Intuitive Geometric Demonstration of Trigonometric Angle Transformations
2020-11-08 13:24
This applet provides an intuitive geometric demonstration of angle transformations in trigonometry. By utilizing circles, segments, intersection points, and angle markers, it dynamically illustrates the geometric meanings and transformation relationships of sine, cosine, and tangent functions across different angles. Designed for intuitive understanding, this interactive teaching material helps st
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.672.0" app="classic" platform="w" id="0f985e78-7448-477a-8733-017d86b67d75" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/apps/xsd/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="810" height="474" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.21851851851851853" 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 | 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="false" 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="4097" visible="false" inframe="false" stylebar="true" location="1,1" size="395" window="610,245,700,550" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="177" tab="ALGEBRA" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="406" 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="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="1" visible="true" inframe="false" stylebar="true" location="" size="474" window="100,100,600,400" />
</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="2"/>
<font size="24"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="809" height="473"/>
<coordSystem xZero="365.14285677611457" yZero="199.3119221224256" scale="152.78621152108659" yscale="152.78621152108647"/>
<evSettings axes="true" 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="true" label="" unitLabel="" tickStyle="1" showNumbers="false" tickDistance="1" tickExpression="1"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1" tickExpression="1"/>
</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="">
<command name="Intersect">
<input a0="xAxis" a1="yAxis"/>
<output a0="O"/>
</command>
<element type="point" label="O">
<show object="true" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelOffset x="-24" y="10"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="AxisStepY">
<input/>
<output a0="a"/>
</command>
<element type="numeric" label="a">
<value val="1"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="AxisStepX">
<input/>
<output a0="b"/>
</command>
<element type="numeric" label="b">
<value val="1"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="AxisStepX">
<input/>
<output a0="c"/>
</command>
<element type="numeric" label="c">
<value val="1"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="C" exp="(x(Corner[2]), 0)" type="point" />
<element type="point" label="C">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="2.911631480321698" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="(0, 0)" a1="1"/>
<output a0="d"/>
</command>
<element type="conic" label="d">
<show object="true" label="false" ev="4"/>
<objColor r="160" g="160" b="160" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<lineStyle thickness="5" type="10" 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="d"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0"/>
<layer val="1"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="0.936887517287137" y="0.34963091961602083" z="1"/>
<pointSize val="8"/>
<pointStyle val="0"/>
</element>
<command name="Angle">
<input a0="C" a1="O" a2="A"/>
<output a0="θ"/>
</command>
<element type="angle" label="θ">
<angleStyle val="0"/>
<value val="0.3571771315618569"/>
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="31" y="-4"/>
<labelMode val="0"/>
<decoration type="6"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="50"/>
<selectionAllowed val="false"/>
</element>
<command name="Segment">
<input a0="O" a1="A"/>
<output a0="f"/>
</command>
<element type="segment" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.34963091961602083" y="0.936887517287137" z="0"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="D" exp="(x(Corner[1]), 0)" type="point" />
<element type="point" label="D">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-2.396439136299821" y="0" z="1"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="A" a1="yAxis"/>
