高度な特殊タグ練習

高度な練習（加法定理）

前回は、
$$\require{color}\color{#777777}{\require{enclose}\smash{\rlap{\hskip 0.8em \enclose{circle}{\phantom{\Rule{6em}{3em}{3em}}}}} \smash{\rlap{\Rule{8em}{0.05em}{0.05em}}} \smash{\rlap{\hskip 4em \Rule{0.1em}{4em}{4em}}} \smash{\rlap{\hskip 8.35em \raise -0.35em{x}}} \smash{\rlap{\hskip 4.18em \raise 4.35em{y}}} \smash{\rlap{\hskip 4.15em \raise -0.85em{O}}} \smash{\rlap{\hskip 5.15em \raise 0.4em{\alpha}}} \smash{\rlap{\hskip 6.05em \raise 2.05em{\bullet}}} \smash{\rlap{\hskip 6.65em \raise 2.45em{P(\cos\alpha,\sin\alpha)}}} \smash{\rlap{\hskip 4em \raise 0.2em \enclose{updiagonalstrike}{\phantom{\Rule{1.9em}{1.9em}{0em}}}}} \phantom{\Rule{8em}{4em}{4em}}}$$

\require{color}\color{#777777}{\require{enclose}\smash{\rlap{\hskip 0.8em \enclose{circle}{\phantom{\Rule{6em}{3em}{3em}}}}} \smash{\rlap{\Rule{8em}{0.05em}{0.05em}}} \smash{\rlap{\hskip 4em \Rule{0.1em}{4em}{4em}}} \smash{\rlap{\hskip 8.35em \raise -0.35em{x}}} \smash{\rlap{\hskip 4.18em \raise 4.35em{y}}} \smash{\rlap{\hskip 4.15em \raise -0.85em{O}}} \smash{\rlap{\hskip 5.15em \raise 0.4em{\alpha}}} \smash{\rlap{\hskip 6.05em \raise 2.05em{\bullet}}} \smash{\rlap{\hskip 6.65em \raise 2.45em{P(\cos\alpha,\sin\alpha)}}} \smash{\rlap{\hskip 4em \raise 0.2em \enclose{updiagonalstrike}{\phantom{\Rule{1.9em}{1.9em}{0em}}}}} \phantom{\Rule{8em}{4em}{4em}}}

という職人芸で三角関数っぽい図を得たのだった。

---

「じゃあ、加法定理の説明に入ろうか」

ぼくは黒板の方へ向き直る。

掠れたチョークで図を描く。

$$\require{color}\color{#777777}{\require{enclose}\smash{\rlap{\hskip 0.8em \enclose{circle}{\phantom{\Rule{6em}{3em}{3em}}}}} \smash{\rlap{\Rule{8em}{0.05em}{0.05em}}} \smash{\rlap{\hskip 4em \Rule{0.1em}{4em}{4em}}} \smash{\rlap{\hskip 8.35em \raise -0.35em{x}}} \smash{\rlap{\hskip 4.18em \raise 4.35em{y}}} \smash{\rlap{\hskip 4.15em \raise -0.85em{O}}} \smash{\rlap{\hskip 5.15em \raise 0.4em{\alpha}}} \smash{\rlap{\hskip 6.05em \raise 2.05em{\bullet}}} \smash{\rlap{\hskip 6.65em \raise 2.45em{P(\cos\alpha,\sin\alpha)}}} \smash{\rlap{\hskip 4em \raise 0.2em \enclose{updiagonalstrike}{\phantom{\Rule{1.9em}{1.9em}{0em}}}}} \phantom{\Rule{8em}{4em}{4em}}}$$

