Sine & Cosine of a Sum: Discovery
2015-12-23 12:31
This interactive applet explores the sine and cosine addition formulas. Through geometric constructions and dynamic puzzles, it visually demonstrates their geometric meaning. Using points, lines, and polygons with angle measurements, it helps learners understand the derivation of trigonometric identities, serving as a great resource for teaching trigonometry.
In collections
Parallelogram Properties and Theorems
Loading the math board and drawing, please wait…
<?xml version="1.0" encoding="utf-8"?>
<geogebra format="5.0" version="5.0.560.0" app="classic" platform="w" id="e16a7ee0-d540-4b0a-94a8-30afaa4af0e0" xsi:noNamespaceSchemaLocation="http://www.geogebra.org/ggb.xsd" xmlns="" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" >
<gui>
<window width="798" height="646" />
<perspectives>
<perspective id="tmp">
<panes>
<pane location="" divider="0.15413533834586465" orientation="1" />
</panes>
<views>
<view id="4097" visible="false" inframe="false" stylebar="true" location="1,1,1,1" size="400" window="631,187,700,550" />
<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="796" window="100,100,600,400" />
<view id="2" visible="false" inframe="false" stylebar="false" location="3" size="123" 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" toolbar="0" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="128" visible="false" inframe="false" stylebar="false" location="1" size="480" window="50,50,500,500" />
<view id="70" toolbar="0 || 2020 || 2021 || 2022" visible="false" inframe="false" stylebar="true" location="1" size="150" window="50,50,500,500" />
<view id="43" visible="false" inframe="false" stylebar="false" location="1" size="450" window="50,50,500,500" />
</views>
<toolbar show="false" 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="false" cmd="true" top="algebra" />
<dockBar show="false" east="false" />
</perspective>
</perspectives>
<labelingStyle val="0"/>
<font size="18"/>
</gui>
<euclidianView>
<viewNumber viewNo="1"/>
<size width="795" height="643"/>
<coordSystem xZero="355.2460081156681" yZero="457.8156638042652" scale="19.38923827738674" yscale="19.389238277386802"/>
<evSettings axes="false" grid="false" gridIsBold="false" pointCapturing="3" rightAngleStyle="1" checkboxSize="26" gridType="0"/>
<bgColor r="188" g="212" b="230"/>
<axesColor r="0" g="0" b="0"/>
<gridColor r="192" g="192" b="192"/>
<lineStyle axes="1" grid="0"/>
<axis id="0" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<axis id="1" show="false" label="" unitLabel="" tickStyle="1" showNumbers="true"/>
<grid distX="1" distY="1" 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>
<tableview min="-2" max="2" step="1"/>
<scripting blocked="false" disabled="false"/>
<construction title="" author="" date="">
<element type="line" label="a">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="1" y="0" z="-6.891781626004642"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Point">
<input a0="a"/>
<output a0="A"/>
</command>
<element type="point" label="A">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="6.891781626004642" y="4.950551211360383" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<element type="line" label="b">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="0" playing="false"/>
<coords x="1" y="0" z="15.602958547125745"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Point">
<input a0="b"/>
<output a0="B"/>
</command>
<element type="point" label="B">
<show object="false" label="false" ev="4"/>
<objColor r="102" g="153" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="-15.602958547125745" y="-7.080687562504867" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<command name="OrthogonalLine">
<input a0="B" a1="a"/>
<output a0="f"/>
</command>
<element type="line" label="f">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="1" z="7.080687562504867"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Point">
<input a0="a"/>
<output a0="D"/>
</command>
<element type="point" label="D">
<show object="false" label="true" ev="4"/>
<objColor r="125" g="125" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="6.891781626004642" y="16.744648949950847" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Ray">
<input a0="A" a1="D"/>
<output a0="g"/>
</command>
<element type="ray" label="g">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-11.794097738590462" y="0" z="81.28234609012065"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="g"/>
<output a0="E"/>
</command>
<element type="point" label="E">
<show object="false" label="false" ev="4"/>
<objColor r="102" g="153" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<animation step="1" speed="1" type="1" playing="false"/>
<coords x="6.891781626004642" y="21.5768565521225" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="B" a1="E"/>
<output a0="h"/>
</command>
