(*^ ::[ Information = "This is a Mathematica Notebook file. It contains ASCII text, and can be transferred by email, ftp, or other text-file transfer utility. It should be read or edited using a copy of Mathematica or MathReader. If you received this as email, use your mail application or copy/paste to save everything from the line containing (*^ down to the line containing ^*) into a plain text file. On some systems you may have to give the file a name ending with ".ma" to allow Mathematica to recognize it as a Notebook. The line below identifies what version of Mathematica created this file, but it can be opened using any other version as well."; FrontEndVersion = "Macintosh Mathematica Notebook Front End Version 2.2"; MacintoshStandardFontEncoding; fontset = title, inactive, noPageBreakBelow, nohscroll, preserveAspect, cellOutline, groupLikeTitle, center, M18, O486, R65535, e8, 24, "Calculus"; fontset = subtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, bold, R21845, G21845, B21845, e6, 12, "Calculus"; fontset = subsubtitle, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeTitle, center, M18, O486, R21845, G21845, B21845, e6, 12, "Calculus"; fontset = section, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, grayBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = subsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, blackBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = subsubsection, inactive, noPageBreakBelow, nohscroll, preserveAspect, groupLikeSection, whiteBox, M18, O486, bold, R21845, G21845, B21845, a10, 12, "Calculus"; fontset = text, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Calculus"; fontset = smalltext, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, B65535, 12, "Calculus"; fontset = input, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeInput, M36, N23, O486, bold, L-5, 12, "Courier"; fontset = output, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M36, N23, O486, L-5, 12, "Courier"; fontset = message, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, R65535, L-5, 12, "Courier"; fontset = print, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, L-5, 12, "Courier"; fontset = info, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeOutput, M18, N23, O486, B65535, L-5, 12, "Courier"; fontset = postscript, PostScript, formatAsPostScript, output, inactive, noPageBreakInGroup, nowordwrap, preserveAspect, groupLikeGraphics, M18, O486, l19, o2, w378, h214, 12, "Courier"; fontset = name, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, italic, 10, "Geneva"; fontset = header, inactive, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = leftheader, inactive, M18, O486, L2, 12, "Times"; fontset = footer, inactive, noKeepOnOnePage, preserveAspect, center, M18, O486, 12, "Times"; fontset = leftfooter, inactive, M18, O486, L2, 12, "Times"; fontset = help, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 10, "Times"; fontset = clipboard, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = completions, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special1, inactive, nohscroll, noKeepOnOnePage, preserveAspect, whiteBox, M18, O486, bold, R21845, G21845, B21845, 12, "Calculus"; fontset = special2, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, R21845, G21845, B21845, 12, "Calculus"; fontset = special3, inactive, nohscroll, noKeepOnOnePage, preserveAspect, M18, O486, 12, "Times"; fontset = special4, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, 10, "Courier"; fontset = special5, inactive, nohscroll, noKeepOnOnePage, preserveAspect, center, M18, O486, 10, "Courier"; automaticGrouping; currentKernel; ] :[font = title; inactive; preserveAspect; startGroup] Trig Functions Sine, Cosine, Tangent ;[s] 4:0,1;14,0;15,2;36,0;37,-1; 3:2,45,32,Calculus,0,24,65535,0,0;1,27,21,New York,0,24,65535,0,0;1,22,17,New York,0,18,21844,21844,21844; :[font = subsection; inactive; preserveAspect; startGroup] Overview :[font = text; inactive; preserveAspect; endGroup] The subject matter of TRIGONOMETRY is based upon six trigonometric functions. In this section we will introduce the first three--the sine, cosine and tangent functions (abbreviated sin, cos and tan). ;[s] 15:0,0;22,1;34,0;134,2;138,0;140,2;146,0;151,2;158,0;182,2;185,0;187,2;190,0;195,2;198,0;201,-1; 3:8,26,19,Calculus,0,12,0,0,0;1,26,19,Calculus,0,12,65535,0,0;6,26,19,Calculus,1,12,0,0,0; :[font = subsection; inactive; preserveAspect; startGroup] Definitions :[font = text; inactive; preserveAspect; endGroup] 1. Let q be any angle. 2. Place q in standard position (vertex at the origin) 3. Choose any point P = (x,y), other than (0,0), on q. 4. Draw a perpendicular from P to the x-axis forming a right triangle. 5. Let r be the distance from P to the origin. 