「ここまではいい？」

「……まあ、そうだね。単位円でしょう？」

「そうだね。望むなら軸と円の交点の座標を明記してもいいけど」

「右側の1個だけでいいよ」

　ごもっとも。

$$\require{color}\color{#777777}{\require{enclose}\smash{\rlap{\hskip 0.8em \enclose{circle}{\phantom{\Rule{6em}{3em}{3em}}}}} \smash{\rlap{\Rule{8em}{0.05em}{0.05em}}} \smash{\rlap{\hskip 4em \Rule{0.1em}{4em}{4em}}} \smash{\rlap{\hskip 8.35em \raise -0.35em{x}}} \smash{\rlap{\hskip 7.35em \raise 0.35em{(1,0)}}} \smash{\rlap{\hskip 4.18em \raise 4.35em{y}}} \smash{\rlap{\hskip 4.15em \raise -0.85em{O}}} \smash{\rlap{\hskip 5.15em \raise 0.4em{\alpha}}} \smash{\rlap{\hskip 6.05em \raise 2.05em{\bullet}}} \smash{\rlap{\hskip 6.65em \raise 2.45em{P(\cos\alpha,\sin\alpha)}}} \smash{\rlap{\hskip 4em \raise 0.2em \enclose{updiagonalstrike}{\phantom{\Rule{1.9em}{1.9em}{0em}}}}} \phantom{\Rule{8em}{4em}{4em}}}$$

　少し字が被ってしまった。

　まあ説明に支障はない。

「もうひとつ角度を書き込もう」

$$\require{color}\color{#777777}{\require{enclose}\smash{\rlap{\hskip 0.8em \enclose{circle}{\phantom{\Rule{6em}{3em}{3em}}}}} \smash{\rlap{\Rule{8em}{0.05em}{0.05em}}} \smash{\rlap{\hskip 4em \Rule{0.1em}{4em}{4em}}} \smash{\rlap{\hskip 8.35em \raise -0.35em{x}}} \smash{\rlap{\hskip 7.35em \raise 0.35em{(1,0)}}} \smash{\rlap{\hskip 4.18em \raise 4.35em{y}}} \smash{\rlap{\hskip 4.15em \raise -0.85em{O}}} \smash{\rlap{\hskip 5.15em \raise 0.4em{\alpha}}} \smash{\rlap{\hskip 6.05em \raise 2.05em{\bullet}}} \smash{\rlap{\hskip 2.5em \raise 2.6em{\bullet}}} \smash{\rlap{\hskip 6.65em \raise 2.45em{P(\cos\alpha,\sin\alpha)}}} \smash{\rlap{\hskip -8em \raise 3.4em{Q(\cos(\alpha+\beta),\sin(\alpha+\beta))}}} \smash{\rlap{\hskip 4.1em \raise 1.1em{\beta}}} \smash{\rlap{\hskip 4em \raise 0.2em \enclose{updiagonalstrike}{\phantom{\Rule{1.9em}{1.9em}{0em}}}}} \smash{\rlap{\hskip 2.6em \raise 0.2em \enclose{downdiagonalstrike}{\phantom{\Rule{1.04em}{2.6em}{0em}}}}} \phantom{\Rule{8em}{4em}{4em}}}$$

「ふんふん……それで？」

「ここで垂線を下ろす」

$$\require{color}\color{#777777}{\require{enclose}\smash{\rlap{\hskip 0.8em \enclose{circle}{\phantom{\Rule{6em}{3em}{3em}}}}} \smash{\rlap{\Rule{8em}{0.05em}{0.05em}}} \smash{\rlap{\hskip 4em \Rule{0.1em}{4em}{4em}}} \smash{\rlap{\hskip 8.35em \raise -0.35em{x}}} \smash{\rlap{\hskip 7.35em \raise 0.35em{(1,0)}}} \smash{\rlap{\hskip 4.18em \raise 4.35em{y}}} \smash{\rlap{\hskip 4.15em \raise -0.85em{O}}} \smash{\rlap{\hskip 5.15em \raise 0.4em{\alpha}}} \smash{\rlap{\hskip 6.05em \raise 2.05em{\bullet}}} \smash{\rlap{\hskip 2.5em \raise 2.6em{\bullet}}} \smash{\rlap{\hskip 6.65em \raise 2.45em{P(\cos\alpha,\sin\alpha)}}} \smash{\rlap{\hskip -8em \raise 3.4em{Q(\cos(\alpha+\beta),\sin(\alpha+\beta))}}} \smash{\rlap{\hskip 4.1em \raise 1.1em{\beta}}} \smash{\rlap{\hskip 4em \raise 0.2em \enclose{updiagonalstrike}{\phantom{\Rule{1.9em}{1.9em}{0em}}}}} \smash{\rlap{\hskip 6.21em \Rule{0.1em}{2.21em}{0em}}} \smash{\rlap{\hskip 2.7em \Rule{0.1em}{2.8em}{0em}}} \smash{\rlap{\hskip 2.6em \raise 0.2em \enclose{downdiagonalstrike}{\phantom{\Rule{1.04em}{2.6em}{0em}}}}} \phantom{\Rule{8em}{4em}{4em}}}$$