<element type="segment" label="h">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-28.657544114627367" y="22.49474017313039" z="-287.86424591729525"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="E" a1="b"/>
<output a0="i"/>
</command>
<element type="line" label="i">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="1" z="-21.5768565521225"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="a" a1="f"/>
<output a0="F"/>
</command>
<element type="point" label="F">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="6.891781626004642" y="-7.080687562504867" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="A" a1="F"/>
<output a0="j"/>
</command>
<element type="segment" label="j">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="12.031238773865251" y="0" z="-82.91667031979915"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="B" a1="F"/>
<output a0="k"/>
</command>
<element type="segment" label="k">
<show object="true" label="false" ev="4"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="22.49474017313039" z="159.2782269656629"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="E" a1="b"/>
<output a0="l"/>
</command>
<element type="line" label="l">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="1" z="-21.5768565521225"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="b" a1="l"/>
<output a0="H"/>
</command>
<element type="point" label="H">
<show object="false" label="true" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-15.602958547125745" y="21.5768565521225" z="1"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="B" a1="H"/>
<output a0="n"/>
</command>
<element type="segment" label="n">
<show object="true" label="false" ev="4"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-80" y="-2"/>
<labelMode val="3"/>
<coords x="-28.657544114627367" y="0" z="-447.14247288295815"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="sin(α + β)"/>
</element>
<command name="Angle">
<input a0="E" a1="F" a2="B"/>
<output a0="β"/>
</command>
<element type="angle" label="β">
<angleStyle val="1"/>
<value val="1.5707963267948966"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="4" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<command name="Segment">
<input a0="A" a1="E"/>
<output a0="s"/>
</command>
<element type="segment" label="s">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-16.626305340762116" y="0" z="114.58486565580719"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E" a1="F"/>
<output a0="t"/>
</command>
<element type="segment" label="t">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="28.657544114627367" y="0" z="-197.50153597560634"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Point">
<input a0="t"/>
<output a0="C"/>
</command>
<element type="point" label="C">
<show object="true" label="false" ev="4"/>
<objColor r="51" g="255" b="204" alpha="0"/>
<layer val="0"/>
<labelOffset x="11" y="25"/>
<labelMode val="3"/>
<animation step="0.1" speed="1" type="1" playing="false"/>
<coords x="6.891781626004642" y="8" z="1"/>
<pointSize val="9"/>
<pointStyle val="0"/>
<caption val="Drag Me!"/>
</element>
<command name="Segment">
<input a0="B" a1="C"/>
<output a0="c"/>
</command>
<element type="segment" label="c">
<show object="false" label="false" ev="4"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-15.080687562504867" y="22.49474017313039" z="-76.02511593425533"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="OrthogonalLine">
<input a0="C" a1="c"/>
<output a0="d"/>
</command>
<element type="line" label="d">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-22.49474017313039" y="-15.080687562504867" z="275.6743375069674"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="OrthogonalLine">
<input a0="E" a1="t"/>
<output a0="e"/>
</command>
<element type="line" label="e">
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="28.657544114627367" z="-618.339718497437"/>
<lineStyle thickness="2" type="0" typeHidden="1"/>
<eqnStyle style="implicit"/>
</element>
<command name="Intersect">
<input a0="d" a1="e"/>
<output a0="G"/>
</command>
<element type="point" label="G">
<show object="false" label="false" ev="4"/>
<objColor r="68" g="68" b="68" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-1424.8386137704674" y="13909.391306326434" z="644.6440088585646"/>
<pointSize val="3"/>
<pointStyle val="0"/>
</element>
<command name="Segment">
<input a0="B" a1="G"/>
<output a0="m"/>
</command>
<element type="segment" label="m">
<show object="false" label="false" ev="4"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-28.657544114627367" y="13.392686530026959" z="-352.3130439412698"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="G" a1="C"/>
<output a0="p"/>
</command>
<element type="segment" label="p">
<show object="false" label="false" ev="4"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="13.5768565521225" y="9.102053643103428" z="-166.385159669646"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="G" a1="C" a2="B"/>
