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4P0>05_WM`30RaL001@002010028mA800;"], "Graphics", ImageSize->{21, 21}], " " }], "Text", Evaluatable->False, AspectRatioFixed->True, FontSize->18, CellTags->"Section 5.4"], Cell[TextData[{ "\tBased on the success we had in using power series to define the complex \ exponential, we have reason to believe this approach will be fruitful for \ other elementary functions as well. The power series expansions for the \ real-valued sine and cosine functions are \n\n\t\t", Cell[BoxData[ \(cos \((x)\) = \[Sum]\+\(n = 0\)\%\[Infinity]\(\((\(-1\))\)\^n\ x\^\(2\ \ n\)\)\/\(\((2\ n)\)!\)\)]], " and ", Cell[BoxData[ RowBox[{\(sin \((x)\)\), " ", "=", " ", StyleBox[\(\[Sum]\+\(n = 0\)\%\[Infinity]\(\((\(-1\))\)\^n\ x\^\(2 n \ + 1\)\)\/\(\((2 n + 1)\)!\)\), FontSize->12]}]]], ". \n\nThus, it is natural to make the following definitions." }], "Text"], Cell[TextData[{ " ", StyleBox["\nDefinition 5.5, Page 182.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " The series for ", ButtonBox["Sine", ButtonData:>{ URL[ "http://mathworld.wolfram.com/Sine.html"], None}, ButtonStyle->"Hyperlink"], " and ", ButtonBox["Cosine", ButtonData:>{ URL[ "http://mathworld.wolfram.com/Cosine.html"], None}, ButtonStyle->"Hyperlink"], " are ", StyleBox["\n", FontColor->RGBColor[1, 0, 1]], "\n\t\t", Cell[BoxData[ \(cos \((z)\) = \[Sum]\+\(n = 0\)\%\[Infinity]\(\((\(-1\))\)\^n\ z\^\(2\ \ n\)\)\/\(\((2\ n)\)!\)\)]], " and ", Cell[BoxData[ RowBox[{\(sin \((z)\)\), " ", "=", " ", StyleBox[\(\[Sum]\+\(n = 0\)\%\[Infinity]\(\((\(-1\))\)\^n\ z\^\(2 n \ + 1\)\)\/\(\((2 n\ + \ 1)\)!\)\), FontSize->12]}]]], ". \n\n\tClearly, these definitions agree with their real counterparts \ when z is real. Additionally, it is easy to show that cos(z) and sin(z) \ are entire functions. (We leave the proof as an exercise.)\n\t\n\tWith \ these definitions in place, it is now easy to create the other complex \ trigonometric functions, provided the denominators in the following \ expressions do not equal zero." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ StyleBox["\n", FontColor->RGBColor[1, 0, 0]], StyleBox["Exploration (i), Page 182.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Investigate the series ", Cell[BoxData[ \(cos \((z)\) = \[Sum]\+\(n = 0\)\%\[Infinity]\(\((\(-1\))\)\^n\ z\^\(2\ \ n\)\)\/\(\((2\ n)\)!\)\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution for Exploration (i), Page 182.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Consider cos(z) and use ", StyleBox["Mathematica", FontSlant->"Italic"], " to find a Taylor polynomial expanded about z = 0, the remainder is \ expressed in the Big \"O\" notation." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[a, z];\)\ \), "\[IndentingNewLine]", \(\(Clear[cos, k, n, S, S12];\)\ \), "\n", \(\(cos[z_]\ = \ Series[Cos[z], \ {z, 0, 12}];\)\ \), "\n", \(\(Print["\", cos[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell["\<\ Use the general term in the series for cos(z) and sum the infinite \ series.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(a\_n_\ = \ \((\(-1\))\)\^n\/\(\((2 n)\)!\);\)\ \ \), "\n", \(\(S12 = \[Sum]\+\(n = 0\)\%6\ \(\((\(-1\))\)\^n\ z\^\(2\ n\)\)\/\(\((2\ \ n)\)!\);\)\ \), "\n", \(\(S = \[Sum]\+\(n = 0\)\%\[Infinity]\(\((\(-1\))\)\^n\ z\^\(2\ \ n\)\)\/\(\((2\ n)\)!\);\)\ \), "\n", \(\(Print[\*"\"\< \!\(a\_n\) = \>\"", a\_n];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\<\!\(S\_12\)[z] = \!\(\[Sum]\+\(n = 0\)\%6\) \ \!\(\(\(\((\(-1\))\)\^n\) z\^\(2 n\)\)\/\(\((2 n)\)!\)\)\>\""];\)\ \), "\n\ ", \(\(Print[\*"\"\<\!\(S\_12\)[z] = \>\"", S12];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\< \!\(S\_\[Infinity]\)[z] = \!\(\[Sum]\+\(n = 0\)\%\ \[Infinity]\)\!\(\(\(\((\(-1\))\)\^n\) z\^\(2 n\)\)\/\(\((2 \ n)\)!\)\)\>\""];\)\ \), "\n", \(\(Print[\*"\"\< \!\(S\_\[Infinity]\)[z] = \>\"", S];\)\ \), "\n", \(\(Print[\*"\"\< \!\(S\_\[Infinity]\)[z] = \>\"", PowerExpand[S]];\)\ \)}], "Input"], Cell[TextData[{ "\nUse ", StyleBox["Mathematica", FontSlant->"Italic"], " to plot some partial sums for cos(z)." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Clear[f, m, n, s, z];\)\ \), "\[IndentingNewLine]", \(\(f[z_] = Cos[z];\)\ \), "\[IndentingNewLine]", \(\(For\ [m = 2, m \[LessEqual] 5, \(m++\), \[IndentingNewLine]s[ z_] = \[Sum]\+\(n = 0\)\%m\ \(\((\(-1\))\)\^n\ z\^\(2\ \ n\)\)\/\(\((2\ n)\)!\); \ \[IndentingNewLine]CartesianMap[ s, {0, \[Pi], \[Pi]\/16}, {\(-1\), 1, 0.25}, PlotRange \[Rule] {{\(-1.7\), 1.7}, {\(-1.2\), 1.2}}, AspectRatio \[Rule] 1.2\/1.7, Ticks \[Rule] {Range[\(-2\), 2, 1], Range[\(-2\), 2, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Blue, Cyan}]; \ \ \[IndentingNewLine]Print["\", 2 m, "\<] = \>", s[z]];\ ];\)\ \), "\[IndentingNewLine]", \(\(CartesianMap[f, {0, \[Pi], \[Pi]\/16}, {\(-1\), 1, 0.25}, PlotRange \[Rule] {{\(-1.7\), 1.7}, {\(-1.2\), 1.2}}, AspectRatio \[Rule] 1.2\/1.7, Ticks \[Rule] {Range[\(-2\), 2, 1], Range[\(-2\), 2, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Magenta, Red}];\)\ \), "\n", \(\(Print["\< The mapping w = \>", f[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"] }, Closed]], Cell[TextData[{ StyleBox["\n", FontColor->RGBColor[1, 0, 0]], StyleBox["Exploration (ii), Page 182.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Investigate the series ", Cell[BoxData[ RowBox[{\(sin \((z)\)\), " ", "=", " ", StyleBox[\(\[Sum]\+\(n = 0\)\%\[Infinity]\(\((\(-1\))\)\^n\ z\^\(2 n \ + 1\)\)\/\(\((2 n + 1)\)!\)\), FontSize->12]}]]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution for Exploration (ii), Page 182.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Consider sin(z) and use ", StyleBox["Mathematica", FontSlant->"Italic"], " to find a Taylor polynomial expanded about z = 0, the remainder is \ expressed in the Big \"O\" notation." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[b, z];\)\ \), "\[IndentingNewLine]", \(\(Clear[k, n, sin, S, S13];\)\ \), "\n", \(\(sin[z_]\ = \ Series[Sin[z], \ {z, 0, 11}];\)\ \), "\n", \(\(Print["\", sin[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell["\<\ Use the general term in the series for sin(z) and sum the infinite \ series.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(b\_n_\ = \ \((\(-1\))\)\^n\/\(\((2 n + \ 1)\)!\);\)\ \), "\n", \(\(S13\ = \ \[Sum]\+\(n = 0\)\%6\(\((\(-1\))\)\^n\ z\^\(2 n + \ 1\)\)\/\(\((2 n + 1)\)!\);\)\ \), "\n", \(\(S\ = \ \[Sum]\+\(n = 0\)\%\[Infinity]\(\((\(-1\))\)\^n\ z\^\(2 n + \ 1\)\)\/\(\((2 n + 1)\)!\);\)\ \), "\n", \(\(Print[\*"\"\< \!\(b\_n\) = \>\"", b\_n];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\<\!\(S\_13\)[z] = \!\(\[Sum]\+\(n = \ 0\)\%6\)\!\(\(\(\((\(-1\))\)\^n\) z\^\(2 n + 1\)\)\/\(\((2 n + \ 1)\)!\)\)\>\""];\)\ \), "\n", \(\(Print[\*"\"\<\!\(S\_13\)[z] = \>\"", S13];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\< \!\(S\_\[Infinity]\)[z] = \!\(\[Sum]\+\(n = 0\)\%\ \[Infinity]\)\!\(\(\(\((\(-1\))\)\^n\) z\^\(2 n + 1\)\)\/\(\((2 n + 1)\)!\)\ \)\>\""];\)\ \), "\n", \(\(Print[\*"\"\< \!\(S\_\[Infinity]\)[z] = \>\"", S];\)\ \), "\n", \(\(Print[\*"\"\< \!\(S\_\[Infinity]\)[z] = \>\"", PowerExpand[S]];\)\ \)}], "Input"], Cell[TextData[{ "\nUse ", StyleBox["Mathematica", FontSlant->"Italic"], " to plot some partial sums for sin(z)." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Clear[f, m, n, s, z];\)\ \), "\[IndentingNewLine]", \(\(f[z_] = Sin[z];\)\ \), "\[IndentingNewLine]", \(\(For\ [m = 1, m \[LessEqual] 3, \(m++\), \[IndentingNewLine]s[ z_] = \[Sum]\+\(n = 0\)\%m\(\((\(-1\))\)\^n\ z\^\(2 n + \ 1\)\)\/\(\((2 n + 1)\)!\); \ \[IndentingNewLine]CartesianMap[ s, {\(-\[Pi]\)\/2, \[Pi]\/2, \[Pi]\/16}, {\(-1\), 1, 0.25}, PlotRange \[Rule] {{\(-1.7\), 1.7}, {\(-1.2\), 1.2}}, AspectRatio \[Rule] 1.2\/1.7, Ticks \[Rule] {Range[\(-2\), 2, 1], Range[\(-2\), 2, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {DarkGreen, Green}]; \ \ \[IndentingNewLine]Print["\", 2 m + 1, "\<] = \>", s[z]];\ ];\)\ \), "\[IndentingNewLine]", \(\(CartesianMap[ f, {\(-\[Pi]\)\/2, \[Pi]\/2, \[Pi]\/16}, {\(-1\), 1, 0.25}, PlotRange \[Rule] {{\(-1.7\), 1.7}, {\(-1.2\), 1.2}}, AspectRatio \[Rule] 1.2\/1.7, Ticks \[Rule] {Range[\(-2\), 2, 1], Range[\(-2\), 2, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Magenta, Red}];\)\ \), "\n", \(\(Print["\< The mapping w = \>", f[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"] }, Closed]], Cell[TextData[{ " ", StyleBox["\nDefinition 5.6, Page 183.