诱导公式五(90°+α)的几何演示
2020-03-14 03:54
本GeoGebra范例通过几何作图与单位圆动态演示了三角函数的诱导公式五,即角度为90°+α时的正弦、余弦和正切值的变化规律。利用旋转、对称及坐标几何构造,直观展示了sin(90°+α)=cosα,cos(90°+α)=-sinα等核心公式的几何意义,帮助学习者深入理解三角函数在不同象限的符号与数值转换关系。
所属专题
中职数学GeoGebra互动教学资源库
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.574.0" app="classic" platform="w" id="c7693de7-9ae0-4bfc-b122-b817edb3f8f2" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="721" height="545" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.4868238557558946" orientation="1" />
</panes>
<views>
<view id="512" toolbar="0 | 1 501 5 19 , 67 | 2 15 45 18 , 7 37 | 514 3 9 , 13 44 , 47 | 16 51 | 551 550 11 , 20 22 21 23 , 55 56 57 , 12 | 69 | 510 511 , 512 513 | 533 531 , 534 532 , 522 523 , 537 536 , 535 | 521 520 | 36 , 38 49 560 | 571 30 29 570 31 33 | 17 | 540 40 41 42 , 27 28 35 , 6 , 502" visible="false" inframe="false" stylebar="false" location="1,1,1" size="500" window="100,100,600,400" />
<view id="4" toolbar="0 || 2020 , 2021 , 2022 , 66 || 2001 , 2003 , 2002 , 2004 , 2005 || 2040 , 2041 , 2042 , 2044 , 2043" visible="false" inframe="false" stylebar="true" location="1,1" size="300" window="100,100,600,400" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="3" size="351" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="true" location="3" size="104" window="100,100,600,400" />
<view id="16" visible="true" inframe="false" stylebar="true" location="1" size="362" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="64" toolbar="0" visible="false" inframe="false" stylebar="false" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="4097" visible="false" inframe="false" stylebar="true" location="1" size="482" window="100,100,700,550" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
</views>
<toolbar show="false" items="0 39 73 62 | 1 501 67 , 5 19 , 72 75 76 | 2 15 45 , 18 65 , 7 37 | 4 3 8 9 , 13 44 , 58 , 47 | 16 51 64 , 70 | 10 34 53 11 , 24 20 22 , 21 23 | 55 56 57 , 12 | 36 46 , 38 49 50 , 71 14 68 | 30 29 54 32 31 33 | 25 17 26 60 52 61 | 40 41 42 , 27 28 35 , 6" position="1" help="false" />
<input show="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="351" height="545"/>
<coordSystem xZero="163.60383462662534" yZero="287.0434684646682" scale="132.064253264953" yscale="132.06425326495298"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="0" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1" tickExpression="1"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1" tickExpression="1"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<size width="362" height="545"/>
<coordSystem xZero="184.00000000000014" yZero="319.00000000000006" scale="50.00000000000002" yscale="50.00000000000002"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="3"/>
<bgColor r="255" g="255" b="255"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<kernel>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label="rp'" exp="rp'(x) = 0.3 + (0.7 * (1 - ℯ^(((-abs(x))) / ((12 * π)))))" type="function"/>
<element type="function" label="rp'">
<show object="false" label="true" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<element type="numeric" label="count">
<value val="0"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<slider min="-10" max="10" absoluteScreenLocation="true" width="200" x="71" y="73" fixed="false" horizontal="true" showAlgebra="true"/>
<lineStyle thickness="10" type="0" typeHidden="1"/>
<animation step="1" speed="1" type="0" playing="false"/>
</element>
<element type="angle" label="k'">
<angleStyle val="3"/>
<value val="0.9906545455622089"/>
<slider min="0" max="6.283185307179586" absoluteScreenLocation="true" width="180" x="60" y="40" fixed="false" horizontal="true" showAlgebra="true"/>
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="10" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<animation speed="1" type="0" playing="false"/>
</element>
<expression label="theta" exp="k' + (360° * count)" />
<element type="angle" label="theta">
<angleStyle val="3"/>
<value val="0.9906545455622089"/>
</element>
<expression label="c" exp="x^(2) + y^(2) = 1" type="conic"/>
<element type="conic" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1" y0="0" z0="1.0" x1="0" y1="1" z1="1.0"/>
<matrix A0="1" A1="1" A2="-1" A3="0" A4="0" A5="0"/>
<eqnStyle style="specific"/>
</element>
<command name="Point">
<input a0="c"/>
<output a0="P"/>
</command>
<element type="point" label="P">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="true"/>