<output a0="A'"/>
</command>
<element type="point" label="A'">
<show object="true" label="false" ev="4"/>
<condition showObject="a180s"/>
<objColor r="204" g="102" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.936887517287137" y="0.34963091961602094" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="A" a1="xAxis"/>
<output a0="A'_{1}"/>
</command>
<element type="point" label="A'_{1}">
<show object="true" label="false" ev="4"/>
<condition showObject="an"/>
<objColor r="0" g="0" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.936887517287137" y="-0.34963091961602083" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Mirror">
<input a0="A" a1="O"/>
<output a0="A'_{2}"/>
</command>
<element type="point" label="A'_{2}">
<show object="true" label="false" ev="4"/>
<condition showObject="a180"/>
<objColor r="204" g="102" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.936887517287137" y="-0.34963091961602083" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="O" a1="A'"/>
<output a0="g"/>
</command>
<element type="segment" label="g">
<show object="true" label="false" ev="4"/>
<condition showObject="a180s"/>
<objColor r="204" g="102" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.34963091961602094" y="-0.936887517287137" z="0"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="O" a1="A'_{2}"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="true" label="false" ev="4"/>
<condition showObject="a180"/>
<objColor r="204" g="102" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.34963091961602083" y="-0.936887517287137" z="0"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="O" a1="A'_{1}"/>
<output a0="i"/>
</command>
<element type="segment" label="i">
<show object="true" label="false" ev="4"/>
<condition showObject="an"/>
<objColor r="0" g="0" b="204" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.34963091961602083" y="0.936887517287137" z="0"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="B" exp="(x(A), 0)" type="point" />
<element type="point" label="B">
<show object="true" label="false" ev="4"/>
<objColor r="204" g="153" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.936887517287137" y="0" z="1"/>
<pointSize val="2"/>
<pointStyle val="0"/>
</element>
<expression label="E" exp="((-x(A)), 0)" type="point" />
<element type="point" label="E">
<show object="true" label="false" ev="4"/>
<condition showObject="a180 ∨ a180s"/>
<objColor r="204" g="153" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.936887517287137" y="0" z="1"/>
<pointSize val="2"/>
<pointStyle val="0"/>
</element>
<expression label="F" exp="(0, y(A))" type="point" />
<element type="point" label="F">
<show object="true" label="false" ev="4"/>
<objColor r="102" g="153" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="0.34963091961602083" z="1"/>
<pointSize val="2"/>
<pointStyle val="0"/>
</element>
<expression label="G" exp="(0, (-y(A)))" type="point" />
<element type="point" label="G">
<show object="true" label="false" ev="4"/>
<condition showObject="an ∨ a180"/>
<objColor r="102" g="153" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="-0.34963091961602083" z="1"/>
<pointSize val="2"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.34963091961602083" y="0" z="-0.32756484424587234"/>
<lineStyle thickness="6" type="15" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="F"/>
<output a0="k"/>
</command>
<element type="segment" label="k">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="-0.936887517287137" z="0.32756484424587234"/>
<lineStyle thickness="6" type="15" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="F" a1="A'"/>
<output a0="l"/>
</command>
<element type="segment" label="l">
<show object="true" label="false" ev="4"/>
<condition showObject="a180s"/>
<objColor r="102" g="153" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-1.1102230246251565e-16" y="-0.936887517287137" z="0.32756484424587234"/>
<lineStyle thickness="6" type="15" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A'" a1="E"/>
<output a0="m"/>
</command>
<element type="segment" label="m">
<show object="true" label="false" ev="4"/>
<condition showObject="a180s"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.34963091961602094" y="0" z="0.32756484424587246"/>