<output a0="α"/>
</command>
<element type="angle" label="α">
<angleStyle val="1"/>
<value val="1.5707963267948966"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-43" y="-4"/>
<labelMode val="1"/>
<lineStyle thickness="6" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<command name="Angle">
<input a0="B" a1="H" a2="G"/>
<output a0="γ"/>
</command>
<element type="angle" label="γ">
<angleStyle val="1"/>
<value val="1.5707963267948966"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="6" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<command name="Segment">
<input a0="H" a1="E"/>
<output a0="q"/>
</command>
<element type="segment" label="q">
<show object="false" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-35" y="0"/>
<labelMode val="0"/>
<coords x="0" y="22.49474017313039" z="-485.3657818929016"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Angle">
<input a0="F" a1="B" a2="C"/>
<output a0="δ"/>
</command>
<element type="angle" label="δ">
<angleStyle val="1"/>
<value val="0.5905893231659374"/>
<show object="true" label="true" ev="4"/>
<objColor r="204" g="0" b="102" alpha="0.550000011920929"/>
<layer val="0"/>
<labelOffset x="43" y="-8"/>
<labelMode val="3"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="60"/>
<caption val="α"/>
</element>
<command name="Angle">
<input a0="C" a1="B" a2="G"/>
<output a0="ε"/>
</command>
<element type="angle" label="ε">
<angleStyle val="1"/>
<value val="0.5430307938757541"/>
<show object="true" label="true" ev="4"/>
<objColor r="0" g="51" b="204" alpha="0.5"/>
<layer val="0"/>
<labelOffset x="33" y="-40"/>
<labelMode val="3"/>
<decoration type="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="60"/>
<caption val="β"/>
</element>
<command name="Angle">
<input a0="H" a1="G" a2="B"/>
<output a0="ζ"/>
</command>
<element type="angle" label="ζ">
<angleStyle val="1"/>
<value val="1.1336201170416917"/>
<show object="true" label="true" ev="4"/>
<objColor r="102" g="0" b="204" alpha="0.5"/>
<layer val="0"/>
<labelOffset x="-75" y="27"/>
<labelMode val="3"/>
<decoration type="2"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="60"/>
<caption val="α + β"/>
</element>
<command name="Angle">
<input a0="E" a1="C" a2="G"/>
<output a0="η"/>
</command>
<element type="angle" label="η">
<angleStyle val="1"/>
<value val="0.5905893231659374"/>
<show object="true" label="true" ev="4"/>
<objColor r="204" g="0" b="102" alpha="0.550000011920929"/>
<layer val="0"/>
<labelOffset x="-10" y="-38"/>
<labelMode val="3"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="60"/>
<caption val="α"/>
</element>
<command name="Angle">
<input a0="H" a1="E" a2="F"/>
<output a0="θ"/>
</command>
<element type="angle" label="θ">
<angleStyle val="1"/>
<value val="1.5707963267948966"/>
<show object="true" label="false" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="1"/>
<lineStyle thickness="6" type="0" typeHidden="2"/>
<arcSize val="50"/>
</element>
<command name="Segment">
<input a0="H" a1="G"/>
<output a0="r"/>
</command>
<element type="segment" label="r">
<show object="false" label="false" ev="4"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="-43" y="1"/>
<labelMode val="3"/>
<coords x="0" y="13.392686530026959" z="-288.97207610593495"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
<caption val="cos(α +β )"/>
</element>
<command name="Segment">
<input a0="G" a1="E"/>
<output a0="a_1"/>
</command>
<element type="segment" label="a_1">
<show object="false" label="false" ev="4"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="9.102053643103428" z="-196.39370578696668"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E" a1="C"/>
<output a0="b_1"/>
</command>
<element type="segment" label="b_1">
<show object="false" label="false" ev="4"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="13.5768565521225" y="0" z="-93.56873052481858"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="C" a1="F"/>
<output a0="c_1"/>
</command>
<element type="segment" label="c_1">
<show object="false" label="false" ev="4"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="15.080687562504867" y="0" z="-103.93280545078777"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Distance">
<input a0="B" a1="C"/>
<output a0="distanceBC"/>
</command>
<element type="numeric" label="distanceBC">
<value val="27.082106136977885"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<command name="Distance">
<input a0="B" a1="G"/>
<output a0="distanceBG"/>
</command>
<element type="numeric" label="distanceBG">
<value val="31.632560553540067"/>
<objColor r="0" g="0" b="0" alpha="0.10000000149011612"/>
</element>
<expression label="TextBG" exp=""1""/>
<element type="text" label="TextBG">
<show object="true" label="true" ev="4"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<font serif="false" sizeM="4" size="54" style="0"/>
<startPoint exp="Midpoint[B, G]"/>
<labelOffset x="-4" y="-13"/>
</element>
<command name="Angle">
