Inequality of the Sum of Squared Cosines of Triangle Angles
2017-10-13 06:42
This applet demonstrates the inequality for the sum of squared cosines of a triangle's interior angles. For any triangle ABC, it satisfies 3/4 ≤ cos²A+cos²B+cos²C ≤ 3. Through dynamic geometric construction in GeoGebra, it visually illustrates the conditions and boundary cases, aiding the understanding of trigonometric functions in triangle geometry.
In collections
不等式
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.389.0" app="classic" platform="w" id="e27cfc93-54ba-4b5b-b647-67c8d39093d5" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="893" height="709" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.26539753639417696" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="593,137,700,550" />
<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="false" 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="237" window="100,100,250,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="474" 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="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="true" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" 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" />
<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="891" 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="3"/>
<font size="16"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="777" height="490"/>
<coordSystem xZero="343.00000000000006" yZero="247.9999999999999" scale="50.00000000000003" yscale="49.99999999999998"/>
<evSettings axes="false" grid="true" gridIsBold="false" pointCapturing="2" 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="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
</euclidianView>
<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="-1.3815205191430813" yZero="-1.565229594582576" zZero="-1.5905885442930239" scale="96.43697038706986" xAngle="17" zAngle="300"/>
<evSettings axes="true" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" gridType="3"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<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="1"/>
<projection type="1"/>
</euclidianView3D>
<construction title="" author="" date="">
<expression label ="A" exp="(π / 3, π / 3)" type="point"/>
<element type="point" label="A">
<show object="true" label="false" ev="4"/>
<objColor r="96" g="96" b="96" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<isShape val="false"/>
<coords x="1.0471975511965976" y="1.0471975511965976" z="1"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<expression label ="inDomain" exp="inDomain(x, y) = (((x ≥ 0) * (y ≥ 0)) * (x + y ≤ π))"/>
<element type="functionNVar" label="inDomain">
<show object="false" label="false" ev="4"/>
<objColor r="102" g="102" b="102" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label ="g" exp="g(x, y) = cos(x)^(2) + cos(y)^(2) + cos(x + y)^(2)"/>
<element type="functionNVar" label="g">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label="h" exp="h(x, y) = g(x, y) / inDomain(x, y)" />
<element type="functionNVar" label="h">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label="fmin" exp="fmin(x, y) = 3 / 4 / inDomain(x, y)" />
<element type="functionNVar" label="fmin">
<show object="true" label="false" ev="4"/>
<objColor r="204" g="0" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label="fmax" exp="fmax(x, y) = 3 / inDomain(x, y)" />
<element type="functionNVar" label="fmax">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label ="a" exp="z = 0" type="plane"/>
<element type="plane3d" label="a">
<show object="true" label="false" ev="4"/>
<objColor r="204" g="255" b="102" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<coords x="0" y="0" z="1" w="0"/>
<fading val="0"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label="B" exp="(x(A), y(A), g(A))" type="point" />
<element type="point3d" label="B">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<coords x="1.0471975511965976" y="1.0471975511965976" z="0.75" w="1"/>
<pointSize val="5"/>
</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"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text1" exp=""(\frac{π}{3}, \frac{π}{3}, \frac{3}{4})""/>
<element type="text" label="text1">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="B"/>
<labelOffset x="1" y="7"/>
</element>
<expression label ="C" exp="(π, 0, 3)" type="point"/>
<element type="point3d" label="C">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="255" alpha="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="0" z="3" w="1"/>
<pointSize val="5"/>
</element>
<expression label ="D" exp="(0, π, 3)" type="point"/>
<element type="point3d" label="D">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<coords x="0" y="3.141592653589793" z="3" w="1"/>
<pointSize val="5"/>
</element>
<expression label ="E" exp="(0, 0, 3)" type="point"/>
<element type="point3d" label="E">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="153" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<coords x="0" y="0" z="3" w="1"/>
<pointSize val="5"/>
</element>
<expression label="text2" exp=""(0,π,3)""/>
<element type="text" label="text2">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="D"/>
<labelOffset x="1" y="-31"/>
</element>
<expression label="text3" exp=""(0,0,3)""/>
<element type="text" label="text3">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="E"/>
<labelOffset x="-73" y="-32"/>
</element>
<expression label="text4" exp=""(π,0,3)""/>
<element type="text" label="text4">
<show object="true" label="false" ev="20"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<startPoint exp="C"/>
<labelOffset x="2" y="-33"/>
</element>
<expression label="text5" exp=""0≤A,B,C≤π,\; A+B+C=π\\ \frac{3}{4}≤\cos^2A+\cos^2B+\cos^2C≤3""/>
<element type="text" label="text5">
<show object="true" label="false" ev="37"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="250" b="205" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1" size="0" style="0"/>
<absoluteScreenLocation x="597" y="18"/>
</element>
</construction>
</geogebra>