「そしてもう一本」

ぼくは点Qを通る線分OPの垂線を引く。

$$\require{color}\color{#777777}{\require{enclose}\smash{\rlap{\hskip 0.8em \enclose{circle}{\phantom{\Rule{6em}{3em}{3em}}}}} \smash{\rlap{\Rule{8em}{0.05em}{0.05em}}} \smash{\rlap{\hskip 4em \Rule{0.1em}{4em}{4em}}} \smash{\rlap{\hskip 8.35em \raise -0.35em{x}}} \smash{\rlap{\hskip 7.35em \raise 0.35em{(1,0)}}} \smash{\rlap{\hskip 4.18em \raise 4.35em{y}}} \smash{\rlap{\hskip 4.15em \raise -0.85em{O}}} \smash{\rlap{\hskip 5.15em \raise 0.4em{\alpha}}} \smash{\rlap{\hskip 6.05em \raise 2.05em{\bullet}}} \smash{\rlap{\hskip 2.5em \raise 2.6em{\bullet}}} \smash{\rlap{\hskip 6.65em \raise 2.45em{P(\cos\alpha,\sin\alpha)}}} \smash{\rlap{\hskip -8em \raise 3.4em{Q(\cos(\alpha+\beta),\sin(\alpha+\beta))}}} \smash{\rlap{\hskip 4.1em \raise 1.1em{\beta}}} \smash{\rlap{\hskip 4em \raise 0.2em \enclose{updiagonalstrike}{\phantom{\Rule{1.9em}{1.9em}{0em}}}}} \smash{\rlap{\hskip 6.21em \Rule{0.1em}{2.21em}{0em}}} \smash{\rlap{\hskip 2.7em \Rule{0.1em}{2.8em}{0em}}} \smash{\rlap{\hskip 2.6em \raise 0.2em \enclose{downdiagonalstrike}{\phantom{\Rule{1.04em}{2.6em}{0em}}}}} \smash{\rlap{\hskip 2.7em \raise 1.0em \enclose{downdiagonalstrike}{\phantom{\Rule{1.7em}{1.7em}{0em}}}}} \phantom{\Rule{8em}{4em}{4em}}}$$

