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 behavior of the complex number $exp(−ix)$ and the varying value of $\cos(x)$.
Web references:
// Define the ids of your boards in BOARDID0, BOARDID1,...
// First board
const board1 = JXG.JSXGraph.initBoard(BOARDID0, {boundingbox: [-10, 1.5, 10, -1.5], axis:true});
// Glider on x-axis
var x = board1.create('glider', [-9, 0, board1.defaultAxes.x], {name:'x'});
var y = board1.create('point', [() => x.X(), () => Math.cos(x.X())], {size:1, name:'', color:'green'});
var x1 = board1.create('segment', [x,y], {strokeColor:'red'});
var f = board1.create('functiongraph', [(x) => Math.cos(x)]);
board1.create('text',[
() => x.X()+0.3,
() => y.Y()*0.5,
'cos'], {});
// Second board
const board2 = JXG.JSXGraph.initBoard(BOARDID1, {boundingbox: [-1.5, 1.5, 1.5, -1.5], axis:true});
// First board triggers updates on second board
board1.addChild(board2);
// Fixed circle
var p0 = board2.create('point', [0, 0], {fixed:true,visible:false});
var p1 = board2.create('point', [1, 0], {name:'', visible:false, fixed:true});
var c = board2.create('circle', [p0, p1], {dash:2, strokeWidth:1, strokeOpacity:0.6});
// Point on circle depends on point x in first board
var p2 = board2.create('point', [() => Math.cos(x.X()), ()=> Math.sin(x.X())], {
name:'exp(ix)',
fixed:true,
size:1, color:'green'
});
// Helper points and cos segment
var p3 = board2.create('point', [() => p2.X(), 0.0], {visible:false, name:''});
var p4 = board2.create('point', [0.0, () => p2.Y()], {visible:false, name:''});
board2.create('segment', [p2,p4], {strokeColor:'red'}); // cos
board2.create('text',[
() => (p2.X() + p4.X()) * 0.3,
() => p2.Y()+0.05,
'cos'
], {});
This example is licensed under a Creative Commons Attribution ShareAlike 4.0 International License.
Please note that you have to mention The Center of Mobile Learning with Digital Technology in the credits.