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " \n\n\t", Cell[BoxData[ \(Tan[z] = Sin[z]\/Cos[z]\)]], ", ", Cell[BoxData[ \(Cot[z] = Cos[z]\/Sin[z]\)]], ", ", Cell[BoxData[ \(Sec[z] = 1\/Cos[z]\)]], ", and ", Cell[BoxData[ \(Csc[z] = 1\/Sin[z]\)]], ". " }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Exploration for Def. 5.6.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Tan[z] \[Equal] Sin[z]\/Cos[z]\)], "Input"], Cell[BoxData[ \(Cot[z] \[Equal] Cos[z]\/Sin[z]\)], "Input"], Cell[BoxData[ \(Sec[z] \[Equal] 1\/Cos[z]\)], "Input"], Cell[BoxData[ \(Csc[z] \[Equal] 1\/Sin[z]\)], "Input"] }, Closed]], Cell["\<\ \tSince the series for the complex sine and cosine agree with the real sine \ and cosine when z is real, the remaining complex trigonometric functions \ likewise agree with their real counterparts. What additional properties are \ common? For starters, we have:\ \>", "Text"], Cell[TextData[{ StyleBox["\nTheorem 5.4, Page 183.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " ", Cell[BoxData[ \(Sin[z]\)]], " and ", Cell[BoxData[ \(Cos[z]\)]], " are entire functions, with ", Cell[BoxData[ \(\[PartialD]\_z\ Sin[z] = Cos[z]\)]], " and ", Cell[BoxData[ \(\[PartialD]\_z\ Cos[z] = \(-Sin[z]\)\)]], "." }], "Text"], Cell[TextData[StyleBox["Proof of Theorem 5.4, see text Page 125.", FontWeight->"Bold", FontColor->RGBColor[0, 1, 1]]], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Exploration for Thm. 5.4.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\[PartialD]\_z\ Sin[z] \[Equal] Cos[z]\)], "Input"], Cell[BoxData[ \(\[PartialD]\_z\ Cos[z] \[Equal] \(-Sin[z]\)\)], "Input"] }, Closed]], Cell[TextData[{ "\n\tFor all complex numbers z, the following identities hold: \n\n\t\t\t\ ", Cell[BoxData[ \(Sin[\(-z\)] = \(-Sin[z]\)\)]], ", \n\n\t\t\t", Cell[BoxData[ \(Cos[\(-z\)] = Cos[z]\)]], ", \n\t\t\t\n\t\t\t", Cell[BoxData[ \(Cos[z]\^2 + Sin[z]\^2 = 1\)]], "." }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Exploration for identities.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(Sin[\(-z\)] \[Equal] \(-Sin[z]\)\)], "Input"], Cell[BoxData[ \(Cos[\(-z\)] \[Equal] Cos[z]\)], "Input"], Cell[BoxData[ \(eqn = Cos[z]\^2 + Sin[z]\^2 \[Equal] 1\)], "Input"], Cell[BoxData[ \(Simplify[eqn]\)], "Input"] }, Closed]], Cell[TextData[{ "\n", StyleBox["A series exploration (i)", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[" the derivative of ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["cos(z) ", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[" is ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["-sin(z)", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution of series exploration (i).", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Construct the Maclaurin series for cos(z) and sin(z) using ", StyleBox["Mathematica", FontSlant->"Italic"], "'s \"Series\" procedure." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[z];\)\ \), "\[IndentingNewLine]", \(\(Clear[cos, sin];\)\ \), "\n", \(\(cos[z_]\ = \ Series[Cos[z], \ {z, 0, 12}];\)\ \), "\[IndentingNewLine]", \(\(sin[z_]\ = \ Series[Sin[z], \ {z, 0, 11}];\)\ \), "\[IndentingNewLine]", \(\(Print["\", cos[z]];\)\ \), "\n", \(\(Print["\", sin[z]];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print["\< Cos'[z] = \>", \(cos'\)[z]];\)\ \), "\n", \(\(Print["\<-Sin[z] = \>", \(-sin[z]\)];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print["\", \(cos'\)[ z]\ \[Equal] \ \(-\ sin[z]\)];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True] }, Closed]], Cell[TextData[{ "\n", StyleBox["A series exploration (ii)", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[" the derivative of ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["sin(z) ", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], StyleBox[" is ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], StyleBox["cos(z)", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution of series exploration (ii).", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[TextData[{ "Construct the Maclaurin series for cos(z) and sin(z) using ", StyleBox["Mathematica", FontSlant->"Italic"], "'s \"Series\" procedure." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[z];\)\ \), "\[IndentingNewLine]", \(\(Clear[cos, sin];\)\ \), "\n", \(\(cos[z_]\ = \ Series[Cos[z], \ {z, 0, 12}];\)\ \), "\[IndentingNewLine]", \(\(sin[z_]\ = \ Series[Sin[z], \ {z, 0, 13}];\)\ \), "\n", \(\(Print["\", sin[z]];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print["\", \(sin'\)[z]];\)\ \), "\n", \(\(Print["\< Cos[z] = \>", cos[z]];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print["\", \(sin'\)[z]\ \[Equal] \ cos[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True] }, Closed]], Cell[TextData[{ "\n\tFor all complex numbers z for which the expressions are defined, we \ have\n\n\t\t\t", Cell[BoxData[ \(\[PartialD]\_z\ Tan[z] = Sec[z]\^2\)]], ", \n\n\t\t\t", Cell[BoxData[ \(\[PartialD]\_z\ Cot[z] = \(-Csc[z]\^2\)\)]], ", \n\n\t\t\t", Cell[BoxData[ \(\[PartialD]\_z\ Sec[z] = Sec[z]\ Tan[z]\)]], ", \n\n\t\t\t", Cell[BoxData[ \(\[PartialD]\_z\ Csc[z] = \(-Cot[z]\)\ Csc[z]\)]], ". .\n\t\t\t" }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Exploration for more identities.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[ \(\[PartialD]\_z\ Tan[z] \[Equal] Sec[z]\^2\)], "Input"], Cell[BoxData[ \(\[PartialD]\_z\ Cot[z] \[Equal] \(-Csc[z]\^2\)\)], "Input"], Cell[BoxData[ \(\[PartialD]\_z\ Sec[z] \[Equal] Sec[z]\ Tan[z]\)], "Input"], Cell[BoxData[ \(\[PartialD]\_z\ Csc[z] \[Equal] \(-Cot[z]\)\ Csc[z]\)], "Input"] }, Closed]], Cell["", "Text"], Cell[TextData[{ "\tTo establish additional properties, it will be useful to express cos z \ and sin z in the Cartesian form u+iv. (Additionally, the applications in \ Chapters 9 and 10 will use these formulas.) We begin by observing that the \ argument given to prove part (iii) in Theorem 5.1 easily generalizes to the \ complex case with the aid of Definition 5.4. That is, \n\n\t\t\t", Cell[BoxData[ \(\[ExponentialE]\^\(\[ImaginaryI]\ z\) = Cos[z] + \[ImaginaryI]\ Sin[z]\)]], ",\n\nfor all z, whether z is real or complex. Hence, \n\n\t\t\t", Cell[BoxData[ \(\[ExponentialE]\^\(\(-\[ImaginaryI]\)\ z\) = Cos[\(-z\)] + \[ImaginaryI]\ Sin[\(-z\)]\)]], ", \n\t\t\t", Cell[BoxData[ \(\(\(\ \ \ \ \ \ \ \ \)\(\(=\)\(Cos[ z] - \[ImaginaryI]\ Sin[z]\)\)\)\)]], ". " }], "Text"], Cell[TextData[{ "\tSubtracting the second equation from the first and solving for z gives\n\ \n\t\t\t", Cell[BoxData[ \(Sin[ z] = \(\[ExponentialE]\^\(\[ImaginaryI]\ z\)\ - \ \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ z\)\)\/\(2 \[ImaginaryI]\)\)]], ", \n\nwhich can then be used to obtain the important identity\n\n\t\t\t", Cell[BoxData[ \(Sin[x + \[ImaginaryI]\ y] = Cosh[y]\ Sin[x] + \[ImaginaryI]\ Cos[x]\ Sinh[y]\)]], ". \n\nSimilarly,\n\n\t\t\t", Cell[BoxData[ \(Cos[x + \[ImaginaryI]\ y] = Cos[x]\ Cosh[y] - \[ImaginaryI]\ Sin[x]\ Sinh[y]\)]], ". " }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Exploration for real and imaginary parts of Sin and \ Cos.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\(Print[\ \[ExponentialE]\^\(\[ImaginaryI]\ z\) \[Equal] Cos[z] + \[ImaginaryI]\ Sin[z]\ ];\)\ \), "\[IndentingNewLine]", \(ComplexExpand[\ \[ExponentialE]\^\(\[ImaginaryI]\ z\) \[Equal] Cos[z] + \[ImaginaryI]\ Sin[z]\ ]\)}], "Input"], Cell[BoxData[{ \(\(Print[\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ z\) \[Equal] Cos[\(-z\)] + \[ImaginaryI]\ Sin[\(-z\)]\ ];\)\ \), "\ \[IndentingNewLine]", \(ComplexExpand[\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ z\) \[Equal] Cos[\(-z\)] + \[ImaginaryI]\ Sin[\(-z\)]\ ]\)}], "Input"], Cell[BoxData[{ \(\(Print[\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ z\) \[Equal] Cos[z] - \[ImaginaryI]\ Sin[z]\ ];\)\ \), "\[IndentingNewLine]", \(ComplexExpand[\ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ z\) \[Equal] Cos[z] - \[ImaginaryI]\ Sin[z]\ ]\)}], "Input"], Cell[BoxData[{ \(\(Print[\ Sin[z] \[Equal] \(\[ExponentialE]\^\(\[ImaginaryI]\ z\) - \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ z\)\)\/\(2 \[ImaginaryI]\)];\)\ \), "\ \[IndentingNewLine]", \(ComplexExpand[\ Sin[z] \[Equal] \(\[ExponentialE]\^\(\[ImaginaryI]\ z\) - \ \[ExponentialE]\^\(\(-\[ImaginaryI]\)\ z\)\)\/\(2 \[ImaginaryI]\)]\)}], \ "Input"], Cell[BoxData[{ \(\(Print[ Sin[x + \[ImaginaryI]\ y] == ComplexExpand[\ Sin[x + \[ImaginaryI]\ y]]\ ];\)\ \), "\[IndentingNewLine]", \(\(Print[ ComplexExpand[ Sin[x + \[ImaginaryI]\ y] == \ Sin[x + \[ImaginaryI]\ y]]\ ];\)\ \)}], "Input"], Cell[BoxData[{ \(\(Print[ Cos[x + \[ImaginaryI]\ y] == ComplexExpand[\ Cos[x + \[ImaginaryI]\ y]]\ ];\)\ \), "\[IndentingNewLine]", \(\(Print[ ComplexExpand[ Cos[x + \[ImaginaryI]\ y] == \ Cos[x + \[ImaginaryI]\ y]]\ ];\)\ \)}], "Input"] }, Closed]], Cell[TextData[{ "\n\tFor all complex numbers ", Cell[BoxData[ \(z = x + \[ImaginaryI]\ y\)]], ", the following identities hold \n\n\t\t\t", Cell[BoxData[ \(Sin[z + 2 \[Pi]] = Sin[z]\)]], ", \n\n\t\t\t", Cell[BoxData[ \(Cos[z + 2 \[Pi]] = Cos[z]\)]], ", \n\n\t\t\t", Cell[BoxData[ \(Sin[z + \[Pi]] = \(-Sin[z]\)\)]], ", \n\n\t\t\t", Cell[BoxData[ \(Cos[z + \[Pi]] = \(-Cos[z]\)\)]], ", \n\n\t\t\t", Cell[BoxData[ \(Tan[z + \[Pi]] = Tan[z]\)]], ", \n\n\t\t\t", Cell[BoxData[ \(Cot[z + \[Pi]] = Cot[z]\)]], ". " }], "Text"], Cell[CellGroupData[{ Cell[TextData[StyleBox["Exploration for trigonometric identities.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(Sin[z + 2 \[Pi]]\), "\[IndentingNewLine]", \(Simplify[Sin[z + 2 n\ \[Pi]], Element[n, Integers]]\)}], "Input"], Cell[BoxData[{ \(Cos[z + 2 \[Pi]]\), "\[IndentingNewLine]", \(Simplify[Cos[z + 2 n\ \[Pi]], Element[n, Integers]]\)}], "Input"], Cell[BoxData[{ \(Sin[z + \[Pi]]\), "\[IndentingNewLine]", \(Simplify[Sin[z + \((2 n + 1)\)\ \[Pi]], Element[n, Integers]]\)}], "Input"], Cell[BoxData[{ \(Cos[z + \[Pi]]\), "\[IndentingNewLine]", \(Simplify[Cos[z + \((2 n + 1)\)\ \[Pi]], Element[n, Integers]]\)}], "Input"], Cell[BoxData[{ \(Tan[z + \[Pi]]\), "\[IndentingNewLine]", \(Simplify[Tan[z + \((2 n + 1)\)\ \[Pi]], Element[n, Integers]]\)}], "Input"], Cell[BoxData[{ \(Cot[z + \[Pi]]\), "\[IndentingNewLine]", \(Simplify[Cot[z + \((2 n + 1)\)\ \[Pi]], Element[n, Integers]]\)}], "Input"] }, Closed]], Cell[TextData[{ "\n", StyleBox["Extra Trig. Example for cos(z), Page 187.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Verify the identity cos(z) = 0, if and only if ", Cell[BoxData[ \(z = \((n + 1\/2)\) \[Pi]\)]], " for n an integer." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution for extra trig. example for cos(z).", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell[TextData[{ "First, use ", StyleBox["Mathematica", FontSlant->"Italic"], "'s \"Solve\" procedure to find some of the solutions to cos(z) = 0. " }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[x, y, z];\)\ \), "\[IndentingNewLine]", \(\(Clear[cosZ, eq1, eq2];\)\ \), "\n", \(\(cosZ\ = \ ComplexExpand[ Cos[x\ + \ \[ImaginaryI]\ y]];\)\ \), "\[IndentingNewLine]", \(\(eq1\ = \ ComplexExpand[Re[cosZ]]\ \[Equal] \ 0;\)\ \), "\n", \(\(eq2\ = \ ComplexExpand[Im[cosZ]]\ \[Equal] \ 0;\)\ \), "\[IndentingNewLine]", \(\(Print[Cos[x\ + \ \[ImaginaryI]\ y]\ \[Equal] \ 0];\)\ \), "\n", \(\(Print[cosZ\ \[Equal] \ 0];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[eq1\ ];\)\ \), "\n", \(\(Print[eq2];\)\ \), "\n", \(\(Print[Solve[{eq1, eq2}, {x, y}]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\n", StyleBox["Remark.", FontWeight->"Bold", FontColor->RGBColor[0, 1, 0]], " It is assumed that both x and y are real numbers. Hence, the only two \ valid solution in the above list are ", Cell[BoxData[ \({y \[Rule] 0, x \[Rule] \(-\(\[Pi]\/2\)\)}, {y \[Rule] 0, x \[Rule] \[Pi]\/2}\)]], ". \n\nThis is another way to solve the equation cos(z) = 0." