Cosine - Revision history

Bookofproofs at 13:21, 6 March 2016

2016-03-06T13:21:15Z

← Older revision		Revision as of 13:21, 6 March 2016
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	The cosine is a projection of the complex number exp(−ix), which ~~lies~~ on the unit circle in the complex plane, to the real axis on the complex plane. ~~The behavior of the cosine function can be studied in~~ the following interactive figure, ~~where~~ you can drag the point x on the real axis and observe the behaviour of the complex number exp(−ix) and the value of cosine(x).		The cosine is a projection of the complex number exp(−ix) (which is a point on the unit circle in the complex plane) to the real axis on the complex plane. In the following interactive figure, you can drag the point x on the real axis and observe the behaviour of the complex number exp(−ix) and the varying value of cosine(x).

	{\|		{\|

Bookofproofs at 13:10, 6 March 2016

2016-03-06T13:10:56Z

← Older revision		Revision as of 13:10, 6 March 2016
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	[[Category:Contributions]]		[[Category:Contributions]]
			[[Category:Examples]]

	[http://www.bookofproofs.org/branches/cosine/ read more about cosine on Bookofproofs]		[http://www.bookofproofs.org/branches/cosine/ read more about cosine on Bookofproofs]

Bookofproofs at 13:09, 6 March 2016

2016-03-06T13:09:02Z

← Older revision		Revision as of 13:09, 6 March 2016
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	~~Geometrically, the~~ cosine is a projection of the complex number exp(−ix) to the real axis. The behavior of the cosine function can be studied in the following interactive figure, where you can drag the point x and observe the behaviour of the complex number exp(−ix) and the value of cosine(x).		The cosine is a projection of the complex number exp(−ix), which lies on the unit circle in the complex plane, to the real axis on the complex plane. The behavior of the cosine function can be studied in the following interactive figure, where you can drag the point x on the real axis and observe the behaviour of the complex number exp(−ix) and the value of cosine(x).

	{\|		{\|

Bookofproofs at 13:07, 6 March 2016

2016-03-06T13:07:34Z

← Older revision		Revision as of 13:07, 6 March 2016
Line 1:		Line 1:
	Geometrically, the cosine is a projection of the complex number exp(−ix) to the real axis. The behavior of the cosine function can be studied in the following interactive figure, where you can drag the point x and observe the behaviour of the complex number exp(−ix) and the value of cosine(x).		Geometrically, the cosine is a projection of the complex number exp(−ix) to the real axis. The behavior of the cosine function can be studied in the following interactive figure, where you can drag the point x and observe the behaviour of the complex number exp(−ix) and the value of cosine(x).
	<jsxgraph box="boxR" width="500" height="500">
			{\|
			\|Cosine
			\|Unit Circle on the Complex Plane
			\|-
			\| <jsxgraph box="boxR" width="500" height="500">
	var brd1 = JXG.JSXGraph.initBoard('boxR', {boundingbox: [-10, 1.5, 10, -1.5], axis:true});		var brd1 = JXG.JSXGraph.initBoard('boxR', {boundingbox: [-10, 1.5, 10, -1.5], axis:true});
	var xr = brd1.create('line',[[-9,0],[9,0]],{visible:false});		var xr = brd1.create('line',[[-9,0],[9,0]],{visible:false});
Line 22:		Line 27:

	</jsxgraph>		</jsxgraph>
			\| <jsxgraph box="box" width="500" height="500">
	<jsxgraph box="box" width="500" height="500">
	var brd = JXG.JSXGraph.initBoard('box', {boundingbox: [-1.5, 1.5, 1.5, -1.5], axis:true});		var brd = JXG.JSXGraph.initBoard('box', {boundingbox: [-1.5, 1.5, 1.5, -1.5], axis:true});
	brd1.addChild(brd);		brd1.addChild(brd);
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	</jsxgraph>		</jsxgraph>
			\|}

Bookofproofs at 13:04, 6 March 2016

2016-03-06T13:04:39Z

← Older revision		Revision as of 13:04, 6 March 2016
Line 51:		Line 51:

	[[Category:Contributions]]		[[Category:Contributions]]

			[http://www.bookofproofs.org/branches/cosine/ read more about cosine on Bookofproofs]