<input a0="C" a1="G" a2="E"/>
<output a0="ι"/>
</command>
<element type="angle" label="ι">
<angleStyle val="0"/>
<value val="0.9802070036289592"/>
<show object="false" label="true" ev="4"/>
<objColor r="0" g="0" b="255" alpha="0.25"/>
<layer val="0"/>
<labelOffset x="19" y="15"/>
<labelMode val="3"/>
<decoration type="1"/>
<lineStyle thickness="2" type="0" typeHidden="2"/>
<arcSize val="30"/>
<caption val="β"/>
</element>
<expression label="text1" exp=""\sin\left(\alpha \; + \; \beta \right) ""/>
<element type="text" label="text1">
<show object="true" label="true" ev="40"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="11.187708791984969" y="10.817000247569004" z="1"/>
</element>
<expression label="text1_1" exp=""\cos\left(\alpha \; + \; \beta \right) ""/>
<element type="text" label="text1_1">
<show object="true" label="true" ev="40"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="11.181269327467739" y="-1.1603047931401704" z="1"/>
</element>
<expression label="text2" exp=""\sin\left( \alpha \right) \cos\left( \beta \right)""/>
<element type="text" label="text2">
<show object="true" label="true" ev="40"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="11.174618398720488" y="4.824460569682744" z="1"/>
</element>
<expression label="text2_1" exp=""\cos\left( \alpha \right) \cos\left( \beta \right)""/>
<element type="text" label="text2_1">
<show object="true" label="true" ev="40"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="11.169013707965767" y="7.817973365106724" z="1"/>
</element>
<expression label="text2_2" exp=""\cos\left( \alpha \right) \sin\left( \beta \right)""/>
<element type="text" label="text2_2">
<show object="true" label="true" ev="40"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="11.17751444261234" y="13.839480325580329" z="1"/>
</element>
<expression label="text2_3" exp=""\sin\left( \alpha \right) \sin\left( \beta \right)""/>
<element type="text" label="text2_3">
<show object="true" label="true" ev="40"/>
<condition showObject="o ≟ false"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="11.152752331124836" y="16.821583817467502" z="1"/>
</element>
<command name="Polygon">
<input a0="B" a1="F" a2="C"/>
<output a0="poly1" a1="c_2" a2="b_2" a3="f_1"/>
</command>
<element type="polygon" label="poly1">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="c_2">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0" y="22.49474017313039" z="159.2782269656629"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="b_2">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-15.080687562504867" y="0" z="103.93280545078777"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="f_1">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="15.080687562504867" y="-22.49474017313039" z="76.02511593425533"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Polygon">
<input a0="C" a1="E" a2="G"/>
<output a0="poly2" a1="g_1" a2="c_3" a3="e_1"/>
</command>
<element type="polygon" label="poly2">
<lineStyle thickness="2" type="0" typeHidden="1"/>
<show object="true" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0.75"/>
<layer val="0"/>
<labelMode val="0"/>
</element>
<element type="segment" label="g_1">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="-13.5768565521225" y="0" z="93.56873052481858"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="c_3">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="0" y="-9.102053643103428" z="196.39370578696668"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<element type="segment" label="e_1">
<show object="false" label="false" ev="4"/>
<objColor r="255" g="255" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<auxiliary val="false"/>
<coords x="13.5768565521225" y="9.102053643103428" z="-166.385159669646"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text3" exp=""k, m, n, p, r, c, c_1, a_1, b_1""/>
<element type="text" label="text3">
<show object="false" label="true" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<startPoint x="0.40477392407091006" y="23.06704884608311" z="1"/>
</element>
<command name="Segment">
<input a0="H" a1="B"/>
<output a0="d_1"/>
</command>
<element type="segment" label="d_1">
<show object="true" label="false" ev="4"/>
<condition showObject="o ≠ 0"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="28.657544114627367" y="0" z="447.14247288295815"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="H" a1="G"/>
<output a0="h_1"/>
</command>
<element type="segment" label="h_1">
<show object="true" label="false" ev="4"/>
<condition showObject="o ≠ 0"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="13.392686530026959" z="-288.97207610593495"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="G" a1="E"/>
<output a0="i_1"/>
</command>
<element type="segment" label="i_1">
<show object="true" label="false" ev="4"/>
<condition showObject="o ≠ 0"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="9.102053643103428" z="-196.39370578696668"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="E" a1="C"/>