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[z];\)\ \), "\[IndentingNewLine]", \(\(Clear[solset];\)\ \), "\n", \(\(solset = Solve[Cos[z] \[Equal] 0, z];\)\ \), "\n", \(\(Print[solset];\)\ \), "\n", \(\(Print["\"];\)\ \), "\n", \(\(Print["\<2 n \[Pi] + \>", \ solset\_\(\(\[LeftDoubleBracket]\)\(2, 1, \ 2\)\(\[RightDoubleBracket]\)\)];\)\ \), "\n", \(\(Print["\<2 n \[Pi] \>", solset\_\(\(\[LeftDoubleBracket]\)\(1, 1, \ 2\)\(\[RightDoubleBracket]\)\)];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"], Cell["\<\ We can also list some of the solutions.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Clear[n];\)\ \), "\[IndentingNewLine]", \(\(For[n = 0, n \[LessEqual] 5, \(n++\), \[IndentingNewLine]Print[\*"\"\\"", n, "\< \[Pi]] = \>", Cos[\[Pi]\/2 + n\ \[Pi]]]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"], Cell["\<\ Or by showing that the system of equations is satisfied.\ \>", "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Print[\*"\"\\"", Cos[\[Pi]\/2] Cosh[0], \n\*"\"\<, -Sin[\!\(\[Pi]\/2\)]Sinh[0] = \>\"", \(-Sin[\ \[Pi]\/2]\) Sinh[0]];\)\ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print[\*"\"\\"", Cos[\[Pi]\/2 + \[Pi]] Cosh[0], \n\*"\"\<, -Sin[\!\(\[Pi]\/2\)+\[Pi]]Sinh[0] = \>\"", \ \(-Sin[\[Pi]\/2 + \[Pi]]\) Sinh[0]];\)\ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print[\*"\"\\"", Cos[\[Pi]\/2 + 2 \[Pi]] Cosh[0], \n\*"\"\<, -Sin[\!\(\[Pi]\/2\)+2\[Pi]]Sinh[0] = \>\"", \ \(-Sin[\[Pi]\/2 + 2 \[Pi]]\) Sinh[0]];\)\ \), "\n", \(\(Print["\< \>"];\)\ \), "\n", \(\(Print[\*"\"\\"", Cos[\[Pi]\/2 + 3 \[Pi]] Cosh[0], \n\*"\"\<, -Sin[\!\(\[Pi]\/2\)+3\[Pi]]Sinh[0] = \>\"", \ \(-Sin[\[Pi]\/2 + 3 \[Pi]]\) Sinh[0]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"], Cell[TextData[{ "\nFinally, we could just let ", StyleBox["Mathematica", FontSlant->"Italic"], " do it. " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(Cos[\((n + 1\/2)\) \[Pi]]\), "\[IndentingNewLine]", \(Simplify[Cos[\((n + 1\/2)\) \[Pi]], Element[n, Integers]]\)}], "Input"] }, Closed]], Cell[TextData[{ "\n", StyleBox["Extra Example for trig. identities (i), Page 187.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Establish the trigonometric identity for complex numbers ", Cell[BoxData[ \(cos \((z\_1 + z\_2)\) = cos \((z\_1)\) cos \((z\_2)\)\ - \ sin \((z\_1)\) sin \((z\_2)\)\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution for extra example for trig identities (i).", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell[TextData[{ "Use ", StyleBox["Mathematica", FontSlant->"Italic"], "'s \"ComplexExpand\" procedure and the identities ", Cell[BoxData[ RowBox[{\(Cos[z]\), " ", "=", RowBox[{ RowBox[{\(\(\[ExponentialE]\^\(\[ImaginaryI]\ z\) + \ \ \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ z\)\)\/2\), " ", StyleBox["and", FontFamily->"Times New Roman"], " ", \(Sin[z]\)}], " ", "=", \(\(\[ExponentialE]\^\(\[ImaginaryI]\ \ z\) - \ \ \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ z\)\)\/\(2 \[ImaginaryI]\)\)}]}]]], ". " }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[w, x, y, z];\)\ \), "\n", \(\(z\_1\ = \ x\_1\ + \ \[ImaginaryI]\ y\_1;\)\ \), "\n", \(\(z\_2\ = \ x\_2\ + \ \[ImaginaryI]\ y\_2;\)\ \), "\n", \(\(w\_1\ = \ Cos[z\_1\ + \ z\_2];\)\ \), "\n", \(\(w\_2\ = \ Cos[z\_1] Cos[z\_2]\ - \ Sin[z\_1] Sin[z\_2];\)\ \), "\n", \(\(w\_3\ = \ ReplaceAll[w\_1, Cos[z_]\ \[Rule] \(\[ExponentialE]\^\(\[ImaginaryI]\ z\) + \ \ \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ z\)\)\/2];\)\ \), "\n", \(\(w\_4\ = \ ReplaceAll[ w\_2, {Cos[ z_]\ \[Rule] \(\[ExponentialE]\^\(\[ImaginaryI]\ z\) + \ \ \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ \ z\)\)\/2, \n\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ Sin[ z_]\ \[Rule] \(\[ExponentialE]\^\(\[ImaginaryI]\ \ z\) - \ \ \[ExponentialE]\^\(\(-\ \[ImaginaryI]\)\ z\)\)\/\(2 \[ImaginaryI]\)}];\)\ \ \), "\n", \(\(w\_5\ = \ ComplexExpand[w\_3];\)\ \), "\n", \(\(w\_6\ = \ ComplexExpand[w\_4];\)\ \), "\n", \(\(w\_7\ = \ Apart[w\_5];\)\ \), "\n", \(\(w\_8\ = \ Apart[w\_6];\)\ \), "\n", \(\(w\_9\ = \ Expand[w\_7, Trig \[Rule] True];\)\ \), "\n", \(\(w\_0\ = \ Expand[w\_8, Trig \[Rule] True];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_1\) = \>\"", w\_1];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_2\) = \>\"", w\_2];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_1\) = \>\"", w\_3];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_2\) = \>\"", w\_4];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_1\) = \>\"", w\_5];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_2\) = \>\"", w\_6];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_1\) = \>\"", w\_7];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_2\) = \>\"", w\_8];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_1\) = \>\"", w\_9];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_2\) = \>\"", w\_0];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\\"", Expand[w\_9, Trig \[Rule] True]\ \[Equal] \ \ Expand[w\_0, Trig \[Rule] True]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"], Cell[TextData[{ "\nTherefore, ", Cell[BoxData[ \(cos \((z\_1 + z\_2)\) = cos \((z\_1)\) cos \((z\_2)\)\ - \ sin \((z\_1)\) sin \((z\_2)\)\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True] }, Closed]], Cell[TextData[{ "\n", StyleBox["Extra Example for trig. identities (ii), Page 187.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Establish the trigonometric identity for complex numbers ", Cell[BoxData[ \(cos \((z\_1 + z\_2)\) = cos \((z\_1)\) cos \((z\_2)\)\ - \ sin \((z\_1)\) sin \((z\_2)\)\)]], ". " }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution for extra example for trig identities (ii).", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell[TextData[{ "Use ", StyleBox["Mathematica", FontSlant->"Italic"], "'s \"ComplexExpand\" procedure." }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[v, w, x, y, z];\)\ \), "\n", \(\(v\_1\ = \ ComplexExpand[ Cos[x\_1 + \[ImaginaryI]\ y\_1\ + \ x\_2 + \[ImaginaryI]\ y\_2]];\)\ \), "\n", \(\(v\_2\ = \ ComplexExpand[Cos[\((x\_1 + \[ImaginaryI]\ y\_1)\)]];\)\ \), "\n", \(\(v\_3\ = \ ComplexExpand[Cos[\((x\_2 + \[ImaginaryI]\ y\_2)\)]];\)\ \), "\n", \(\(v\_4\ = \ ComplexExpand[Sin[\((x\_1 + \[ImaginaryI]\ y\_1)\)]];\)\ \), "\n", \(\(v\_5\ = \ ComplexExpand[Sin[\((x\_2 + \[ImaginaryI]\ y\_2)\)]];\)\ \), "\n", \(\(v\_6\ = \ v\_2\ v\_3\ - \ v\_4\ v\_5;\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_1\) = Cos[\!\(z\_1\)+\!\(z\_2\)]\>\""];\)\ \), "\n\ ", \(\(Print[\*"\"\<\!\(w\_2\) = Cos[\!\(z\_1\)]Cos[\!\(z\_2\)]-Sin[\!\(z\_1\ \)]Sin[\!\(z\_2\)]\>\""];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_1\) = \>\"", v\_1];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_2\) = \>\"", v\_6];\)\ \), "\n", \(\(v\_7\ = \ Expand[v\_6, Trig \[Rule] True];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_1\) = \>\"", v\_1];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_2\) = \>\"", v\_7];\)\ \), "\n", \(\(v\_1\ = \ Expand[v\_1, Trig \[Rule] True];\)\ \), "\n", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_1\) = \>\"", v\_1];\)\ \), "\n", \(\(Print[\*"\"\<\!\(w\_2\) = \>\"", v\_7];\)\ \), "\[IndentingNewLine]", \(\(Print["\<\>"];\)\ \), "\n", \(\(Print[\*"\"\\"", v\_1\ \[Equal] \ v\_7];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True], Cell[TextData[{ "\nTherefore, ", Cell[BoxData[ \(cos \((z\_1 + z\_2)\) = cos \((z\_1)\) cos \((z\_2)\)\ - \ sin \((z\_1)\) sin \((z\_2)\)\)]], ". " }], "Text"] }, Closed]], Cell[TextData[{ "\n", StyleBox["Extra Example for Sin(z), Page 187.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Graph the transformation w = sin(z)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution for Page 145.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell["\<\ Enter the formula f[z] = Sin[z] and graph the transformation.