数学 - JavaScript - 解析学 - 各種の初等関数 - 三角関数(正弦、余弦、正接、加法定理、2倍角の公式、半角の公式)

解析入門〈1〉

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax

解析入門〈1〉(松坂 和夫(著)、岩波書店)の第5章(各種の初等関数)、5.3(三角関数)、問題5.3-5.を取り組んでみる。

1. 
   $\begin{array}{l}\sin \left(\frac{\pi }{2}-\theta \right)\\ =\sin \frac{\pi }{2}\cos \theta -\cos \frac{\pi }{2}\sin \theta \\ =\cos \theta \\ \\ \cos \left(\frac{\pi }{2}-\theta \right)\\ =\cos \frac{\pi }{2}\cos \theta +\sin \frac{\pi }{2}\sin \theta \\ =\sin \theta \end{array}$
2. 
   $\begin{array}{l}\sin \left(\pi -\theta \right)\\ =\sin \pi \cos \theta -\cos \pi \sin \theta \\ =\sin \theta \\ \\ \cos \left(\pi -\theta \right)\\ =\cos \pi \cos \theta +\sin \pi \sin \theta \\ =-\cos \theta \end{array}$
3. 
   $\begin{array}{l}\sin 2\alpha \\ =\sin \left(\alpha +\alpha \right)\\ =\sin \alpha \cos \alpha +\cos \alpha \sin \alpha \\ =2\sin \alpha \cos \alpha \\ \\ \cos 2\alpha \\ =\cos \left(\alpha +\alpha \right)\\ =\cos \alpha \cos \alpha -\sin \alpha \sin \alpha \\ ={\cos }^2\alpha -{\sin }^2\alpha \\ ={\cos }^2\alpha -\left(1-{\cos }^2\alpha \right)\\ =2{\cos }^2\alpha -1\\ =2\left(1-{\sin }^2\alpha \right)-1\\ =1-2{\sin }^2\alpha \\ \\ \tan 2\alpha \\ =\tan \left(\alpha +\alpha \right)\\ =\frac{\tan \alpha +\tan \alpha }{1-\tan \alpha \tan \alpha }\\ =\frac{2\tan \alpha }{1-{\tan }^2\alpha }\end{array}$
4. 
   $\begin{array}{l}{\sin }^2\frac{\alpha }{2}\\ =\frac{1}{2}\left(1-\cos 2·\frac{\alpha }{2}\right)\\ =\frac{1-\cos \alpha }{2}\\ \\ {\cos }^2\frac{\alpha }{2}\\ =\frac{\cos 2·\frac{\alpha }{2}+1}{2}\\ =\frac{1+\cos \alpha }{2}\end{array}$
5. 
   $\begin{array}{l}\sin 3\alpha \\ =\sin \left(\alpha +2\alpha \right)\\ =\sin \alpha \cos 2\alpha +\cos \alpha \sin 2\alpha \\ =\sin \alpha \left({\cos }^2\alpha -{\sin }^2\alpha \right)+\cos \alpha \left(\sin \alpha \cos \alpha +\cos \alpha \sin \alpha \right)\\ =\sin \alpha {\cos }^2\alpha -{\sin }^3\alpha +\sin \alpha {\cos }^2\alpha +\sin \alpha {\cos }^2\alpha \\ =3\sin \alpha {\cos }^2\alpha -{\sin }^3\alpha \\ =3\sin \alpha \left(1-{\sin }^2\alpha \right)-{\sin }^3\alpha \\ =3\sin \alpha -4{\sin }^3\alpha \\ \\ \cos 3\alpha \\ =\cos \left(\alpha +2\alpha \right)\\ =\cos \alpha \cos 2\alpha -\sin \alpha \sin 2\alpha \\ =\cos \alpha \left({\cos }^2\alpha -{\sin }^2\alpha \right)-\sin \alpha \left(2\sin \alpha \cos \alpha \right)\\ =\cos \alpha \left({\cos }^2\alpha -\left(1-{\cos }^2\alpha \right)\right)-2{\sin }^2\alpha \cos \alpha \\ =2{\cos }^3\alpha -\cos \alpha -2\left(1-{\cos }^2\alpha \right)\cos \alpha \\ =4{\cos }^3\alpha -3\cos \alpha \end{array}$

コード(Emacs)

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>
<script src="sample1.js"></script>    

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    p = (x) => pre0.textContent += x + '\n';

let f = (x) => Math.sin(3 * x),
    g = (x) => 3 * Math.sin(x) - 4 * Math.sin(x) ** 3;

let draw = () => {
    pre0.textContent = '';    
    let points = [];

    for (let x = -Math.PI; x <= Math.PI; x += 0.001) {
        points.push([x, f(x)]);
    }
    for (let x = -Math.PI; x <= Math.PI; x += 0.001) {
        points.push([x, g(x)]);
    }
    for (let x = -Math.PI; x <= Math.PI; x += 0.001) {
        points.push([x, Math.sin(x)]);
    }
    let xscale = d3.scaleLinear()
        .domain([-Math.PI, Math.PI])
        .range([padding, width - padding]);
    let ys = points.map((a) => a[1]);
    let yscale = d3.scaleLinear()
        .domain([-Math.PI, Math.PI])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    let t1 = points.length / 3,
        t2 = t1 * 2;
    
    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', (d, i) => i < t1 ? 4 : 1)
        .attr('fill', (d, i) =>
              i < t1 ? 'green' :
              i < t2 ? 'blue' : 'red');

    svg.append('g')
        .attr('transform', `translate(0, ${height / 2})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${width / 2}, 0)`)
        .call(yaxis);
}

btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();