数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 正弦と余弦 - 加法公式(倍角、3倍角、整数倍(一般)、帰納法)

解析入門 原書第3版
原著

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
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第2部(微分と基本的な関数)、第4章(正弦と余弦)、3(加法公式)、練習問題4、5、6.を取り組んでみる。

4. 1. 
      $\begin{array}{l}\sin 2x\\ =\sin \left(x+x\right)\\ =\sin x\cos x+\cos x\sin x\\ =2\sin x\cos x\end{array}$
   2. 
      $\begin{array}{l}\cos 2x\\ =\cos \left(x+x\right)\\ =\cos x\cos x-\sin x\sin x\\ ={\cos }^{2}x-{\sin }^{2}x\end{array}$
   3. 
      $\begin{array}{l}\cos 2x={\cos }^{2}x-\left(1-{\cos }^{2}x\right)\\ \cos 2x=2{\cos }^{2}x-1\\ {\cos }^{2}x=\frac{1+\cos 2x}{2}\end{array}$
   4. 
      $\begin{array}{l}\cos 2x=1-{\sin }^{2}x-{\sin }^{2}x\\ \cos 2x=1-2{\sin }^{2}x\\ {\sin }^{2}x=\frac{1-\cos 2x}{2}\end{array}$
5. 
   $\begin{array}{l}\sin 3x\\ =\sin \left(2x+x\right)\\ =\sin 2x\cos x+\cos 2x\sin x\\ =2\sin x{\cos }^{2}x+\left({\cos }^{2}x-{\sin }^{2}x\right)\sin x\\ =3\sin x{\cos }^{2}x-{\sin }^{3}x\\ =3\sin x\left(1-{\sin }^{2}x\right)-{\sin }^{3}x\\ =3\sin x-4{\sin }^{3}x\\ \\ \sin 4x\\ =\sin 2\left(2x\right)\\ =2\sin 2x\cos 2x\\ =2\left(2\sin x\cos x\right)\left({\cos }^{2}x-{\sin }^{2}x\right)\\ =2\left(2\sin x\cos x\right)\left(1-2{\sin }^{2}x\right)\\ =4\sin x\cos x-8\sin x^3\cos x\\ \\ \sin 5x\\ =\sin \left(4x+x\right)\\ =\sin 4x\cos x+\cos 4x\sin x\\ =4\sin x{\cos }^{2}x-8\sin x^3{\cos }^{2}x+\cos 2\left(2x\right)\sin x\\ =4\sin x{\cos }^{2}x-8{\sin }^{3}x{\cos }^{2}x+\left({\cos }^{2}2x-{\sin }^{2}2x\right)\sin x\\ =4\sin {\cos }^{2}x-8{\sin }^{3}x{\cos }^{2}x+\left({\left({\cos }^{2}x-{\sin }^{2}x\right)}^2-4{\sin }^{2}x{\cos }^{2}x\right)\sin x\end{array}$
6. 
   $\begin{array}{l}\sin nx={\displaystyle \sum a_{ij}{\left(\sin x\right)}^i{\left(\cos x\right)}^j}\\ \\ \sin \left(n+1\right)x\\ =\sin nx\cos x+\cos nx\sin x\\ \\ \cos 2kx={\cos }^{2}kx+{\sin }^{2}kx\\ \\ \cos \left(2k+1\right)x\\ =\cos 2kx\cos x+\sin 2kx\sin x\\ \\ \sin \left(n+1\right)x={\displaystyle \sum a_{ij}{\left(\sin x\right)}^i{\left(\cos x\right)}^j}\end{array}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import pprint, symbols, sin, cos, plot, pi

x = symbols('x')
fs = [sin(x),
      cos(x),
      sin(2 * x),
      cos(2 * x),
      cos(x) ** 2,
      sin(x) ** 2]

p = plot(*fs, (x, -pi, pi), show=False, legend=True)

for i, c in enumerate(['red', 'green', 'blue', 'brown', 'purple', 'yellow']):
    p[i].line_color = c

p.save('sample4.svg')

入出力結果(Terminal, IPython)

$ ./sample4.py
$

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<br>
<label for="n0">n = </label>
<input id="n0" type="number" min="0" step="1" value="2">

<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="sample4.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'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    input_n0 = document.querySelector('#n0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
             input_n0],    
    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

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

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value),
        n0 = parseFloat(input_n0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;n
    }

    let points = [],
        lines = [],
        f = (x) => Math.sin(n0 * x);
    
    for (let x = x1; x <= x2; x += dx) {
        let y = Math.sin(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t1 = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        let y = Math.cos(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t2 = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .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);

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', 'black');
    
    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', (d, i) =>
              i < t1 ? 'red' :
              i < t2 ? 'green' : 'blue');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

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

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =
n =