\ \>", \ "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[z];\)\ \), "\[IndentingNewLine]", \(\(Clear[f, wplane];\)\ \), "\n", \(\(Iden[z_]\ = \ z;\)\ \), "\n", \(\(f[z_]\ = \ Sin[z];\)\ \), "\n", \(\(CartesianMap[ Iden, \ {\(-\[Pi]\)\/2, \[Pi]\/2, \[Pi]\/16}, {\(-1\), 1, 0.25}, PlotRange \[Rule] {{\(-1.7\), 1.7}, {\(-1.1\), 1.1}}, AspectRatio \[Rule] 1.1\/1.7, Ticks \[Rule] {Range[\(-2\), 2, 1], Range[\(-1\), 1, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Magenta, Red}];\)\ \), "\[IndentingNewLine]", \(\(Print[\*"\"\\""];\)\ \), "\n", \(\(CartesianMap[ f, {\(-\[Pi]\)\/2, \[Pi]\/2, \[Pi]\/16}, {\(-1\), 1, 0.25}, PlotRange \[Rule] {{\(-1.6\), 1.6}, {\(-1.2\), 1.2}}, AspectRatio \[Rule] 1.2\/1.6, Ticks \[Rule] {Range[\(-2\), 2, 1], Range[\(-2\), 2, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Magenta, Red}];\)\ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\n", \(\(Print["\", f[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True] }, Closed]], Cell[TextData[{ "\n", StyleBox["Identities for ", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], Cell[BoxData[ StyleBox[\(\(|\)\(Sin[x + \[ImaginaryI]y]\)\( | \^2\)\), FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]]]], StyleBox[", on Page 187.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Text"], Cell[CellGroupData[{ Cell[TextData[{ StyleBox["Solution for identities for ", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]], Cell[BoxData[ StyleBox[\(\(|\)\(Sin[x + \[ImaginaryI]y]\)\( | \^2\)\), FontColor->RGBColor[1, 0, 1]]], FontWeight->"Bold"], StyleBox[".", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]] }], "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[w];\)\ \), "\[IndentingNewLine]", \(\(Clear[aw];\)\ \), "\n", \(\(w\ \ = \ ComplexExpand[Sin[x\ + \ \[ImaginaryI]\ y]];\)\ \), "\n", \(\(aw\ = \ ComplexExpand[ Abs[w], \n\ \ \ \ \ \ \ TargetFunctions \[Rule] {Im, Re}];\)\ \), "\[IndentingNewLine]", \(\(aw1 = ReplaceAll[ aw\^2, {Cos[x]\^2 \[Rule] 1 - 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2\[ImaginaryI]] = \>", Cos[2 k\ \[Pi] - 2 \[ImaginaryI]]]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input"] }, Closed]], Cell[TextData[{ "\n", StyleBox["Expansion for tan(x+iy) on Page 188.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]] }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution of expansion of tan(x+iy) on Page 188.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[w, x, y];\)\ \), "\n", \(\(w\_3\ = \ ComplexExpand[Tan[x\ + \ \[ImaginaryI]\ y]];\)\ \), "\n", \(\(Print["\", w\_3];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True] }, Closed]], Cell[TextData[{ "\n", StyleBox["Extra example for tan(z), Page 188.", FontWeight->"Bold", FontColor->RGBColor[1, 0, 0]], " Graph the transformation w = tan(z)." }], "Text", Evaluatable->False, AspectRatioFixed->True], Cell[CellGroupData[{ Cell[TextData[StyleBox["Solution 5.9 for extra example for tan(z).", FontWeight->"Bold", FontColor->RGBColor[1, 0, 1]]], "Text"], Cell["\<\ Enter the formula f[z] = Tan[z] and graph the transformation.\ \>", \ "Text"], Cell[BoxData[{ \(\[IndentingNewLine]\(Remove[z];\)\ \), "\[IndentingNewLine]", \(\(Clear[f, Iden];\)\ \), "\n", \(\(Iden[z_]\ = \ z;\)\ \), "\n", \(\(f[z_]\ = \ Tan[z];\)\ \), "\n", \(\(CartesianMap[ Iden, \ {\(-\[Pi]\)\/4, \[Pi]\/4, \[Pi]\/16}, {\(-3\), 3, 0.25}, PlotRange \[Rule] {{\(-1\), 1}, {\(-3.2\), 3.2}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[\(-\[Pi]\)\/4, \[Pi]\/4, \[Pi]\/4], Range[\(-3\), 3, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Magenta, Pink}];\)\ \ \), "\[IndentingNewLine]", \(\(Print[\*"\"\\""];\)\ \), "\n", \(\(CartesianMap[ f, {\(-\[Pi]\)\/4, \[Pi]\/4, \[Pi]\/16}, {\(-3\), 3, 0.25}, PlotRange \[Rule] {{\(-1.1\), 1.1}, {\(-1.1\), 1.1}}, AspectRatio \[Rule] 1, Ticks \[Rule] {Range[\(-1\), 1, 1], Range[\(-1\), 1, 1]}, AxesLabel \[Rule] {"\", "\"}, PlotStyle \[Rule] {Magenta, Pink}];\)\ \ \), "\[IndentingNewLine]", \(\(Print["\"];\)\ \), "\n", \(\(Print["\", f[z]];\)\ \), "\[IndentingNewLine]", \(\)}], "Input", AspectRatioFixed->True] }, Closed]], Cell[TextData[{ " ", StyleBox["\nDefinition 5.7, Page 189.", FontWeight->"Bold", FontColor->RGBColor[0, 0, 1]], " The hyperbolic cosine and hyperbolic sine functions are \n\n\t\t", Cell[BoxData[ \(Cosh[ z] = \(1\/2\) \((\[ExponentialE]\^z + \ \[ExponentialE]\^\(-z\))\)\)]], " and ", Cell[BoxData[ \(Sinh[ z] = \(1\/2\) \((\[ExponentialE]\^z - 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(******************************************************************* End of Mathematica Notebook file. *******************************************************************)