<output a0="j_1"/>
</command>
<element type="segment" label="j_1">
<show object="true" label="false" ev="4"/>
<condition showObject="o ≠ 0"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="13.5768565521225" y="0" z="-93.56873052481858"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="C" a1="F"/>
<output a0="k_1"/>
</command>
<element type="segment" label="k_1">
<show object="true" label="false" ev="4"/>
<condition showObject="o ≠ 0"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="15.080687562504867" y="0" z="-103.93280545078777"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="F" a1="B"/>
<output a0="l_1"/>
</command>
<element type="segment" label="l_1">
<show object="true" label="false" ev="4"/>
<condition showObject="o ≠ 0"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="0" y="-22.49474017313039" z="-159.2782269656629"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="B" a1="C"/>
<output a0="m_1"/>
</command>
<element type="segment" label="m_1">
<show object="true" label="false" ev="4"/>
<condition showObject="o ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-15.080687562504867" y="22.49474017313039" z="-76.02511593425533"/>
<lineStyle thickness="3" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="C" a1="G"/>
<output a0="n_1"/>
</command>
<element type="segment" label="n_1">
<show object="true" label="false" ev="4"/>
<condition showObject="o ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="-13.5768565521225" y="-9.102053643103428" z="166.385159669646"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<command name="Segment">
<input a0="G" a1="B"/>
<output a0="p_1"/>
</command>
<element type="segment" label="p_1">
<show object="true" label="false" ev="4"/>
<condition showObject="o ≠ 0"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelMode val="0"/>
<coords x="28.657544114627367" y="-13.392686530026959" z="352.3130439412698"/>
<lineStyle thickness="5" type="0" typeHidden="1"/>
<outlyingIntersections val="false"/>
<keepTypeOnTransform val="true"/>
</element>
<expression label="text1_4" exp=""\cos\left(\alpha \; + \; \beta \right) ""/>
<element type="text" label="text1_4">
<show object="true" label="true" ev="40"/>
<condition showObject="o ≠ 0"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="11.155284682374845" y="-1.1831603004287232" z="1"/>
</element>
<expression label="text2_4" exp=""\sin\left( \alpha \right) \cos\left( \beta \right)""/>
<element type="text" label="text2_4">
<show object="true" label="true" ev="40"/>
<condition showObject="o ≠ 0"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="11.18468582865015" y="4.855496508158386" z="1"/>
</element>
<expression label="text1_6" exp=""\sin\left(\alpha \; + \; \beta \right) ""/>
<element type="text" label="text1_6">
<show object="true" label="true" ev="40"/>
<condition showObject="o ≠ 0"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="11.18468582865016" y="10.81855048765589" z="1"/>
</element>
<expression label="text2_5" exp=""\cos\left( \alpha \right) \cos\left( \beta \right)""/>
<element type="text" label="text2_5">
<show object="true" label="true" ev="40"/>
<condition showObject="o ≠ 0"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="11.184685828650164" y="7.809808686702957" z="1"/>
</element>
<expression label="text2_6" exp=""\cos\left( \alpha \right) \sin\left( \beta \right)""/>
<element type="text" label="text2_6">
<show object="true" label="true" ev="40"/>
<condition showObject="o ≠ 0"/>
<objColor r="255" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="11.17709900728613" y="13.839466633674524" z="1"/>
</element>
<expression label="text2_7" exp=""\sin\left( \alpha \right) \sin\left( \beta \right)""/>
<element type="text" label="text2_7">
<show object="true" label="true" ev="40"/>
<condition showObject="o ≠ 0"/>
<objColor r="0" g="0" b="255" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="11.184685828650167" y="16.838405122975058" z="1"/>
</element>
<element type="boolean" label="o">
<value val="false"/>
<show object="true" label="true"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<layer val="0"/>
<labelOffset x="563" y="539"/>
<labelMode val="0"/>
<checkbox fixed="true"/>
<caption val="Show Hint"/>
</element>
<expression label="text4" exp=""\sin\left( \beta \right) ""/>
<element type="text" label="text4">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="11.184685828650137" y="19.8400338236881" z="1"/>
</element>
<expression label="text4_1" exp=""\cos\left( \beta \right) ""/>
<element type="text" label="text4_1">
<show object="true" label="false" ev="40"/>
<objColor r="0" g="0" b="0" alpha="0"/>
<bgColor r="255" g="255" b="255" alpha="255"/>
<layer val="0"/>
<labelMode val="0"/>
<isLaTeX val="true"/>
<font serif="true" sizeM="1.4" size="8" style="1"/>
<startPoint x="11.177939876656438" y="1.8517130513412638" z="1"/>
</element>
</construction>
</geogebra>