097766E-4,null,2,{"$f":"xxxx00","v":19.2},{"$f":"COS(AC5*PI()/180)","v":0.9443763702374811},0.3,0.9999862922474267,0.9994471267324063,-1.1441495124453471E-4,1.1441495124453471E-4]
[3,{"$f":"x000","v":0.74},{"$f":"COS(x000)","v":0.7384685587295879},0.006283185307179587,0.9999802608561371,1.001473128977428,1.4928681212907913E-3,1.4928681212907913E-3,null,3,{"$f":"xx000","v":25},{"$f":"COS(K6*PI()/180)","v":0.9063077870366499},0.4,0.9999756307053947,1.0012698758130512,1.294245107656411E-3,1.294245107656411E-3,null,3,{"$f":"xxx000","v":0.5920000000000001},{"$f":"COS(T6)","v":0.8298262959153482},0.006283185307179587,0.9999802608561371,0.9989801756528323,-2.4699890581847406E-4,2.4699890581847406E-4,null,3,{"$f":"xxxx000","v":32.8},{"$f":"COS(AC6*PI()/180)","v":0.8405666034956844},0.4,0.9999756307053947,0.9994513620195987,-1.0634078573168448E-4,1.0634078573168448E-4]
[{"$nulls":3},0.007853981633974483,0.9999691576447897,1.001432117206443,1.4629595616534319E-3,1.4629595616534319E-3,{"$nulls":4},0.5,0.9999619230641713,1.0012351046439463,1.2731815797749624E-3,1.2731815797749624E-3,null,4,{"$f":"xxx0000","v":0.5880000000000001},{"$f":"COS(T7)","v":0.8320517385233702},0.007853981633974483,0.9999691576447897,0.9989865872123147,-2.3701570610967782E-4,2.3701570610967782E-4,null,4,{"$f":"xxxx0000","v":41.5},{"$f":"COS(AC7*PI()/180)","v":0.7489557207890022},0.5,0.9999619230641713,0.9994523135610802,-9.8444229284289086E-5,9.8444229284289086E-5]
[{"$nulls":3},0.00942477796076938,0.9999555871089498,1.0013888653591085,1.4332782501587094E-3,1.4332782501587094E-3,{"$nulls":4},0.6,0.9999451693655121,1.0011974660986613,0.001252296733149194,0.001252296733149194,{"$nulls":4},0.00942477796076938,0.9999555871089498,0.9989903192611678,-2.2720333768921908E-4,2.2720333768921908E-4,{"$nulls":4},0.6,0.9999451693655121,0.9994499835852877,-9.072332447179754E-5,9.072332447179754E-5]
["MaxError =",{"$f":"MAX(H2:H502)","v":1.6147752978665775E-3},null,0.010995574287564275,0.9999395492821014,1.001343373435425,1.4038241533235185E-3,1.4038241533235185E-3,null,"MaxError =",{"$f":"MAX(Q2:Q452)","v":0.005888921991268337},null,0.7,0.999925369660452,1.0011569601771972,1.2315905167452623E-3,1.2315905167452623E-3,null,"MaxError =",{"$f":"MAX(Z2:Z502)","v":2.8867100836271753E-4},null,0.010995574287564275,0.9999395492821014,0.9989913734600435,-2.1756026166941389E-4,2.1756026166941389E-4,null,"MaxError =",{"$f":"MAX(AI2:AL452)","v":1.7392530246373994E-4},null,0.7,0.999925369660452,0.9994443743206578,-8.3176123142680858E-5,8.3176123142680858E-5]
[{"$nulls":3},0.012566370614359171,0.9999210442038161,1.00129564143539,1.3745972315738486E-3,1.3745972315738486E-3,{"$nulls":4},0.7999999999999999,0.9999025240093042,1.001113586879553,0.001211062870248858,0.001211062870248858,{"$nulls":4},0.012566370614359171,0.9999210442038161,0.9989897514694945,-2.080849452511524E-4,2.080849452511524E-4,{"$nulls":4},0.8,0.9999025240093042,0.9994354879956231,-7.5800686424098984E-5,7.5800686424098984E-5]
[{"$nulls":3},0.014137166941154067,0.9999000719197535,1.0012456693590062,0.001345597439252666,0.001345597439252666,{"$nulls":4},0.8999999999999999,0.9998766324816606,1.0010673462057307,1.1907137240700916E-3,1.1907137240700916E-3,{"$nulls":4},0.014137166941154067,0.9999000719197535,0.9989854549501729,-1.9877586172223349E-4,1.9877586172223349E-4,{"$nulls":4},0.9,0.9998766324816606,0.9994233268386202,-6.8595084721234656E-5,6.8595084721234656E-5]
["Point",null,null,0.015707963267948963,0.9998766324816606,1.0011934572062722,1.3168247246115872E-3,1.3168247246115872E-3,null,"Point",null,null,0.9999999999999999,0.9998476951563913,1.0010182381557264,1.1705429993351668E-3,1.1705429993351668E-3,null,"Point",null,null,0.015707963267948963,0.9998766324816606,0.9989784855627022,-1.8963149045803096E-4,1.8963149045803096E-4,null,"Point",null,null,1,0.9998476951563913,0.9994078930780848,-6.1557397718403628E-5,6.1557397718403628E-5]
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