<lineStyle thickness="6" type="15" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E" a1="A'_{2}"/>
<output a0="n"/>
</command>
<element type="segment" label="n">
<show object="true" label="false" ev="4"/>
<condition showObject="a180"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.34963091961602083" y="0" z="0.32756484424587234"/>
<lineStyle thickness="6" type="15" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A'_{2}" a1="G"/>
<output a0="p"/>
</command>
<element type="segment" label="p">
<show object="true" label="false" ev="4"/>
<condition showObject="a180"/>
<objColor r="102" g="153" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="0.936887517287137" z="0.32756484424587234"/>
<lineStyle thickness="6" type="15" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="G" a1="A'_{1}"/>
<output a0="q"/>
</command>
<element type="segment" label="q">
<show object="true" label="false" ev="4"/>
<condition showObject="an"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="0.936887517287137" z="0.32756484424587234"/>
<lineStyle thickness="6" type="15" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A'_{1}" a1="B"/>
<output a0="r"/>
</command>
<element type="segment" label="r">
<show object="true" label="false" ev="4"/>
<condition showObject="an"/>
<objColor r="255" g="127" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="-0.34963091961602083" y="0" z="0.32756484424587234"/>
<lineStyle thickness="6" type="15" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="boolean" label="an">
<value val="true"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="572" y="367"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="If(an,SetValue[a90,false])
If(an,SetValue[a360,false])
If(an,SetValue[a180,false])
If(an,SetValue[a180s,false])"/>
<caption val="-θ(負角)"/>
</element>
<element type="boolean" label="a180">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="33" y="381"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="If(a180,SetValue[a90,false])
If(a180,SetValue[an,false])
If(a180,SetValue[a360,false])
If(a180,SetValue[a180s,false])"/>
<caption val="θ±180°(對向)"/>
</element>
<element type="boolean" label="a180s">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="48" y="42"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="If(a180s,SetValue[a90,false])
If(a180s,SetValue[an,false])
If(a180s,SetValue[a180,false])
If(a180s,SetValue[a360,false])"/>
<caption val="180°-θ"/>
</element>
<command name="Angle">
<input a0="A" a1="O" a2="A'_{2}"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="3.141592653589793"/>
<show object="true" label="true" ev="4"/>
<condition showObject="a180"/>
<objColor r="255" g="0" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-50" y="-32"/>
<labelMode val="3"/>
<decoration type="6"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="80"/>
<selectionAllowed val="false"/>
<caption val="+180°"/>
</element>
<command name="Angle">
<input a0="A'_{2}" a1="O" a2="A"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="0"/>
<value val="3.141592653589793"/>
<show object="true" label="true" ev="4"/>
<condition showObject="a180"/>
<objColor r="255" g="153" b="153" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="3"/>
<decoration type="7"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="70"/>
<selectionAllowed val="false"/>
<caption val="-180°"/>
</element>
<command name="Angle">
<input a0="A'_{1}" a1="O" a2="C"/>
<output a0="γ"/>
</command>
<element type="angle" label="γ">
<angleStyle val="0"/>
<value val="0.3571771315618569"/>
<show object="true" label="true" ev="4"/>
<condition showObject="an"/>
<objColor r="0" g="0" b="204" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="32" y="15"/>
<labelMode val="3"/>
<decoration type="7"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="55"/>
<selectionAllowed val="false"/>
<caption val="-θ"/>
</element>
<expression label="text1" exp=""(\textcolor{0,0,255}{\cosθ},\textcolor{153,51,0}{\sinθ})""/>
<element type="text" label="text1">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.85" size="-4" style="1"/>
<startPoint exp="A"/>
<labelOffset x="5" y="-29"/>
</element>
<expression label="text2" exp=""\,\,\,\,\,\,\left(\textcolor{0,0,255}{\cos(-θ)},\textcolor{255,127,0}{\sin(-θ)}\right)\\ =\,\,\,\,\,\,\,\,(\textcolor{0,0,255}{\cosθ},\,-\textcolor{153,51,0}{\sinθ})""/>