6. By the Pythagorean theorem: rÛ = xÛ + yÛ 7. sin q = y/r cos q = x/r tan q = y/x ;[s] 13:0,0;8,1;9,0;34,1;35,0;133,1;134,0;310,1;311,0;326,1;327,0;342,1;343,0;350,-1; 2:7,26,19,Calculus,0,12,0,0,0;6,18,13,Symbol,0,12,0,0,0; :[font = subsection; inactive; preserveAspect; startGroup] Selection of specific point P not critical :[font = text; inactive; preserveAspect; endGroup] It does not matter which point P on the terminal side of q is chosen. The value of sin q, cos q and tan q are independent of the point P. An similar triangles argument will easily show that sin q depends only upon q and not on which point P was chosen. Note: For this reason we will choose our point P on the unit circle making r = 1. The coordinates of P will thus be (cos q, sin q). ;[s] 16:0,0;59,1;60,0;90,1;91,0;97,1;98,0;107,1;108,0;198,1;199,0;218,1;219,0;258,2;259,3;392,0;393,-1; 4:8,26,19,Calculus,0,12,0,0,0;6,18,13,Symbol,0,12,0,0,0;1,26,19,Calculus,0,12,21844,21844,21844;1,26,19,Calculus,0,12,0,0,65535; :[font = subsection; inactive; preserveAspect; startGroup] The Sine function :[font = text; inactive; preserveAspect] The y-coordinate of the point on the terminal side of an angle q intersecting with the unit circle sin q. ;[s] 5:0,0;65,1;67,0;105,1;106,0;109,-1; 2:3,26,19,Calculus,0,12,0,0,0;2,18,13,Symbol,0,12,0,0,0; :[font = input; preserveAspect; startGroup] (* SIN curve display *) Clear[pCircle,pTrig,pAxes,s1,s2,x,y]; num=4; x[t_]= Cos[t]; y[t_]= Sin[t]; pAxes=Graphics[{Thickness[0.002], RGBColor[0,0,0], Line[{{-1.5,-1.1},{-1.5,1.1}}]}]; pCircle=ParametricPlot[{x[t]-1.5,y[t]},{t,0,2 Pi}, DisplayFunction->Identity, PlotStyle->{RGBColor[0,1,0],Thickness[0.005]}, Axes->True, Ticks->{{0,Pi/2,Pi,3Pi/2,2Pi},None}]; pTrig:=Plot[y[t],{t,0,2Pi}, DisplayFunction->Identity, Ticks->{{0,Pi/2,Pi,3Pi/2,2Pi}}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}, PlotRange->{{-1,2Pi},{-1,1}}]; Do[s1=Graphics[{Thickness[0.005], RGBColor[0,0,1], Line[{{-1.5,0},{x[theta]-1.5,y[theta]}}], RGBColor[0,.5,.4], Line[{{x[theta]-1.5,y[theta]},{theta,y[theta]}}], Line[{{x[theta]-1.5,y[theta]},{x[theta]-1.5,0}}], Line[{{theta,y[theta]},{theta,0}}], PointSize[0.015], RGBColor[1,0,0], Point[{x[theta]-1.5,y[theta]}], Point[{theta,y[theta]}] }]; Show[pCircle,pTrig,s1,pAxes, AspectRatio->Automatic, DisplayFunction->$DisplayFunction], {theta,0,2Pi,Pi/6} ] (* SIN curve display *) :[font = postscript; PICT; formatAsPICT; output; inactive; Cclosed; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; startGroup; pictureID = 17581] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 1216] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 10689] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 15917] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 30095] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 17668] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 2525] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 14666] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 26554] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 20355] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 23628] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 15861] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; endGroup; endGroup; endGroup; pictureID = 13970] :[font = subsection; inactive; preserveAspect; startGroup] The Cosine function :[font = text; inactive; preserveAspect] The x-coordinate of the point on the terminal side of an angle q intersecting with the unit circle cos q. ;[s] 5:0,0;65,1;67,0;105,1;106,0;110,-1; 2:3,26,19,Calculus,0,12,0,0,0;2,18,13,Symbol,0,12,0,0,0; :[font = input; preserveAspect; startGroup] (* COS curve display *) Clear[pCircle,pTrig,l1,t, p1,p2,p3,p4,p5,p6,s1,x,y,theta,num,n,org]; num=6; org={-1.5,0}; x[t_]=Cos[t]; y[t_]=Sin[t]; pAxes=Graphics[{Thickness[0.002], RGBColor[0,0,0], Line[{{-1.5,-1.1},{-1.5,1.1}}]}]; pCircle=ParametricPlot[{x[t]-1.5,y[t]},{t,0,2 Pi}, DisplayFunction->Identity, PlotStyle->{RGBColor[0,1,0],Thickness[0.005]}, Axes->True, Ticks->{{0,Pi/2,Pi,3Pi/2,2Pi},None}]; pTrig=Plot[x[t],{t,0,2Pi}, DisplayFunction->Identity, Ticks->{{Pi/2,Pi,3Pi/2,2Pi},None}, PlotStyle->{RGBColor[1,0,0],Thickness[0.01]}, PlotRange->{{-1,2Pi},{-1,1}}]; Do[p1={x[theta]-1.5,y[theta]}; p2={-1.5,y[theta]}; p3={theta,0}; p4={theta,x[theta]}; s1=Graphics[{Thickness[0.005], RGBColor[0,0,1], Line[{org,p1}], RGBColor[0,.9,.9], Thickness[0.001], Line[{p1,p2}], Line[{p3,p4}], PointSize[0.01], RGBColor[1,0,1], Point[p1], Point[p4]}]; Do[p5=p1+(p3-p1) n; p6=p2+(p4-p2) n; l1=Graphics[{Thickness[0.001], RGBColor[0,.5,.5], Line[{p5,p6}]}]; Show[pCircle,pTrig,s1,l1,pAxes, AspectRatio->Automatic, DisplayFunction->$DisplayFunction];, {n,1/num,1-1/num,1/num} ], {theta,0,2Pi-Pi/6,Pi/6} ] (* COS curve display *) :[font = postscript; PICT; formatAsPICT; output; inactive; Cclosed; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; startGroup; pictureID = 12091] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 754] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 22717] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 29606] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 20775] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 19125] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 6197] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 2415] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 8982] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 26857] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 27946] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 11867] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 5347] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 7705] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 21315] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 7491] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 26865] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 12637] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 25551] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 