(*********************************************************************** Mathematica-Compatible Notebook This notebook can be used on any computer system with Mathematica 3.0, MathReader 3.0, or any compatible application. The data for the notebook starts with the line of stars above. To get the notebook into a Mathematica-compatible application, do one of the following: * Save the data starting with the line of stars above into a file with a name ending in .nb, then open the file inside the application; * Copy the data starting with the line of stars above to the clipboard, then use the Paste menu command inside the application. Data for notebooks contains only printable 7-bit ASCII and can be sent directly in email or through ftp in text mode. Newlines can be CR, LF or CRLF (Unix, Macintosh or MS-DOS style). NOTE: If you modify the data for this notebook not in a Mathematica- compatible application, you must delete the line below containing the word CacheID, otherwise Mathematica-compatible applications may try to use invalid cache data. For more information on notebooks and Mathematica-compatible applications, contact Wolfram Research: web: http://www.wolfram.com email: info@wolfram.com phone: +1-217-398-0700 (U.S.) Notebook reader applications are available free of charge from Wolfram Research. ***********************************************************************) (*CacheID: 232*) (*NotebookFileLineBreakTest NotebookFileLineBreakTest*) (*NotebookOptionsPosition[ 709451, 24967]*) (*NotebookOutlinePosition[ 710505, 25003]*) (* CellTagsIndexPosition[ 710461, 24999]*) (*WindowFrame->Normal*) Notebook[{ Cell[TextData["Name: Joe Darschewski"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[TextData["Title: Parametric Equations"], "Section", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Description: This project introduces parametric equations showing\ntheir \ relationship to the unit circle and the Sine and Cosine functions. 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Let's put these values in a \ntable with \ another set of ordered pairs. "], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "tbl4= N[ Table[{x[t],y[t]},{t,0,Pi,Pi/4}],4];\n \ tbl6=N[Table[{x[t],y[t]},{t,0,Pi,Pi/6}],4];\n\nClear[tlist]\n\ tlist=Union[tbl4,tbl6];\ntbl1=\nTableForm[tlist,\nTableHeadings->{None,{value \ x,value y}}];\ntbl2=\nTableForm[{{tlist,tlist}},\n \ TableHeadings->{None,\n {\"value x value y\",\"Cos[x] Sin[x]\ \"}}]"], "Input", AspectRatioFixed->True], Cell[OutputFormData[ "\<\ TableForm[{{{{-1., 0}, {-0.8660254037844386468, 0.5}, {-0.7071067811865475244, 0.7071067811865475244}, {-0.5, 0.8660254037844386468}, {0, 1.}, {0.5, 0.8660254037844386468}, {0.7071067811865475244, 0.7071067811865475244}, {0.8660254037844386468, 0.5}, {1., 0}}, {{-1., 0}, {-0.8660254037844386468, 0.5}, {-0.7071067811865475244, 0.7071067811865475244}, {-0.5, 0.8660254037844386468}, {0, 1.}, {0.5, 0.8660254037844386468}, {0.7071067811865475244, 0.7071067811865475244}, {0.8660254037844386468, 0.5}, {1., 0}}}}, TableHeadings -> {None, {\"value x value y\", \"Cos[x] Sin[x]\"}}]\ \>", "\<\ value x value y Cos[x] Sin[x] -1. 0 -1. 0 -0.866 0.5 -0.866 0.5 -0.7071 0.7071 -0.7071 0.7071 -0.5 0.866 -0.5 0.866 0 1. 0 1. 0.5 0.866 0.5 0.866 0.7071 0.7071 0.7071 0.7071 0.866 0.5 0.866 0.5 1. 0 1. 0\ \>"], "Output", Evaluatable->False, LineSpacing->{0.6, 0}, AspectRatioFixed->True]}, Open]], Cell[TextData[ "The values of the ordered pairs of the unit circle \nare the same as the \ values of the Cosine and Sine \nfunctions. The Cosine is the x-coordinate \ and the \nSine is the y-coordinate. To understand this you \nhave to \ remember your right triangle trigonometry.\nSine is y/r and Cosine is x/r. \ Look at a right \ntriangle inscribed in the unit circle. Remember\nthe unit \ circle has a radius of one."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{Cell[TextData[ "Clear[trngl]\ntrngl=Graphics[{Thickness[0.005],RGBColor[0,0,1],\n \ Line[{{0,0},{x[Pi/4],y[Pi/4]}}],\n Line[{{0,0},{x[Pi/4],0}}],\n\ Line[{{x[Pi/4],0},{x[Pi/4],y[Pi/4]}}],\n \ Text[{N[x[Pi/4],4],N[y[Pi/4],4]},\n {0.9,0.707},{-1,0}],\n \ Text[\"x\",{0.5,-0.05}],\n Text[\"y\",{0.86,0.35}],\n \ Text[\"r\",{0.35,0.5}],\n Text[\"a\",{0.2,0.1}]}];\nShow[crcl,trngl,\n\ AspectRatio->Automatic,\n PlotRange->{{-2,2},{-1.5,1.5}},\n \ AxesLabel->None,\n Ticks->None];"], "Input", AspectRatioFixed->True], Cell[GraphicsData["PostScript", "\<\ %! %%Creator: Mathematica %%AspectRatio: .75 MathPictureStart %% Graphics /Courier findfont 10 scalefont setfont % Scaling calculations 0.5 0.25 0.375 0.25 [ [ -0.001 -0.001 0 0 ] [ 1.001 .751 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath [ ] 0 setdash 0 g p p .002 w 0 .375 m 1 .375 L s P p .002 w 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Open]], Cell[TextData[ "If we assume this is the unit circle, then r=1.\nUsing right triangle \ trigonometry, we find:\n\ Sin[a]=\[RegisteredTrademark]\[IAcute]\[CapitalAAcute]=y and Cos[a]=\ \[RegisteredTrademark]\[IAcute]\[CapitalUGrave]=x\nIt follows that for any \ angle, the sine is y and the\ncosine is x."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "The equation of a circle; x\[Currency]+y\[Currency]=r\[Currency] helps \ verify this.\nIf Cosine=x and Sine=y, then substituting gives the equation\n\ Cos^2\[Currency]+Sin\[Currency]=r\[Currency]. Remember that r=1. Then the \ equation \nlooks familiar: Cos\[Currency]+Sin\[Currency]=1. This we know is \ true because\nit is one of the basic trigonometric identities."], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[ "Now lets see the progress of this triangle as it is graphed\naround the unit \ circle. 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(*********************************************************************** End of Mathematica Notebook file. ***********************************************************************)