Inverse Trigonometric Functions and Restricted Ranges

2015-09-15 19:36

This GeoGebra math applet visually demonstrates the restricted ranges of the inverse trigonometric functions arcsin(x), arccos(x), and arctan(x). Utilizing geometric constructions like the Mirror command, it dynamically illustrates the symmetric relationship between the original trigonometric functions and their inverses. It clearly explains the mathematical principle of restricting ranges to sati

#trigonometry #function #graphing #transform

In collections Mathematical Analysis Lecture Notes

<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.146.0" id="074febfb-c0c0-42d0-8c6e-5c10663553f7"  xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
	<window width="1942" height="1242" />
	<perspectives>
<perspective id="tmp">
	<panes>
		<pane location="" divider="0.10298661174047374" orientation="1" />
	</panes>
	<views>
		<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="610,335,700,550" />
		<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="1704" window="100,100,600,400" />
		<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="200" window="100,100,250,400" />
		<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" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
		<view id="70" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
	</views>
	<toolbar show="true" items="0 39 | 1 501 67 , 5 19 , 72 | 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 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 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="0"/>
	<font  size="20"/>
	<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
	<viewNumber viewNo="1"/>
	<size  width="1704" height="988"/>
	<coordSystem xZero="510.49999999999994" yZero="463.7499999999998" scale="112.49999999999997" yscale="112.49999999999991"/>
	<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="0"/>
	<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.5707963267948966"/>
	<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.5707963267948966"/>
</euclidianView>
<kernel>
	<continuous val="false"/>
	<usePathAndRegionParameters val="true"/>
	<decimals val="2"/>
	<angleUnit val="degree"/>
	<algebraStyle val="0"/>
	<coordStyle val="0"/>
	<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label ="f" exp="f(x) = sin(x)"/>
<element type="function" label="f">
	<show object="true" label="true" ev="4"/>
	<objColor r="0" g="100" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<animation step="0.1" speed="1" type="0" playing="false"/>
	<lineStyle thickness="3" type="0" typeHidden="1"/>
</element>
<element type="line" label="a">
	<show object="false" label="true"/>
	<objColor r="255" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<animation step="1" speed="1" type="0" playing="false"/>
	<coords x="-1.0" y="1.0" z="0.0"/>
	<lineStyle thickness="2" type="15" typeHidden="1"/>
	<eqnStyle style="explicit"/>
</element>
<command name="Mirror">
	<input a0="f" a1="a"/>
	<output a0="f_1"/>
</command>
<element type="curvecartesian" label="f_1">
	<show object="false" label="true"/>
	<objColor r="127" g="0" b="255" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<lineStyle thickness="3" type="0" typeHidden="1"/>
</element>
</construction>
</geogebra>

<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.146.0" id="074febfb-c0c0-42d0-8c6e-5c10663553f7"  xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
	<window width="1942" height="1242" />
	<perspectives>
<perspective id="tmp">
	<panes>
		<pane location="" divider="0.10298661174047374" orientation="1" />
	</panes>
	<views>
		<view id="4097" visible="false" inframe="true" stylebar="true" location="1,1,1,1" size="400" window="610,335,700,550" />
		<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="1704" window="100,100,600,400" />
		<view id="2" visible="true" inframe="false" stylebar="false" location="3" size="200" window="100,100,250,400" />
		<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" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
		<view id="70" visible="false" inframe="true" stylebar="true" location="1" size="150" window="50,50,500,500" />
	</views>
	<toolbar show="true" items="0 39 | 1 501 67 , 5 19 , 72 | 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 | 30 29 54 32 31 33 | 17 26 62 73 , 14 68 | 25 52 60 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="0"/>
	<font  size="20"/>
	<graphicsSettings javaLatexFonts="false"/>
</gui>
<euclidianView>
	<viewNumber viewNo="1"/>
	<size  width="1704" height="988"/>
	<coordSystem xZero="510.49999999999994" yZero="463.7499999999998" scale="112.49999999999997" yscale="112.49999999999991"/>
	<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="0"/>
	<axis id="0" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.5707963267948966"/>
	<axis id="1" show="true" label="" unitLabel="" tickStyle="1" showNumbers="true" tickDistance="1.5707963267948966"/>
</euclidianView>
<kernel>
	<continuous val="false"/>
	<usePathAndRegionParameters val="true"/>
	<decimals val="2"/>
	<angleUnit val="degree"/>
	<algebraStyle val="0"/>
	<coordStyle val="0"/>
	<angleFromInvTrig val="false"/>
</kernel>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<expression label ="f" exp="f(x) = sin(x)"/>
<element type="function" label="f">
	<show object="true" label="true" ev="4"/>
	<objColor r="0" g="100" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<animation step="0.1" speed="1" type="0" playing="false"/>
	<lineStyle thickness="3" type="0" typeHidden="1"/>
</element>
<element type="line" label="a">
	<show object="false" label="true"/>
	<objColor r="255" g="0" b="0" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<animation step="1" speed="1" type="0" playing="false"/>
	<coords x="-1.0" y="1.0" z="0.0"/>
	<lineStyle thickness="2" type="15" typeHidden="1"/>
	<eqnStyle style="explicit"/>
</element>
<command name="Mirror">
	<input a0="f" a1="a"/>
	<output a0="f_1"/>
</command>
<element type="curvecartesian" label="f_1">
	<show object="false" label="true"/>
	<objColor r="127" g="0" b="255" alpha="0.0"/>
	<layer val="0"/>
	<labelMode val="0"/>
	<lineStyle thickness="3" type="0" typeHidden="1"/>
</element>
</construction>
</geogebra>