18979] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 5566] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 10658] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 24092] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 24251] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 2668] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 5679] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 26519] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 24385] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 19031] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 3541] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 15844] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 19276] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 10958] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 25469] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 30264] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 6095] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 21539] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 14743] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 11755] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 23267] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 19979] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 19874] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 16390] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 1570] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 18016] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 8087] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 6347] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 29246] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 15253] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 28305] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 27836] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 2107] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 26605] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 27206] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 28325] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 3599] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 21293] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 3777] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; pictureID = 7614] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 378; pictureHeight = 94; endGroup; endGroup; endGroup; pictureID = 1760] :[font = subsection; inactive; preserveAspect; startGroup] The Tangent function :[font = text; inactive; preserveAspect] The y-coordinate of the point where the terminal side of angle q intersects with the vertical line tangent passing through the point (1,0) is the tan q. A proof of the above statement involves similar triangles. ;[s] 7:0,0;65,1;66,0;152,1;153,0;157,2;215,0;216,-1; 3:4,26,19,Calculus,0,12,0,0,0;2,18,13,Symbol,0,12,0,0,0;1,26,19,Calculus,0,12,0,0,65535; :[font = input; preserveAspect; startGroup] (* TAN curve display *) Clear[pCircle,pTrig,pAxes,s1,s2,x,y,LongLine]; shift=2; org={shift,0}; num=4; x[t_]= Cos[t]; y[t_]= Sin[t]; pAxes=Graphics[{Thickness[0.002], RGBColor[0,0,0], Line[{{shift,-1.1},{shift,1.1}}]}]; LongLine=Graphics[{Thickness[0.002], RGBColor[.2,.2,.2], Line[{{shift+1,-6},{shift+1,6}}]}]; pCircle=ParametricPlot[{x[t]+shift,y[t]},{t,-Pi/2, Pi/2}, DisplayFunction->Identity, PlotStyle->{RGBColor[0,1,0], Thickness[0.005]}]; pTrig:=Plot[Tan[t],{t,-Pi/2+0.05,Pi/2-0.05}, DisplayFunction->Identity, Ticks->{{-Pi/2,0,Pi/2}}, PlotStyle->{RGBColor[1,0,0], Thickness[0.01]}, PlotRange->{{-Pi/2-0.01,4.1},{-6,6}}]; Do[p1={x[theta]+shift,y[theta]}; p2={shift+1,Tan[theta]}; p3={theta,Tan[theta]}; p4={theta,0}; s1=Graphics[{Thickness[0.005], RGBColor[0,0,1], Line[{org,p1}], RGBColor[0,.5,.4], Line[{p1,p2}], Line[{p2,p3}], Line[{p3,p4}], PointSize[0.025], RGBColor[1,0,0], Point[p1], Point[p2], Point[p3] }]; Show[pCircle,LongLine,pAxes,s1,pTrig, AspectRatio->Automatic, DisplayFunction->$DisplayFunction, PlotRange->{{-Pi/2-0.01,4.1},{-6,6}}], {theta,-5Pi/12,5Pi/12,Pi/12} ] (* TAN curve display *) :[font = postscript; PICT; formatAsPICT; output; inactive; Cclosed; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 101; pictureHeight = 214; startGroup; pictureID = 9789] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 101; pictureHeight = 214; pictureID = 6044] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 101; pictureHeight = 214; pictureID = 22866] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 101; pictureHeight = 214; pictureID = 14261] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 101; pictureHeight = 214; pictureID = 21335] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 101; pictureHeight = 214; pictureID = 26695] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 101; pictureHeight = 214; pictureID = 13241] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 101; pictureHeight = 214; pictureID = 19952] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 101; pictureHeight = 214; pictureID = 1671] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 101; pictureHeight = 214; pictureID = 30547] :[font = postscript; PICT; formatAsPICT; output; inactive; preserveAspect; pictureLeft = 19; pictureTop = 2; pictureWidth = 101; pictureHeight = 214; endGroup; endGroup; endGroup; endGroup; pictureID = 30755] ^*)