Trangles for geometric derivation of the cosine sum identity

						var board = JXG.JSXGraph.initBoard('box1', {
boundingbox: [-10, 10, 10, -10],showNavigation: false, showcopyright: false, axis: false,
});

var p1 = board.create('point', [-6, -3.0],{fixed:true,label:{offset:[-20,0]}});
var p2 = board.create('point', [8, -3.0],{fixed:true,face:''});
var c1 = board.create('circle', [p1, 13],{visible:false});
var g1 = board.create('glider', [5.0, 3.5, c1],);
var g2 = board.create('glider', [1.1, 9.5, c1]);

var l1 = board.create('segment', [p1, p2]);
var l2 = board.create('segment', [p1, g1],);
var l3 = board.create('segment', [p1, g2],);
var a1 = board.create('angle', [p2, p1, g1],{radius:2.5});
var a2 = board.create('angle', [g1, p1, g2],{radius:2.5,fillcolor:'#1de'});
var pp1 = board.create('perpendicularpoint', [g2, l1]);
var l4 = board.create('segment', [g2, pp1],);
var pp2 = board.create('perpendicularpoint', [g2, l2],{label:{offset:[10,-10]}});
var l5 = board.create('segment', [g2, pp2],);
var pp3 = board.create('perpendicularpoint', [pp2, l1]);
var l5 = board.create('segment', [pp3, pp2],);
var pp4 = board.create('perpendicularpoint', [pp2, l4]);
var l6 = board.create('segment', [pp4, pp2],); 
var i1 = board.create('intersection', [l4, l2],{label:{offset:[10,-10]}});

The image appears in the following posts

  1. There are multiple methods for derivation of cosine difference identity. Once this identity is present, it's easy to derive other identities. For example, the cosine sum identity is another frequently used identities to solve trigonometric problems. Given the cosine difference identity,

    \cos(\alpha -\beta ) = \cos \alpha\cos \beta +\sin \alpha \sin \beta

    Let \beta be the minus of \theta, the above equation is converted to,

    \cos(\alpha -(-\theta) ) = \cos \alpha\cos(-\theta) +\sin \alpha \sin(-\theta)

    Using symmetry identities, the above expression is transformed to the identity for cosine of two angles addition,

    \cos(\alpha +\theta ) = \cos \alpha\cos\theta -\sin \alpha \sin \theta

    What about any method using geometric shapes for proof of cosine sum identity? We will explore the methods in this article.

Triangle posted 1 year ago

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