Bookofproofs at 13:00, 6 March 2016

2016-03-06T13:00:09Z

← Older revision		Revision as of 13:00, 6 March 2016
Line 1:		Line 1:
			Geometrically, the cosine is a projection of the complex number exp(−ix) to the real axis. The behavior of the cosine function can be studied in the following interactive figure, where you can drag the point x and observe the behaviour of the complex number exp(−ix) and the value of cosine(x).
	<jsxgraph box="boxR" width="500" height="500">		<jsxgraph box="boxR" width="500" height="500">
	var brd1 = JXG.JSXGraph.initBoard('boxR', {boundingbox: [-10, 1.5, 10, -1.5], axis:true});		var brd1 = JXG.JSXGraph.initBoard('boxR', {boundingbox: [-10, 1.5, 10, -1.5], axis:true});
	var xr = brd1.create('line',[[-9,0],[9,0]],{visible:false});		var xr = brd1.create('line',[[-9,0],[9,0]],{visible:false});

Bookofproofs at 12:54, 6 March 2016

2016-03-06T12:54:58Z

@@ Line 1: / Line 1: @@
     <jsxgraph box="boxR" width="500" height="500">
@@ Line 23: / Line 22: @@
     </jsxgraph>
 [[Category:Contributions]]

Bookofproofs at 12:46, 6 March 2016

2016-03-06T12:46:29Z

← Older revision		Revision as of 12:46, 6 March 2016
Line 1:		Line 1:
	~~<jsxgraph box="box" width="500" height="500">~~
	~~var brd = JXG.JSXGraph.initBoard('box', {boundingbox: [-1.5, 1.5, 1.5, -1.5], axis:true});~~
	~~var ax = brd.create('line',[[0,0],[1,0]],{visible:false});~~
	~~var ay = brd.create('line',[[0,0],[0,1]],{visible:false});~~

	~~var p0 = brd.create('point',[0,0],{fixed:true,visible:false});~~
	~~var p1 = brd.create('point',[1,0],{name:'',visible:false,fixed:true});~~
	~~var c = brd.create('circle',[p0,p1],{dash:2,strokeWidth:1,strokeOpacity:0.6});~~


	~~var p2 = brd.create('point',[Math.cos(x.X()),Math.sin(x.X())],{name:'exp(ix)',fixed:true,size:1, strokeColor:'green'});~~

	~~var p3 = brd.create('point',[function(){return p2.X();},0.0],{visible:false,name:'',withLabel:false});~~
	~~var p4 = brd.create('point',[0.0,function(){return p2.Y();}],{visible:false,name:'',withLabel:false});~~

	~~brd.create('line',[p2,p4],{straightFirst:false,straightLast:false,strokeColor:'red'}); // cos~~
	~~brd.create('text',[~~
	~~function(){return (p2.X()+p4.X())*0.3;},~~
	~~function(){return p2.Y()+0.05;},~~
	~~'cos'],{});~~



	~~</jsxgraph>~~

	<jsxgraph box="boxR" width="500" height="500">		<jsxgraph box="boxR" width="500" height="500">

Bookofproofs at 12:43, 6 March 2016

2016-03-06T12:43:27Z

@@ Line 1: / Line 1: @@
     <jsxgraph box="box" width="500" height="500">
         var brd = JXG.JSXGraph.initBoard('box', {boundingbox: [-1.5, 1.5, 1.5, -1.5], axis:true});
@@ Line 33: / Line 23: @@
     </jsxgraph>
-</td>
     <jsxgraph box="boxR" width="500" height="500">
         var brd1 = JXG.JSXGraph.initBoard('boxR', {boundingbox: [-10, 1.5, 10, -1.5], axis:true});
@@ Line 57: / Line 46: @@
     </jsxgraph>
 [[Category:Contributions]]

Bookofproofs at 12:41, 6 March 2016

2016-03-06T12:41:07Z

← Older revision		Revision as of 12:41, 6 March 2016
Line 8:		Line 8:
	</tr>		</tr>
	<tr>		<tr>
	<td><~~div id~~="box~~" class="jxgbox" style=~~"width~~:500px; height:500px;"></div></td>~~		<td>
	~~<td><div id="boxR" class~~="~~jxgbox" style=~~"~~width:500px;~~ height~~:500px;"></div></td>~~		<jsxgraph box="box" width="500" height="500">
	~~</tr>~~		var brd = JXG.JSXGraph.initBoard('box', {boundingbox: [-1.5, 1.5, 1.5, -1.5], axis:true});
	~~</table>~~		var ax = brd.create('line',[[0,0],[1,0]],{visible:false});
			var ay = brd.create('line',[[0,0],[0,1]],{visible:false});
	~~<script language="javascript" type~~="~~text/javascript~~">

	~~var brd1 = JXG.JSXGraph.initBoard('boxR', {boundingbox: [-10, 1.5, 10, -1.5], axis:true});~~
	var brd = JXG.JSXGraph.initBoard('box', {boundingbox: [-1.5, 1.5, 1.5, -1.5], axis:true});