<element type="text" label="text2">
<show object="true" label="false" ev="40"/>
<condition showObject="an"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.85" size="-4" style="1"/>
<startPoint exp="A'_{1}"/>
<labelOffset x="-13" y="-7"/>
</element>
<expression label="text3" exp=""\,\,\,\,\,\,\left(\textcolor{102,153,255}{\cos(θ±180°)},\textcolor{255,127,0}{\sin(θ±180°)}\right)\\
=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(-\textcolor{0,0,255}{\cosθ}\,,\,-\textcolor{153,51,0}{\sinθ})""/>
<element type="text" label="text3">
<show object="true" label="false" ev="40"/>
<condition showObject="a180"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.85" size="-4" style="1"/>
<startPoint exp="A'_{2} + (-1.2, 0)"/>
<labelOffset x="-11" y="9"/>
</element>
<expression label="text4" exp=""\,\,\,\,\,\,\left(\textcolor{102,153,255}{\cos(180°-θ)},\textcolor{153,51,0}{\sin(180°-θ)}\right)\\
=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(-\textcolor{0,0,255}{\cosθ}\,,\,\textcolor{153,51,0}{\sinθ})""/>
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<condition showObject="a180s"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.85" size="-4" style="1"/>
<startPoint exp="A' + (-1.2, 0)"/>
<labelOffset x="-22" y="-65"/>
</element>
<command name="Angle">
<input a0="A'" a1="O" a2="D"/>
<output a0="δ"/>
</command>
<element type="angle" label="δ">
<angleStyle val="0"/>
<value val="0.357177131561857"/>
<show object="true" label="true" ev="4"/>
<condition showObject="a180s"/>
<objColor r="204" g="102" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-55" y="-7"/>
<labelMode val="3"/>
<decoration type="7"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="70"/>
<selectionAllowed val="false"/>
<caption val="-θ"/>
</element>
<command name="Angle">
<input a0="C" a1="O" a2="A'"/>
<output a0="ε"/>
</command>
<element type="angle" label="ε">
<angleStyle val="0"/>
<value val="2.784415522027936"/>
<show object="true" label="true" ev="4"/>
<condition showObject="a180s"/>
<objColor r="255" g="153" b="0" alpha="0.05000000074505806"/>
<layer val="0"/>
<labelOffset x="-1" y="-43"/>
<labelMode val="3"/>
<decoration type="6"/>
<lineStyle thickness="6" type="0" typeHidden="2" opacity="127"/>
<arcSize val="85"/>
<selectionAllowed val="false"/>
<caption val="180°-θ"/>
</element>
<command name="Angle">
<input a0="A" a1="O" a2="Rotate[A, ((2 * π) * 0.999), O]"/>
<output a0="ζ"/>
</command>
<element type="angle" label="ζ">
<angleStyle val="0"/>
<value val="6.276902121872407"/>
<show object="true" label="true" ev="4"/>
<condition showObject="a360"/>
<objColor r="0" g="153" b="255" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="64" y="18"/>
<labelMode val="3"/>
<decoration type="6"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="70"/>
<selectionAllowed val="false"/>
<caption val="+360°"/>
</element>
<command name="Angle">
<input a0="Rotate[A, ((-2 * π) * 0.999), O]" a1="O" a2="A"/>
<output a0="η"/>
</command>
<element type="angle" label="η">
<angleStyle val="0"/>
<value val="6.276902121872407"/>
<show object="true" label="true" ev="4"/>
<condition showObject="a360"/>
<objColor r="0" g="204" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-61" y="52"/>
<labelMode val="3"/>
<decoration type="7"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="90"/>
<selectionAllowed val="false"/>
<caption val="-360°"/>
</element>
<element type="boolean" label="a360">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="597" y="157"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="If(a360,SetValue[a90,false])
If(a360,SetValue[an,false])
If(a360,SetValue[a180,false])
If(a360,SetValue[a180s,false])"/>
<caption val="θ±360°(同界角)"/>
</element>
<command name="Mirror">
<input a0="A" a1="y = x"/>
<output a0="A'_{3}"/>
</command>
<element type="point" label="A'_{3}">
<show object="true" label="false" ev="8"/>
<condition showObject="a90"/>
<objColor r="0" g="255" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.3496309196160209" y="0.936887517287137" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label="H" exp="(0, x(A))" type="point" />
<element type="point" label="H">
<show object="true" label="false" ev="4"/>