<ggbscript onUpdate="SetValue[k',k]
"/>
<coords x="0.5481425259889123" y="0.8363849419988948" z="1"/>
<pointSize val="9"/>
<pointStyle val="1"/>
</element>
<command name="Angle">
<input a0="(1, 0)" a1="(0, 0)" a2="P"/>
<output a0="k"/>
</command>
<element type="angle" label="k">
<angleStyle val="0"/>
<value val="0.9906545455622089"/>
<show object="false" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<expression label="pass1" exp="k - k' < (-(200°))" />
<element type="boolean" label="pass1">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue[count,If[pass1,count+1,count]]"/>
</element>
<expression label="pass2" exp="k - k' > 200°" />
<element type="boolean" label="pass2">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="SetValue[count,If[pass2,count-1,count]]"/>
</element>
<expression label="rp" exp="rp(x) = 0.25 + (0.75 * (1 - ℯ^(((-abs(x))) / ((12 * π)))))" type="function"/>
<element type="function" label="rp">
<show object="false" label="true" ev="4"/>
<objColor r="204" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<auxiliary val="true"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="A" exp="(0, 0)" type="point"/>
<element type="point" label="A">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<auxiliary val="true"/>
<coords x="0" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="CurveCartesian">
<input a0="(rp(t) * cos(t))" a1="(rp(t) * sin(t))" a2="t" a3="Min[0, theta]" a4="Max[0, theta]"/>
<output a0="a"/>
</command>
<element type="curvecartesian" label="a">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="2" type="0" typeHidden="1" opacity="178"/>
</element>
<command name="Intersect">
<input a0="c" a1="xAxis" a2="2"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<condition showObject="isTan"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Intersect">
<input a0="xAxis" a1="yAxis"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<expression label="E" exp="(cos(k), 0)" type="point" />
<element type="point" label="E">
<show object="true" label="true" ev="4"/>
<condition showObject="isSin ∨ isCos"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-8" y="26"/>
<labelMode val="0"/>
<coords x="0.5481425259889123" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="P" a1="E" a2="C"/>
<output a0="t1" a1="sinP" a2="cosP" a3="e"/>
</command>
<element type="polygon" label="t1">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="isSin ∨ isCos"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="sinP">
<show object="true" label="false" ev="4"/>
<condition showObject="isSin"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="25" y="10"/>
<labelMode val="3"/>
<auxiliary val="false"/>
<coords x="0.8363849419988948" y="0" z="-0.45845815480636415"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="0.82"/>
</element>
<element type="segment" label="cosP">
<show object="true" label="false" ev="4"/>
<condition showObject="isCos"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelOffset x="-13" y="-1"/>
<labelMode val="3"/>
<auxiliary val="false"/>
<coords x="0" y="-0.5481425259889123" z="0"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="0.57"/>
</element>
<element type="segment" label="e">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-0.8363849419988948" y="0.5481425259889123" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Ray">
<input a0="C" a1="P"/>
<output a0="g"/>
</command>
<element type="ray" label="g">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.8363849419988948" y="0.5481425259889123" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="B" a1="xAxis"/>
<output a0="h"/>
</command>
<element type="line" label="h">
<show object="true" label="true" ev="4"/>
<condition showObject="isTan"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1" y="0" z="1"/>
<lineStyle thickness="5" type="20" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="g" a1="h"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="true" label="true" ev="4"/>
<condition showObject="isTan"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.5481425259889123" y="0.8363849419988948" z="0.5481425259889123"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="B" a1="F"/>
<output a0="tanP"/>
</command>
<element type="segment" label="tanP">
<show object="true" label="false" ev="4"/>
<condition showObject="isTan"/>