	~~var xr = brd1.create('line',[[-9,0],[9,0]],{visible:false});~~
	~~var x = brd1.create('glider',[-9,0,xr],{visible:true, name:'x'});~~
	~~var y = brd1.create('point',[x.X(),Math.cos(x.X())],{size:1,name:'',strokeColor:'green'});~~
	~~console.log("test1");~~
	~~var x1 = brd1.create('segment',[x,y],{visible:true, straightFirst:false,straightLast:false,strokeColor:'red'});~~
	~~x.on('drag', function(){ transform(x);});~~

	~~var f = brd1.create('functiongraph',[function(x){~~
	~~return Math.cos(x);~~
	~~}]);~~

	~~console.log("test2");~~
	var ax = brd.create('line',[[0,0],[1,0]],{visible:false});
	var ay = brd.create('line',[[0,0],[0,1]],{visible:false});

	var p0 = brd.create('point',[0,0],{fixed:true,visible:false});		var p0 = brd.create('point',[0,0],{fixed:true,visible:false});
	var p1 = brd.create('point',[1,0],{name:'',visible:false,fixed:true});		var p1 = brd.create('point',[1,0],{name:'',visible:false,fixed:true});
	var c = brd.create('circle',[p0,p1],{dash:2,strokeWidth:1,strokeOpacity:0.6});		var c = brd.create('circle',[p0,p1],{dash:2,strokeWidth:1,strokeOpacity:0.6});


	var p2 = brd.create('point',[Math.cos(x.X()),Math.sin(x.X())],{name:'exp(ix)',fixed:true,size:1, strokeColor:'green'});		var p2 = brd.create('point',[Math.cos(x.X()),Math.sin(x.X())],{name:'exp(ix)',fixed:true,size:1, strokeColor:'green'});

	var p3 = brd.create('point',[function(){return p2.X();},0.0],{visible:false,name:'',withLabel:false});		var p3 = brd.create('point',[function(){return p2.X();},0.0],{visible:false,name:'',withLabel:false});
	var p4 = brd.create('point',[0.0,function(){return p2.Y();}],{visible:false,name:'',withLabel:false});		var p4 = brd.create('point',[0.0,function(){return p2.Y();}],{visible:false,name:'',withLabel:false});

	brd.create('line',[p2,p4],{straightFirst:false,straightLast:false,strokeColor:'red'}); // cos		brd.create('line',[p2,p4],{straightFirst:false,straightLast:false,strokeColor:'red'}); // cos
	brd.create('text',[		brd.create('text',[
	function(){return (p2.X()+p4.X())*0.3;},		function(){return (p2.X()+p4.X())*0.3;},
	function(){return p2.Y()+0.05;},		function(){return p2.Y()+0.05;},
	'cos'],{});		'cos'],{});

	~~brd1.create('text',[~~
	~~function(){return x.X()+0.3;},~~
	~~function(){return y.Y()*0.5;},~~
	~~'cos'],{});~~

	~~function transform(x) {~~
	~~p2.setPosition(JXG.COORDS_BY_USER,[Math.cos(x.X()),Math.sin(x.X())]);~~
	~~y.setPosition(JXG.COORDS_BY_USER,[x.X(),Math.cos(x.X())]);~~
	~~brd.update();~~
	}

			</jsxgraph>
			</td>
			<td>
			<jsxgraph box="boxR" width="500" height="500">
			var brd1 = JXG.JSXGraph.initBoard('boxR', {boundingbox: [-10, 1.5, 10, -1.5], axis:true});
			var xr = brd1.create('line',[[-9,0],[9,0]],{visible:false});
			var x = brd1.create('glider',[-9,0,xr],{visible:true, name:'x'});
			var y = brd1.create('point',[x.X(),Math.cos(x.X())],{size:1,name:'',strokeColor:'green'});
			var x1 = brd1.create('segment',[x,y],{visible:true, straightFirst:false,straightLast:false,strokeColor:'red'});
			x.on('drag', function(){ transform(x);});
			var f = brd1.create('functiongraph',
			[function(x){
			return Math.cos(x);
			}]);
			brd1.create('text',[
			function(){return x.X()+0.3;},
			function(){return y.Y()*0.5;},
			'cos'],{});
			function transform(x) {
			p2.setPosition(JXG.COORDS_BY_USER,[Math.cos(x.X()),Math.sin(x.X())]);
			y.setPosition(JXG.COORDS_BY_USER,[x.X(),Math.cos(x.X())]);
			brd.update();
			}

	</~~script~~>		</jsxgraph>
			</td>
			</tr>
			</table>
	</body>		</body>

	</html>		</html>
	[[Category:Contributions]]		[[Category:Contributions]]