「これであらかた揃ったんじゃないかな」

ぼくは点Qを通る線分OPの垂線を引く。

$$\require{color}\color{#777777}{\require{enclose}\smash{\rlap{\hskip 0.8em \enclose{circle}{\phantom{\Rule{6em}{3em}{3em}}}}} \smash{\rlap{\Rule{8em}{0.05em}{0.05em}}} \smash{\rlap{\hskip 4em \Rule{0.1em}{4em}{4em}}} \smash{\rlap{\hskip 8.35em \raise -0.35em{x}}} \smash{\rlap{\hskip 7.35em \raise 0.35em{(1,0)}}} \smash{\rlap{\hskip 4.18em \raise 4.35em{y}}} \smash{\rlap{\hskip 4.15em \raise -0.85em{O}}} \smash{\rlap{\hskip 5.15em \raise 0.4em{\alpha}}} \smash{\rlap{\hskip 6.05em \raise 2.05em{\bullet}}} \smash{\rlap{\hskip 2.5em \raise 2.6em{\bullet}}} \smash{\rlap{\hskip 6.65em \raise 2.45em{P(\cos\alpha,\sin\alpha)}}} \smash{\rlap{\hskip -8em \raise 3.4em{Q(\cos(\alpha+\beta),\sin(\alpha+\beta))}}} \smash{\rlap{\hskip 4.1em \raise 1.1em{\beta}}} \smash{\rlap{\hskip 4em \raise 0.2em \enclose{updiagonalstrike}{\phantom{\Rule{1.9em}{1.9em}{0em}}}}} \smash{\rlap{\hskip 6.21em \Rule{0.1em}{2.21em}{0em}}} \smash{\rlap{\hskip 2.7em \Rule{0.1em}{2.8em}{0em}}} \smash{\rlap{\hskip 2.6em \raise 0.2em \enclose{downdiagonalstrike}{\phantom{\Rule{1.04em}{2.6em}{0em}}}}} \smash{\rlap{\hskip 2.7em \raise 1.0em \enclose{downdiagonalstrike}{\phantom{\Rule{1.7em}{1.7em}{0em}}}}} \phantom{\Rule{8em}{4em}{4em}}}$$

「……あとは長さだけか」

「じゃあ埋めていこう」

「よし」

「最初はサインの加法定理にしよう……高さを埋めるね」

$$\require{color}\color{#777777}{\require{enclose}\smash{\rlap{\hskip 0.8em \enclose{circle}{\phantom{\Rule{6em}{3em}{3em}}}}} \smash{\rlap{\Rule{8em}{0.05em}{0.05em}}} \smash{\rlap{\hskip 4em \Rule{0.1em}{4em}{4em}}} \smash{\rlap{\hskip 8.35em \raise -0.35em{x}}} \smash{\rlap{\hskip 7.35em \raise 0.35em{(1,0)}}} \smash{\rlap{\hskip 4.18em \raise 4.35em{y}}} \smash{\rlap{\hskip 4.15em \raise -0.85em{O}}} \smash{\rlap{\hskip 5.15em \raise 0.4em{\alpha}}} \smash{\rlap{\hskip 5.15em \raise 1.1em{1}}} \smash{\rlap{\hskip 6.05em \raise 2.05em{\bullet}}} \smash{\rlap{\hskip 2.5em \raise 2.6em{\bullet}}} \smash{\rlap{\hskip 6.65em \raise 2.45em{P(\cos\alpha,\sin\alpha)}}} \smash{\rlap{\hskip -8em \raise 3.4em{Q(\cos(\alpha+\beta),\sin(\alpha+\beta))}}} \smash{\rlap{\hskip -1.5em \raise 1.4em{\sin(\alpha+\beta)}}} \smash{\rlap{\hskip 4.1em \raise 1.1em{\beta}}} \smash{\rlap{\hskip 3.1em \raise 1.1em{1}}} \smash{\rlap{\hskip 4em \raise 0.2em \enclose{updiagonalstrike}{\phantom{\Rule{1.9em}{1.9em}{0em}}}}} \smash{\rlap{\hskip 6.21em \Rule{0.1em}{2.21em}{0em}}} \smash{\rlap{\hskip 2.7em \Rule{0.1em}{2.8em}{0em}}} \smash{\rlap{\hskip 2.6em \raise 0.2em \enclose{downdiagonalstrike}{\phantom{\Rule{1.04em}{2.6em}{0em}}}}} \smash{\rlap{\hskip 2.7em \raise 1.0em \enclose{downdiagonalstrike}{\phantom{\Rule{1.7em}{1.7em}{0em}}}}} \smash{\rlap{\hskip 7.35em \raise 0.35em{(1,0)}}} \smash{\rlap{\hskip 6.2em \raise 1.2em{\sin\alpha}}} \phantom{\Rule{8em}{4em}{4em}}}$$

「なんかゴチャゴチャしてきたぞ」

「ちょっと一部だけ抜き出してみようか」