<condition showObject="a90"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="0.936887517287137" z="1"/>
<pointSize val="2"/>
<pointStyle val="0"/>
</element>
<expression label="I" exp="(y(A), 0)" type="point" />
<element type="point" label="I">
<show object="true" label="false" ev="4"/>
<condition showObject="a90"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.34963091961602083" y="0" z="1"/>
<pointSize val="2"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="H" a1="A'_{3}"/>
<output a0="f_1"/>
</command>
<element type="segment" label="f_1">
<show object="true" label="false" ev="4"/>
<condition showObject="a90"/>
<objColor r="153" g="51" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0" y="0.3496309196160209" z="-0.3275648442458724"/>
<lineStyle thickness="5" type="15" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A'_{3}" a1="I"/>
<output a0="g_1"/>
</command>
<element type="segment" label="g_1">
<show object="true" label="false" ev="4"/>
<condition showObject="a90"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.936887517287137" y="-5.551115123125783e-17" z="-0.32756484424587234"/>
<lineStyle thickness="5" type="15" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text5" exp=""\,\,\,\,\,\,\left(\textcolor{153,51,0}{\cos(90°-θ)},\textcolor{0,0,255}{\sin(90°-θ)}\right)\\
=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\textcolor{153,51,0}{\sinθ},\,\textcolor{0,0,255}{\cosθ})""/>
<element type="text" label="text5">
<show object="true" label="false" ev="40"/>
<condition showObject="a90"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.85" size="-4" style="1"/>
<startPoint exp="A'_{3} + (0, 0.3)"/>
<labelOffset x="-61" y="0"/>
</element>
<element type="boolean" label="a90">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="1"/>
<labelOffset x="638" y="62"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<ggbscript onUpdate="If(a90,SetValue[a360,false])
If(a90,SetValue[an,false])
If(a90,SetValue[a180,false])
If(a90,SetValue[a180s,false])"/>
<caption val="90°-θ(餘角)"/>
</element>
<command name="Segment">
<input a0="A'_{3}" a1="O"/>
<output a0="h_1"/>
</command>
<element type="segment" label="h_1">
<show object="true" label="false" ev="4"/>
<condition showObject="a90"/>
<objColor r="0" g="255" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<coords x="0.936887517287137" y="-0.3496309196160209" z="0"/>
<lineStyle thickness="9" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="A'_{3}" a1="O" a2="(0, 1)"/>
<output a0="ι"/>
</command>
<element type="angle" label="ι">
<angleStyle val="0"/>
<value val="0.35717713156185693"/>
<show object="true" label="true" ev="4"/>
<condition showObject="a90"/>
<objColor r="0" g="255" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-25" y="-45"/>
<labelMode val="3"/>
<decoration type="7"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="70"/>
<selectionAllowed val="false"/>
<caption val="90°-θ"/>
</element>
<command name="If">
<input a0="a360" a1=""(θ±360°)"" a2="If[a90, "(90°-θ)", If[an, "(-θ)", If[a180, "(θ±180)°", If[a180s, "(180°-θ)", "θ"]]]]"/>
<output a0="textAng"/>
</command>
<element type="text" label="textAng">
<show object="false" label="false" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<font serif="false" sizeM="0.85" size="-4" style="0"/>
</element>
<expression label="text6" exp=""\left\{\begin{array}{rr}
\textcolor{153,51,0}{\sin" + (LaTeX[textAng + (If[a360 ∨ a90 ∨ an ∨ a180 ∨ a180s, "=", ""])]) + "}&\textcolor{153,51,0}{" + (LaTeX[If[an ∨ a180, "-", ""] + (If[a90, "\cosθ", If[a360 ∨ an ∨ a180 ∨ a180s, "\sinθ", ""]])]) + "}\\
\textcolor{0,0,255}{\cos" + (LaTeX[textAng + (If[a360 ∨ a90 ∨ an ∨ a180 ∨ a180s, "=", ""])]) + "}&\textcolor{0,0,255}{" + (LaTeX[If[a180 ∨ a180s, "-", ""] + (If[a90, "\sinθ", If[a360 ∨ an ∨ a180 ∨ a180s, "\cosθ", ""]])]) + "}\\
\textcolor{255,0,255}{\tan" + (LaTeX[textAng + (If[a360 ∨ a90 ∨ an ∨ a180 ∨ a180s, "=", ""])]) + "}&\textcolor{255,0,255}{" + (LaTeX[If[an ∨ a180s, "-", ""] + (If[a90, "\cotθ", If[a360 ∨ an ∨ a180 ∨ a180s, "\tanθ", ""]])]) + "}
\end{array}\right."" />
<element type="text" label="text6">
<show object="true" label="false" ev="8"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="0.85" size="-4" style="1"/>
<absoluteScreenLocation x="242" y="356"/>
</element>
</construction>
</geogebra>