Graph and Properties of the Tangent Function y=tan(x)
2018-10-14 11:56
This applet demonstrates the graph and properties of the tangent function y=tan(x). Users can intuitively observe its periodicity, asymptotes, and monotonic increase across intervals. Combining geometric constructions with function graphs helps deeply understand the core concepts and geometric meanings of trigonometric functions.
Loading the GeoGebra board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.426.0" app="classic" platform="d" id="fe2ee1f6-3884-4a87-a307-90c1ee5c5610" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="1280" height="777" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.15625" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="290,136,700,550" />
<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="false" inframe="false" stylebar="false" location="1,1,1" 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="8" toolbar="1001 | 1002 | 1003 || 1005 | 1004 || 1006 | 1007 | 1010 || 1008 1009 || 66 68 || 6" visible="false" inframe="false" stylebar="false" location="1,3" size="300" window="100,100,600,400" />
<view id="1" visible="true" inframe="false" stylebar="true" location="1" size="1269" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="200" window="100,100,600,400" />
<view id="16" visible="false" inframe="false" stylebar="false" location="1" size="150" 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="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
</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="20"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="1269" height="616"/>
<coordSystem xZero="379.54806579429436" yZero="322.23280563807435" scale="76.83218691764456" yscale="76.83218691764448"/>
<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="20"/>
<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.5707963267948966" tickExpression="π / 2" positiveAxis="true"/>
<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.0" tickExpression="1"/>
<grid distX="1.0" distY="1.0" distTheta="0.5235987755982988"/>
</euclidianView>
<kernel>
<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"/>
<construction title="正切函數 y=Tan(x)" author="" date="">
<command name="Point">
<input a0="xAxis"/>
<output a0="O"/>
</command>
<element type="point" label="O">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-2.0" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Circle">
<input a0="O" a1="1"/>
<output a0="c"/>
</command>
<element type="conic" label="c">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<eigenvectors x0="1.0" y0="0.0" z0="1.0" x1="-0.0" y1="1.0" z1="1.0"/>
<matrix A0="1.0" A1="1.0" A2="3.0" A3="0.0" A4="2.0" A5="-0.0"/>
<eqnStyle style="specific"/>
</element>
<command name="Intersect">
<input a0="c" a1="xAxis" a2="2"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Tangent">
<input a0="A" a1="c"/>
<output a0="f" a1=""/>
</command>
<element type="line" label="f">
<show object="true" label="false" ev="4"/>
<objColor r="153" g="0" b="51" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="1.0" y="0.0" z="1.0"/>
<lineStyle thickness="5" type="10" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<element type="angle" label="θ">
<angleStyle val="3"/>
<value val="0.0"/>
<slider min="0" max="630°" absoluteScreenLocation="true" width="180.0" x="116.0" y="49.0" fixed="true" horizontal="true" showAlgebra="true"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="-33" y="-6"/>
<labelMode val="1"/>
<lineStyle thickness="10" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
<animation step="1°" speed="1" type="0" playing="true"/>
</element>
<command name="Rotate">
<input a0="A" a1="θ" a2="O"/>
<output a0="A'"/>
</command>
<element type="point" label="A'">
<show object="true" label="false" ev="4"/>
<objColor r="255" g="153" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0" y="0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="0"/>
</element>
<command name="Angle">
<input a0="A" a1="O" a2="A'"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="0"/>
<value val="0.0"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.10000000149011612"/>
<layer val="0"/>
<labelOffset x="19" y="-11"/>
<labelMode val="2"/>
<lineStyle thickness="5" type="0" typeHidden="2" opacity="178"/>
<arcSize val="30"/>
</element>
<expression label="a" exp="(θ / (180°) * π)" />
<element type="numeric" label="a">
<value val="0.0"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Line">
<input a0="O" a1="A'"/>
<output a0="g"/>
</command>
<element type="line" label="g">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0.0" y="1.0" z="0.0"/>
<lineStyle thickness="3" type="15" typeHidden="1" opacity="127"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="g" a1="f"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="true" label="true" ev="4"/>
<objColor r="255" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1.0" y="-0.0" z="1.0"/>
<pointSize val="4"/>
<pointStyle val="1"/>
</element>
<expression label="b" exp="y(B)" />
<element type="numeric" label="b">
<value val="-0.0"/>
<symbolic val="true" />
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="P" exp="(a, b)" type="point" />
<element type="point" label="P">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="204" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<trace val="true"/>
<coords x="0.0" y="-0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="1"/>
</element>
<command name="Segment">
<input a0="B" a1="A"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="100" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-0.0" y="0.0" z="-0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="-3.3605147726619817" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Point">
<input a0="xAxis"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="false" label="true"/>
<objColor r="125" g="125" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-0.6347292651858374" y="0.0" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="C" a1="D"/>
<output a0="u"/>
</command>
<element type="vector" label="u">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="2.7257855074761443" y="0.0" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="C"/>
</element>
<element type="point" label="E">
<show object="false" label="true" ev="4"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="-1.9923598847395927" y="-1.2840940796225022" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<element type="point" label="F">
<show object="false" label="true"/>
<objColor r="77" g="77" b="255" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-1.9923598847395927" y="1.4416914278536446" z="1.0"/>
<pointSize val="5"/>
<pointStyle val="0"/>
</element>
<command name="Vector">
<input a0="E" a1="F"/>
<output a0="v"/>
</command>
<element type="vector" label="v">
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<coords x="0.0" y="2.7257855074761466" z="0.0"/>
<lineStyle thickness="5" type="0" typeHidden="1" opacity="178"/>
<coordStyle style="cartesian"/>
<startPoint exp="E"/>
</element>
<expression label ="i" exp="x = π / 2" type="line"/>
<element type="line" label="i">
<show object="true" label="false" ev="4"/>
<condition showObject="θ ≥ 90°"/>
<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"/>
<coords x="1.0" y="0.0" z="-1.5707963267948966"/>
<lineStyle thickness="5" type="15" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<expression label ="j" exp="x = (3 * π / 2)" type="line"/>
<element type="line" label="j">
<show object="false" label="false" ev="4"/>
<condition showObject="θ ≥ 270°"/>
<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"/>
<coords x="1.0" y="0.0" z="-4.71238898038469"/>
<lineStyle thickness="5" type="15" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
<expression label ="p" exp="p: y = tan(x)" type="function"/>
<element type="function" label="p">
<show object="true" label="false" ev="4"/>
<condition showObject="d"/>
<objColor r="204" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelMode val="0"/>
<fixed val="true"/>
<lineStyle thickness="11" type="0" typeHidden="1" opacity="178"/>
</element>
<element type="boolean" label="d">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0.0"/>
<layer val="0"/>
<labelOffset x="111" y="461"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="顯示函數 tan(x)"/>
</element>
<expression label ="k" exp="x = (5 * π / 2)" type="line"/>
<element type="line" label="k">
<show object="false" label="false" ev="4"/>
<condition showObject="θ ≥ 450°"/>
<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"/>
<coords x="1.0" y="0.0" z="-7.853981633974483"/>
<lineStyle thickness="5" type="15" typeHidden="1" opacity="178"/>
<eqnStyle style="implicit"/>
</element>
</construction>
</geogebra>