cos(α) + cos(β) + cos(γ) 的对称性与最值优化
2017-10-14 09:02
本 GeoGebra 范例深入探讨了三角函数表达式 cos(α) + cos(β) + cos(γ) 的对称性质及其在几何与最值优化问题中的应用。通过构建多边形、向量与三维立体图形，动态演示角度变化对余弦和的影响。结合滑块动画与轨迹追踪，直观展示了对称条件下的极值分布，适用于三角函数、解析几何及不等式优化的教学与探究。
所属专题
GeoGebra曲面互动教学资源集
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.389.0" app="classic" platform="d" id="7a8b886e-259f-4fb6-9091-ee5d444b7384" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1024" height="748" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.25" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="740,350,700,550" />
<view id="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3,3" size="300" window="100,100,600,400" />
<view id="1" visible="false" inframe="false" stylebar="true" location="1,3" 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="2" visible="false" inframe="false" stylebar="false" location="3" size="256" window="100,100,250,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="300" window="50,50,500,500" />
<view id="32" visible="false" inframe="false" stylebar="true" location="1" size="300" window="50,50,500,500" />
<view id="64" toolbar="0" 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="900" window="50,50,500,500" />
<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="true" inframe="false" stylebar="true" location="1" size="1013" window="100,100,600,400" />
</views>
<toolbar show="true" 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="true" cmd="true" top="algebra" />
<dockBar show="true" east="true" />
</perspective>
</perspectives>
<labelingStyle val="3"/>
<font size="12"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="500" height="596"/>
<coordSystem xZero="209.0000000000001" yZero="399.99999999999994" scale="50.00000000000002" yscale="49.99999999999996"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<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"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<euclidianView>
<viewNumber viewNo="2"/>
<coordSystem xZero="215.0" yZero="315.0" scale="50.0" yscale="50.0"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<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"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<algebraView>
<mode val="3"/>
</algebraView>
<kernel>
<uses3D val="true"/>
<continuous val="false"/>
<usePathAndRegionParameters val="true"/>
<decimals val="2"/>
<angleUnit val="degree"/>
<algebraStyle val="0" spreadsheet="0"/>
<coordStyle val="0"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<euclidianView3D>
<coordSystem xZero="-1.7643720596449124" yZero="-1.634831762066033" zZero="-1.7367887483121788E-4" scale="69.64960959097951" xAngle="21.0" zAngle="-63.0"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="3"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="3.141592653589793" tickExpression="π"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="3.141592653589793" tickExpression="π"/>
<axis id="2" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<plate show="false"/>
<bgColor r="255" g="255" b="255"/>
<clipping use="false" show="false" size="2"/>
<projection type="1"/>
</euclidianView3D>
<construction title="" author="test" date="">
<expression label ="f" exp="f(x, y) = cos(x) + cos(y) + cos((-x) - y)"/>
<element type="functionNVar" label="f">
<show object="false" label="false" ev="4"/>
<condition showObject="showGlobalF"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label ="g" exp="g(x, y) = cos(x) + cos(y) + cos(π - x - y)"/>
<element type="functionNVar" label="g">
<show object="false" label="false" ev="4"/>
<condition showObject="showGlobalG"/>
<objColor r="102" g="204" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label ="h" exp="x + y = π" type="line"/>
<element type="line" label="h">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="1.0" y="1.0" z="-3.141592653589793"/>
<lineStyle thickness="2" type="0" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<expression label ="r" exp="π"/>
<element type="numeric" label="r">
<value val="3.141592653589793"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<animation step="1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
</element>
<expression label="inD1" exp="inD1(x, y) = (((abs(x) ≤ r) * (abs(y) ≤ r)) * (x + y ≤ π))" />
<element type="functionNVar" label="inD1">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label="inD2" exp="inD2(x, y) = (((abs(x - π) ≤ r) * (abs(y - π) ≤ r)) * (x + y ≥ π))" />
<element type="functionNVar" label="inD2">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="85" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label="f1" exp="f1(x, y) = f(x, y) / inD1(x, y)" />
<element type="functionNVar" label="f1">
<show object="true" label="false" ev="4"/>
<condition showObject="¬showGlobalF"/>
<objColor r="153" g="51" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label="g1" exp="g1(x, y) = g(x, y) / inD2(x, y)" />
<element type="functionNVar" label="g1">
<show object="true" label="false" ev="4"/>
<condition showObject="¬showGlobalG"/>
<objColor r="0" g="100" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label ="xyPlane" exp="z = 0" type="plane"/>
<element type="plane3d" label="xyPlane">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="204" b="255" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.0" y="0.0" z="1.0" w="0.0"/>
<fading val="0.0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="PointIn">
<input a0="xyPlane"/>
<output a0="A1"/>
</command>
<element type="point3d" label="A1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<coords x="3.141592653589793" y="-3.141592653589793" z="0.0" w="1.0"/>
<pointSize val="5"/>
</element>
<expression label="B1" exp="(π - y(A1), π - x(A1), 0)" type="point" />
<element type="point3d" label="B1">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="6.283185307179586" y="0.0" z="0.0" w="1.0"/>
<pointSize val="5"/>
</element>
<expression label="A" exp="(x(A1), y(A1), f(A1))" type="point" />
