Reduction Formula 3: Symmetry across the x-axis in Trigonometry
2020-03-14 02:16
This GeoGebra applet demonstrates the third trigonometric reduction formula, showing the relationship between trig values of angles symmetric across the x-axis. Using points on the unit circle and their reflections, it visually presents the sign and value changes of sine, cosine, and tangent under x-axis symmetry, helping students understand the parity and symmetric properties of trig functions.
In collections
Secondary Vocational Mathematics
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<condition showObject="isTan"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="NaN" y="NaN" z="NaN"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="B" a1="H"/>
<output a0="tanP2"/>
</command>
<element type="segment" label="tanP2">
<show object="true" label="false" ev="4"/>
<condition showObject="isTan"/>
<objColor r="128" g="0" b="128" alpha="0"/>
<layer val="0"/>
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<labelMode val="3"/>
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<lineStyle thickness="7" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="?"/>
</element>
<element type="boolean" label="isSin">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="40" y="81"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="正弦"/>
</element>
<element type="boolean" label="isCos">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="37" y="177"/>
<labelMode val="3"/>
<checkbox fixed="true"/>
<caption val="余弦"/>
</element>
<element type="boolean" label="isTan">
<value val="true"/>
<show object="true" label="true" ev="3"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="35" y="266"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="正切"/>
</element>
<command name="If">
<input a0="theta ≥ 0°" a1="LaTeX[theta]" a2="LaTeX[(("(" * theta) * ")")]"/>
<output a0="txtTheta"/>
</command>
<element type="text" label="txtTheta">
<show object="false" label="true" ev="11"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="false" sizeM="1" size="0" style="0"/>
<startPoint x="0" y="0" z="1"/>
</element>
<command name="Polygon">
<input a0="P'" a1="E" a2="C"/>
<output a0="t2" a1="sinP'" a2="cosP'" a3="e_1"/>
</command>
<element type="polygon" label="t2">
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<show object="true" label="false" ev="4"/>
<condition showObject="isSin ∨ isCos"/>
<objColor r="153" g="51" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="sinP'">
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<condition showObject="isSin"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-11" y="1"/>
<labelMode val="3"/>
<auxiliary val="false"/>
<coords x="-0.7810115390679803" y="0" z="0.48775466604164863"/>
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<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="-0.82"/>
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<element type="segment" label="cosP'">
<show object="true" label="false" ev="4"/>
<condition showObject="isCos"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
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<labelMode val="3"/>
<auxiliary val="false"/>
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<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="0.57"/>
</element>
<element type="segment" label="e_1">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="51" b="0" alpha="0"/>
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<labelMode val="0"/>
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<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="CurveCartesian">
<input a0="(rp'(t) * cos(t))" a1="(rp'(t) * sin(t))" a2="t" a3="Min[0, (-theta)]" a4="Max[0, (-theta)]"/>
<output a0="d"/>
</command>
<element type="curvecartesian" label="d">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="true"/>
<lineStyle thickness="1" type="0" typeHidden="1" opacity="178"/>
</element>
<expression label="txtSins" exp=""\sin(" + (LaTeX[theta]) + ")=" + (LaTeX[sin(theta)]) + "\\
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<element type="text" label="txtSins">
<show object="true" label="true" ev="11"/>
<condition showObject="isSin"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="-2.0800000000000023" y="4.18" z="1"/>
</element>
<expression label="txtCoss" exp=""\cos(" + (LaTeX[theta]) + ")=" + (LaTeX[cos(theta)]) + "\\
\cos(-" + (LaTeX[theta]) + ")=" + (LaTeX[cos((-theta))]) + """ />
<element type="text" label="txtCoss">
<show object="true" label="true" ev="11"/>
<condition showObject="isCos"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="-2.200000000000002" y="2.34" z="1"/>
</element>
<expression label="txtTans" exp=""\tan(" + (LaTeX[theta]) + ")=" + (LaTeX[tan(theta)]) + "\\
\tan(-" + (LaTeX[theta]) + ")=" + (LaTeX[tan((-theta))]) + """ />
<element type="text" label="txtTans">
<show object="true" label="true" ev="11"/>
<condition showObject="isTan"/>
<objColor r="128" g="0" b="128" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="5" style="1"/>
<startPoint x="-2.2200000000000015" y="0.580000000000001" z="1"/>
</element>
</construction>
</geogebra>