<objColor r="128" g="0" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<coords x="-1.5258530443153628" y="0" z="1.5258530443153628"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="1.44"/>
</element>
<expression label="text4" exp=""\red{\sin(90°+α)=\cosα}\\
\blue{\cos(90°+α)=-\sinα}\\
\purple{\tan(90°+α) =-\frac{1}{\tanα}}\\""/>
<element type="text" label="text4">
<show object="true" label="true" ev="43"/>
<condition showObject="b"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="-2.98" y="-2.3799999999999955" z="1"/>
</element>
<element type="boolean" label="b">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="35" y="394"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="公式"/>
</element>
<expression label="text1" exp=""诱导公式:180°+α与α""/>
<element type="text" label="text1">
<show object="true" label="true" ev="43"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="-3.2199999999999998" y="5.860000000000001" z="1"/>
</element>
<command name="Rotate">
<input a0="P" a1="90°" a2="C"/>
<output a0="P'"/>
</command>
<element type="point" label="P'">
<show object="true" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.8363849419988948" y="0.5481425259889123" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Ray">
<input a0="C" a1="P'"/>
<output a0="f"/>
</command>
<element type="ray" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.5481425259889123" y="-0.8363849419988948" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="h" a1="f"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="true" label="true" ev="4"/>
<condition showObject="isTan"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="NaN" y="NaN" z="NaN"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="G" a1="B"/>
<output a0="tanP'2"/>
</command>
<element type="segment" label="tanP'2">
<show object="true" label="false" ev="4"/>
<condition showObject="isTan"/>
<objColor r="128" g="0" b="128" alpha="0"/>
<layer val="0"/>
<labelOffset x="24" y="2"/>
<labelMode val="3"/>
<coords x="NaN" y="NaN" z="NaN"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="-1.44"/>
</element>
<element type="boolean" label="isSin">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="40" y="81"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="$sin(90°+α)==>cosα$"/>
</element>
<element type="boolean" label="isCos">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="37" y="177"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="$cos(90°+α)==>sinα$"/>
</element>
<element type="boolean" label="isTan">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="35" y="266"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="$tan(90°+α)==>tanα$"/>
</element>
<command name="If">
<input a0="theta ≥ 0°" a1="LaTeX[theta]" a2="LaTeX[(("(" * theta) * ")")]"/>
<output a0="txtTheta"/>
</command>
<element type="text" label="txtTheta">
<show object="false" label="true" ev="11"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1" size="0" style="0"/>
<startPoint x="0" y="0" z="1"/>
</element>
<expression label="E'" exp="(x(P'), 0)" type="point" />
<element type="point" label="E'">
<show object="true" label="true" ev="4"/>
<condition showObject="isSin ∨ isCos"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.8363849419988948" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="C" a1="E'" a2="P'"/>
<output a0="t2" a1="p'" a2="c_1" a3="e'"/>
</command>
<element type="polygon" label="t2">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="isSin ∨ isCos"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="p'">
<show object="true" label="false" ev="4"/>
<condition showObject="isCos"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0" y="-0.8363849419988948" z="0"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c_1">
<show object="true" label="false" ev="4"/>
<condition showObject="isSin"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-0.5481425259889123" y="0" z="-0.45845815480636415"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="e'">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0.5481425259889123" y="0.8363849419988948" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="angle" label="α">
<angleStyle val="3"/>
<value val="0"/>
<slider min="0" max="90°" absoluteScreenLocation="true" width="180" x="32" y="54" fixed="false" horizontal="true" showAlgebra="true"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="10" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<animation speed="1" type="0" playing="false"/>
</element>
<command name="Rotate">
<input a0="P" a1="α" a2="C"/>
