Trigonometric Inequality: -1/8 ≤ cosα cosβ cosγ ≤ 1
2017-10-16 09:04
This GeoGebra applet dynamically demonstrates the trigonometric inequality -1/8 ≤ cosα cosβ cosγ ≤ 1. By constructing points, vectors, and polygons, users can visually observe the range of the product of cosines as the angles α, β, and γ vary. Combining function graphing with geometric construction, this model helps learners deeply understand trigonometric identities and optimization problems.
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<input a0="level" a1="f"/>
<output a0="leveCurve"/>
</command>
<element type="implicitpoly" label="leveCurve">
<show object="true" label="true" ev="4"/>
<condition showObject="showLevel"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="5" type="0" typeHidden="2"/>
<userinput show="false"/></element>
<command name="Segment">
<input a0="PinGround" a1="PinLevel"/>
<output a0="segLevel"/>
</command>
<element type="segment3d" label="segLevel">
<show object="true" label="false" ev="4"/>
<condition showObject="showLevel"/>
<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>
<expression label="txtLevel" exp=""z=" + (LaTeX[z(PinLevel)]) + """ />
<element type="text" label="txtLevel">
<show object="true" label="false" ev="20"/>
<condition showObject="showLevel"/>
<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"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="PinLevel"/>
<labelOffset x="-28" y="-45"/>
</element>
<expression label="fmax" exp="fmax(x, y) = zmax / inPadMax(x, y)" />
<element type="functionNVar" label="fmax">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="Vector">
<input a0="B2" a1="PinMin"/>
<output a0="vecB"/>
</command>
<element type="vector3d" label="vecB">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<isShape val="false"/>
<coords x="0.0" y="0.0" z="0.7026288783054024" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<startPoint exp="B2"/>
</element>
<expression label ="inHexagon" exp="inHexagon(x, y) = (((abs(x) ≤ π / 2) * (abs(y) ≤ π / 2)) * (abs(x + y) ≤ π / 2))"/>
<element type="functionNVar" label="inHexagon">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label="f0" exp="f0(x, y) = f(x, y) / inHexagon(x, y)" />
<element type="functionNVar" label="f0">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="204" b="0" alpha="0.5"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label ="inD3" exp="inD3(x, y) = (((x ≥ ((-π)) / 2) * (y ≥ ((-π)) / 2)) * (x + y ≤ ((-π)) / 2))"/>
<element type="functionNVar" label="inD3">
<show object="false" label="false" ev="4"/>
<objColor r="153" g="51" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label ="off" exp="0.02"/>
<element type="numeric" label="off">
<value val="0.02"/>
<symbolic val="true" />
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="1"/>
<animation step="0.1" speed="1" type="0" playing="false"/>
</element>
<expression label="f1" exp="f1(x, y) = (f(x, y) + off) / inD(x, y)" />
<element type="functionNVar" label="f1">
<show object="true" label="false" ev="4"/>
<objColor r="204" g="102" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label="f3" exp="f3(x, y) = (f(x, y) + off) / inD3(x, y)" />
<element type="functionNVar" label="f3">
<show object="true" label="false" ev="4"/>
<objColor r="51" g="51" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<expression label="fmin" exp="fmin(x, y) = zmin / inPadMin(x, y)" />
<element type="functionNVar" label="fmin">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="153" b="0" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<element type="boolean" label="showLevel">
<value val="false"/>
<show object="false" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
</element>
<expression label="btnShowLevel" exp=""▷ Level Plane ""/>
<element type="text" label="btnShowLevel">
<show object="true" label="false" ev="37"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<bgColor r="255" g="250" b="205" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<absoluteScreenLocation x="16" y="84"/>
<ggbscript val="SetValue(showLevel, !showLevel)"/>
</element>
<command name="PolyLine">
<input a0="(π / 2, π / 2, 0)" a1="(((-π)) / 2, π / 2, 0)" a2="(((-π)) / 2, ((-π)) / 2, 0)" a3="(π / 2, ((-π)) / 2, 0)" a4="(π / 2, π / 2, 0)"/>
<output a0="contour1"/>
</command>
<element type="polyline3d" label="contour1">
<lineStyle thickness="5" type="0" typeHidden="1" 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"/>
</element>
<command name="PolyLine">
<input a0="(π / 2, 0, 0)" a1="(0, π / 2, 0)"/>
<output a0="contour2"/>
</command>
<element type="polyline3d" label="contour2">
<lineStyle thickness="5" type="0" typeHidden="1" 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"/>
</element>
<command name="PolyLine">
<input a0="(((-π)) / 2, 0, 0)" a1="(0, ((-π)) / 2, 0)"/>
<output a0="contour3"/>
</command>
<element type="polyline3d" label="contour3">
<lineStyle thickness="5" type="0" typeHidden="1" 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"/>
</element>
<expression label="f2" exp="f2(x, y) = (f(x, y) + off) / inD2(x, y)" />
<element type="functionNVar" label="f2">
<show object="true" label="false" ev="4"/>
<objColor r="51" g="51" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
<selectionAllowed val="false"/>
<isShape val="false"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
</construction>
</geogebra>