Trigonometric Inequality Visualization: -3 ≤ cosA+cosB+cosC ≤ 3/2
2017-10-14 06:43
This GeoGebra applet interactively demonstrates the trigonometric inequality -3 ≤ cosA + cosB + cosC ≤ 3/2. Users can adjust angles A, B, and C via draggable points and parameters to observe real-time changes in the cosine sum, verifying the upper and lower bounds. The visualization incorporates vectors, contour lines, and extremum annotations, intuitively illustrating that the maximum value of 3/
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<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint exp="Q"/>
<labelOffset x="-28" y="-45"/>
</element>
<expression label="f1" exp="f1(x, y) = (f(x, y) + 0.02) / 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"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label="fmax" exp="fmax(x, y) = z0 / inPad(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"/>
<isShape val="false"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
<command name="PolyLine">
<input a0="(π, 0, 1)" a1="(0, 0, 1)" a2="(0, π, 1)" a3="(π, 0, 1)"/>
<output a0="contour1"/>
</command>
<element type="polyline3d" label="contour1">
<lineStyle thickness="4" 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, 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)"/>
<output a0="contour2"/>
</command>
<element type="polyline3d" label="contour2">
<lineStyle thickness="4" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="showGlobal"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<command name="Vector">
<input a0="B2" a1="PinBottom"/>
<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="1.0" w="0.0"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<startPoint exp="B2"/>
</element>
<expression label="f2" exp="f2(x, y) = (f(x, y) + 0.04) / 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"/>
<isShape val="false"/>
<lineStyle thickness="1" type="0" typeHidden="1"/>
</element>
<expression label="fmin" exp="fmin(x, y) = -3 / inPad2(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"/>
<isShape val="false"/>
<lineStyle thickness="0" type="0" typeHidden="1"/>
</element>
</construction>
</geogebra>