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\((\[Theta])\)\)]\)]", { Sin[2 $CellContext`\[Theta]], {{-1, 1}, {-1, 1}}, {0, 2 Pi}} -> "r = sin(2\[Theta])", { Cos[3 $CellContext`\[Theta]], {{-1, 1}, {-1, 1}}, {0, 2 Pi}} -> "r = cos(3\[Theta])", { 1 - Sin[$CellContext`\[Theta]], {{-2, 2}, {-2, 2}}, {0, 2 Pi}} -> "r = 1 - sin(\[Theta])", { 1 + 2 Sin[4 $CellContext`\[Theta]], {{-3, 3}, {-3, 3}}, {0, 2 Pi}} -> "r = 1 + 2 sin(4\[Theta])", { 1 + 4 Cos[5 $CellContext`\[Theta]], {{-5, 5}, {-5, 5}}, {0, 2 Pi}} -> "r = 1 + 4 cos(5\[Theta])", { Sin[Rational[1, 4] $CellContext`\[Theta]], {{-1, 1}, {-1, 1}}, { 0, 8 Pi}} -> "r = sin(\[Theta]/4)", { Sin[Rational[8, 5] $CellContext`\[Theta]], {{-1, 1}, {-1, 1}}, { 0, 10 Pi}} -> "r = sin(8\[Theta]/5)", { Sin[3^Rational[1, 2] $CellContext`\[Theta]], {{-1, 1}, {-1, 1}}, { 0, 10 Pi}} -> "r = sin(\[Theta]\!\(\*SqrtBox[\(3\)]\))", { Sin[$CellContext`\[Theta]] + Sin[Rational[5, 2] $CellContext`\[Theta]]^3, {{-2, 2}, {-2, 2}}, { 0, 4 Pi}} -> "r = sin(\[Theta])+\!\(\*SuperscriptBox[\(sin\), \ \(3\)]\)(5\[Theta]/2)"}, BaseStyle -> { FontSize -> 9}}, {$CellContext`mode$$, {$CellContext`trace, $CellContext`dot, \ $CellContext`segment}}, {$CellContext`places$$, {2, 3, 4, 5, 6}, ControlType -> PopupMenu, BaseStyle -> {FontSize -> 9}}, {{$CellContext`\[Theta]0$$, 0, "\[Theta]"}, Dynamic[ First[ Last[$CellContext`f$$]]], Dynamic[ Last[ Last[$CellContext`f$$]]]}}, "Options" :> {AutorunSequencing -> {4}}, "DefaultOptions" :> {}], ImageSizeCache->{446., {241., 246.}}, SingleEvaluation->True], Deinitialization:>None, DynamicModuleValues:>{}, Initialization:>({$CellContext`PolarTrace[ Pattern[$CellContext`f, Blank[]], { Pattern[$CellContext`var, Blank[]], Pattern[$CellContext`a, Blank[]], Pattern[$CellContext`b, Blank[]]}, Pattern[$CellContext`\[Theta]0, Blank[]], Pattern[$CellContext`mode, Blank[]], Pattern[$CellContext`plotrange, Blank[]], Pattern[$CellContext`places, Blank[]]] := Module[{$CellContext`pl, $CellContext`dt, $CellContext`mx, \ $CellContext`ln1, $CellContext`ln2, $CellContext`tx, $CellContext`gr, \ $CellContext`grobjs, $CellContext`r, $CellContext`\[Theta], $CellContext`x0, \ $CellContext`y0}, $CellContext`r = $CellContext`Sub[$CellContext`f, \ $CellContext`var, $CellContext`\[Theta]0]; $CellContext`x0 = $CellContext`r Cos[$CellContext`\[Theta]0]; $CellContext`y0 = $CellContext`r Sin[$CellContext`\[Theta]0]; If[$CellContext`a < $CellContext`b, $CellContext`pl = If[ Or[$CellContext`mode === $CellContext`dot, \ $CellContext`\[Theta]0 > $CellContext`a], First[ PolarPlot[$CellContext`f, Evaluate[ Switch[$CellContext`mode, $CellContext`dot, \ {$CellContext`var, $CellContext`a, $CellContext`b}, $CellContext`trace, \ {$CellContext`var, $CellContext`a, $CellContext`\[Theta]0}, \ $CellContext`segment, {$CellContext`var, $CellContext`\[Theta]0 - 0.1, $CellContext`\[Theta]0 + 0.1}]], Axes -> False, PlotStyle -> ColorData["HTML", "SlateBlue"]]], {}]; $CellContext`dt = { PointSize[Medium], Style[ Point[{$CellContext`x0, $CellContext`y0}], RGBColor[1, 0.47, 0]]}; $CellContext`mx = Max[ Select[ Join[ Flatten[$CellContext`plotrange], {$CellContext`r}], NumberQ]]; $CellContext`ln1 = Style[ Line[{{0, 0}, $CellContext`mx { Cos[$CellContext`\[Theta]0], Sin[$CellContext`\[Theta]0]}}], Gray]; $CellContext`grobjs = {$CellContext`pl, $CellContext`dt, \ $CellContext`ln1}; If[$CellContext`r < 0, $CellContext`ln2 = Style[ Line[{{0, 0}, (-$CellContext`mx) { Cos[$CellContext`\[Theta]0], Sin[$CellContext`\[Theta]0]}}], Orange]; AppendTo[$CellContext`grobjs, $CellContext`ln2]]; \ {$CellContext`r, $CellContext`\[Theta], $CellContext`x0, $CellContext`y0} = Map[NumberForm[ Chop[ N[#, $CellContext`places + 1]], {$CellContext`places + 1, $CellContext`places}, NumberPadding -> {" ", "0"}, ExponentFunction -> ( Null& )]& , {$CellContext`r, $CellContext`\[Theta]0, \ $CellContext`x0, $CellContext`y0}]; $CellContext`tx = Text[ Style[ Row[{"\[Theta]:", $CellContext`\[Theta], " r:", $CellContext`r, " x:", $CellContext`x0, " y:", $CellContext`y0}], 10]]; $CellContext`gr = Graphics[$CellContext`grobjs, Axes -> True, AspectRatio -> Automatic, AxesOrigin -> {Automatic, 0}, ImageSize -> {400, 300}, PlotRange -> $CellContext`plotrange]]; Grid[{{$CellContext`tx}, {$CellContext`gr}}]], $CellContext`Sub[ Pattern[$CellContext`expr, Blank[]], Pattern[$CellContext`var, Blank[]], PatternTest[ Pattern[$CellContext`vals, Blank[]], ListQ]] := Transpose[ Map[$CellContext`Sub[$CellContext`expr, $CellContext`var, #]& , \ $CellContext`vals]], $CellContext`Sub[ Pattern[$CellContext`expr, Blank[]], Pattern[$CellContext`var, Blank[]], Pattern[$CellContext`val, Blank[]]] := ReplaceAll[$CellContext`expr, $CellContext`var -> $CellContext`val], Attributes[PlotRange] = {ReadProtected}}; Typeset`initDone$$ = True), SynchronousInitialization->True, UnsavedVariables:>{Typeset`initDone$$}, UntrackedVariables:>{Typeset`size$$}], "Manipulate", Deployed->True, StripOnInput->False], Manipulate`InterpretManipulate[1]]], "Output", CellChangeTimes->{{3.402856701326871*^9, 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FontColor->GrayLevel[0.6], CellID->1031938007], Cell[TextData[{ "A full-function Wolfram ", StyleBox["Mathematica", FontSlant->"Italic"], " 6 system is required to edit or run this notebook.\[IndentingNewLine](", StyleBox["Mathematica Player", FontSlant->"Italic"], " runs only Demonstrations published on this site.)\n", StyleBox[ButtonBox["GET WOLFRAM MATHEMATICA 6 \[RightGuillemet]", BaseStyle->"Hyperlink", ButtonData->{ URL["http://www.wolfram.com/products/mathematica/"], None}, ButtonNote->"http://www.wolfram.com/products/mathematica/"], FontFamily->"Helvetica", FontWeight->"Bold", FontSlant->"Italic", FontColor->RGBColor[1, 0.42, 0]] }], "Text", CellFrame->True, CellMargins->{{48, 68}, {8, 28}}, CellFrameMargins->12, CellFrameColor->RGBColor[0.865507, 0.90634, 0.680751], CellChangeTimes->{3.3750111182355957`*^9}, ParagraphSpacing->{1., 1.}, FontFamily->"Verdana", FontSize->10, FontColor->GrayLevel[0.411765], Background->RGBColor[0.986023, 0.991363, 0.969818]], Cell[TextData[{ 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