<output a0="P'_1"/>
</command>
<element type="point" label="P'_1">
<show object="true" label="true"/>
<objColor r="255" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.5481425259889123" y="0.8363849419988948" z="1"/>
<pointSize val="9"/>
<pointStyle val="1"/>
</element>
<command name="Rotate">
<input a0="E" a1="α" a2="C"/>
<output a0="E'_1"/>
</command>
<element type="point" label="E'_1">
<show object="true" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.5481425259889123" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Rotate">
<input a0="C" a1="α" a2="C"/>
<output a0="C'"/>
</command>
<element type="point" label="C'">
<show object="true" label="true"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Polygon">
<input a0="P'_1" a1="E'_1" a2="C'"/>
<output a0="t1'" a1="c'" a2="p'_1" a3="e'_1"/>
</command>
<element type="polygon" label="t1'">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="c'">
<show object="true" label="false"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<auxiliary val="false"/>
<coords x="0.8363849419988948" y="0" z="-0.45845815480636415"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="p'_1">
<show object="true" label="false"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="3"/>
<auxiliary val="false"/>
<coords x="0" y="-0.5481425259889123" z="0"/>
<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="e'_1">
<show object="true" label="false"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-0.8363849419988948" y="0.5481425259889123" z="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="CurveCartesian">
<input a0="(rp'(t) * cos(t))" a1="(rp'(t) * sin(t))" a2="t" a3="Min[0, 90° + theta]" a4="Max[0, 90° + theta]"/>
<output a0="d"/>
</command>
<element type="curvecartesian" label="d">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="1" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="txtSins" exp=""\sin(" + (LaTeX[theta]) + ")=" + (LaTeX[sin(theta)]) + "\\
\cos(90°+" + (LaTeX[theta]) + ")=" + (LaTeX[cos(90° + theta)]) + """ />
<element type="text" label="txtSins">
<show object="true" label="true" ev="11"/>
<condition showObject="isCos"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="-2.160000000000002" y="2.2600000000000007" z="1"/>
</element>
<expression label="txtCoss" exp=""\cos(" + (LaTeX[theta]) + ")=" + (LaTeX[cos(theta)]) + "\\
\sin(90°+" + (LaTeX[theta]) + ")=" + (LaTeX[sin(90° + theta)]) + """ />
<element type="text" label="txtCoss">
<show object="true" label="true" ev="11"/>
<condition showObject="isSin"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="-2.140000000000002" y="4.159999999999998" z="1"/>
</element>
<command name="Mirror">
<input a0="P'" a1="C"/>
<output a0="P''"/>
</command>
<element type="point" label="P''">
<show object="true" label="true" ev="4"/>
<condition showObject="x(P') < 0 ∧ isTan"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.8363849419988948" y="-0.5481425259889123" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Ray">
<input a0="C" a1="P''"/>
<output a0="i"/>
</command>
<element type="ray" label="i">
<show object="true" label="false" ev="4"/>
<condition showObject="x(P') < 0 ∧ isTan"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.5481425259889123" y="0.8363849419988948" z="0"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="h" a1="i"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="true" label="true" ev="4"/>
<condition showObject="x(P') < 0 ∧ isTan"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.8363849419988948" y="-0.5481425259889123" z="0.8363849419988948"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="B" a1="D"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="true" label="false" ev="4"/>
<condition showObject="x(P') < 0 ∧ isTan"/>
<objColor r="128" g="0" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.6553711078046126" y="0" z="-0.6553711078046126"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="txtTans" exp=""\tan(" + (LaTeX[theta]) + ")=" + (LaTeX[tan(theta)]) + "\\
\tan(90°+" + (LaTeX[theta]) + ")=" + (LaTeX[tan(90° + theta)]) + "\\" + (LaTeX[tan(90° + theta)]) + "=-\frac{1}{" + (LaTeX[tan(theta)]) + "}"" />
<element type="text" label="txtTans">
<show object="true" label="true" ev="11"/>
<condition showObject="isTan"/>
<objColor r="128" g="0" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="-2.2200000000000015" y="0.580000000000001" z="1"/>
</element>
</construction>
</geogebra>