<element type="point3d" label="A">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="3.141592653589793" y="-3.141592653589793" z="-1.0" w="1.0"/>
<pointSize val="5"/>
</element>
<expression label="B" exp="(x(B1), y(B1), g(B1))" type="point" />
<element type="point3d" label="B">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="6.283185307179586" y="0.0" z="1.0" w="1.0"/>
<pointSize val="5"/>
</element>
<command name="PolyLine">
<input a0="(π, 0, -1)" a1="(π, (-π), -1)" a2="(0, (-π), -1)" a3="((-π), 0, -1)" a4="((-π), π, -1)" a5="(0, π, -1)" a6="(π, 0, -1)"/>
<output a0="contour1"/>
</command>
<element type="polyline3d" label="contour1">
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="PolyLine">
<input a0="(π, 0, 1)" a1="((2 * π), 0, 1)" a2="((2 * π), π, 1)" a3="(π, (2 * π), 1)" a4="(0, (2 * π), 1)" a5="(0, π, 1)" a6="(π, 0, 1)"/>
<output a0="contour2"/>
</command>
<element type="polyline3d" label="contour2">
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
</element>
<command name="Vector">
<input a0="A1" a1="A"/>
<output a0="vecA"/>
</command>
<element type="vector3d" label="vecA">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.0" y="0.0" z="-1.0" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<startPoint exp="A1"/>
</element>
<command name="Vector">
<input a0="B1" a1="B"/>
<output a0="vecB"/>
</command>
<element type="vector3d" label="vecB">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.0" y="0.0" z="1.0" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<startPoint exp="B1"/>
</element>
<command name="Segment">
<input a0="A1" a1="B1"/>
<output a0="segAB1"/>
</command>
<element type="segment3d" label="segAB1">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="A" a1="B"/>
<output a0="segAB"/>
</command>
<element type="segment3d" label="segAB">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Intersect">
<input a0="h" a1="segAB1"/>
<output a0="C"/>
</command>
<element type="point3d" label="C">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="4.71238898038469" y="-1.5707963267948966" z="0.0" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Polygon">
<input a0="{C, A1, A}"/>
<output a0="tri1"/>
</command>
<element type="polygon3d" label="tri1">
<lineStyle thickness="0" type="0" typeHidden="2" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="153" g="102" b="255" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
</element>
<command name="Polygon">
<input a0="{C, B1, B}"/>
<output a0="tri2"/>
</command>
<element type="polygon3d" label="tri2">
<lineStyle thickness="0" type="0" typeHidden="2" opacity="178"/>
<show object="true" label="false" ev="4"/>
<objColor r="153" g="102" b="255" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
</element>
<command name="Segment">
<input a0="(π + r, (-r), 0)" a1="(0 - r, π + r, 0)"/>
<output a0="segSym"/>
</command>
<element type="segment3d" label="segSym">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="6" type="0" typeHidden="2"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp=""Drag Here !""/>
<element type="text" label="text1">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<startPoint exp="A1"/>
<labelOffset x="-27" y="-25"/>
</element>
<expression label ="Pin1" exp="(0, 0, 3)" type="point"/>
<element type="point3d" label="Pin1">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.0" y="0.0" z="3.0" w="1.0"/>
<pointSize val="5"/>
</element>
<expression label ="Pin2" exp="(π, (2 * π), 1)" type="point"/>
<element type="point3d" label="Pin2">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="3.141592653589793" y="6.283185307179586" z="1.0" w="1.0"/>
<pointSize val="5"/>
</element>
<command name="Point">
<input a0="Ray[Pin1, Pin1 + (1, 1, 2)]"/>
<output a0="Pin11"/>
</command>
<element type="point3d" label="Pin11">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="0.25210549909972924" y="0.25210549909972924" z="3.5042109981994587" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Vector">
<input a0="Pin11" a1="Pin1"/>
<output a0="vecPin1"/>
</command>
<element type="vector3d" label="vecPin1">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="-0.25210549909972924" y="-0.25210549909972924" z="-0.5042109981994587" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<startPoint exp="Pin11"/>
</element>
<command name="Point">
<input a0="Ray[Pin2, Pin2 + (1, 1, 2)]"/>
<output a0="Pin21"/>
</command>
<element type="point3d" label="Pin21">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="127" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="3.707063918324516" y="6.84865657191431" z="2.130942529469446" w="1.0"/>
<pointSize val="3"/>
</element>
<command name="Vector">
<input a0="Pin21" a1="Pin2"/>
<output a0="vecPi21"/>
</command>
<element type="vector3d" label="vecPi21">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<coords x="-0.5654712647347231" y="-0.5654712647347235" z="-1.1309425294694462" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<startPoint exp="Pin21"/>
</element>
<expression label="txtButtonF" exp=""▷ f(x,y)=\cos(x) + \cos(y) + \cos(-x-y)""/>
<element type="text" label="txtButtonF">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint exp="Pin11"/>
<labelOffset x="2" y="-35"/>
<ggbscript val="SetValue(showGlobalF, !showGlobalF)"/>
</element>
<expression label="txtButtonG" exp=""▶g(x,y)=\cos(x) + \cos(y) + \cos(π-x-y)""/>
<element type="text" label="txtButtonG">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="255" b="0" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.0" size="0" style="0"/>
<startPoint exp="Pin21"/>
<labelOffset x="-71" y="-38"/>
<ggbscript val="SetValue(showGlobalG, !showGlobalG)"/>
</element>
<element type="boolean" label="showGlobalF">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="4"/>
<checkbox fixed="true"/>
</element>
<element type="boolean" label="showGlobalG">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="4"/>
<checkbox fixed="true"/>
</element>
</construction>
</geogebra>