TeX記法 数学I

概要

数学Iで使用する公式のTex記法を下記に示す。

TeX記法 数学I

展開
(ax + b)(cx + d) = acx^2 + (ad + bc)x + bd

(ax + b)(cx + d) = acx^2 + (ad + bc)x + bd

３次式の展開
(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 \\
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3

３次式因数分解
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
a^3 - b^3 = (a - b)(a^2 + ab + b^2)

a^3 + b^3 = (a + b)(a^2 - ab + b^2) \\
a^3 - b^3 = (a - b)(a^2 + ab + b^2)

有理化
\frac{b}{\sqrt{a}} = \frac{b\sqrt{a}}{a}

\frac{b}{\sqrt{a}} = \frac{b\sqrt{a}}{a}

二重根号
\sqrt{a + b \pm 2\sqrt{ab}} = \sqrt{a} \pm \sqrt{b}

\sqrt{a + b \pm 2\sqrt{ab}} = \sqrt{a} \pm \sqrt{b}

解の公式
x = \frac{-b \pm\sqrt{b^2 - 4ac}}{2a}

x = \frac{-b \pm\sqrt{b^2 - 4ac}}{2a}

判別式
ax^2 + bx + c = 0の判別式D = b^2 - 4ac

ax^2 + bx + c = 0 \\
D = b^2 - 4ac \\
D > 0 の時異なる２つの実数解 \\
D = 0 の時重解 \\
D < 0 の時実数解なし

三角比の相互関係
\sin^2\theta + \cos^2\theta = 1
\tan\theta = \frac{\sin\theta}{\cos\theta}
1 + \tan^2\theta = \frac{1}{\cos^2\theta}

\sin^2\theta + \cos^2\theta = 1 \\
\tan\theta = \frac{\sin\theta}{\cos\theta} \\
1 + \tan^2\theta = \frac{1}{\cos^2\theta}

正弦定理
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}  = 2R

余弦定理
a^2 = b^2 + c^2 - 2bc\cos A

a^2 = b^2 + c^2 - 2bc\cos A

三角形の面積
S = \frac{1}{2} ab\sin C

S = \frac{1}{2} ab\sin C

ヘロンの公式
S = \sqrt{s(s - a)(s - b)(s - c)}

S = \sqrt{s(s - a)(s - b)(s - c)}

内接円の半径と面積
S = \frac{1}{2}(a + b + c)r

S = \frac{1}{2}(a + b + c)r

平均値
\overline{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}

\overline{x} = \frac{x_1 + x_2 + \cdots + x_n}{n}

分散
s^2 = \frac{1}{n}\{(x_1 - \overline{x})^2 + (x_2 - \overline{x})^2 \cdots + (x_n - \overline{x})^2\}

s^2 = \frac{1}{n}\{(x_1 - \overline{x})^2 + (x_2 - \overline{x})^2 \cdots + (x_n - \overline{x})^2\}

標準偏差
s = \sqrt{{分散}} = \sqrt{\frac{1}{n}\{(x_1 - \overline{x})^2 + (x_2 - \overline{x})^2 \cdots + (x_n - \overline{x})^2}\}

s = \sqrt{{分散}} = \sqrt{\frac{1}{n}\{(x_1 - \overline{x})^2 + (x_2 - \overline{x})^2 \cdots + (x_n - \overline{x})^2}\}

共分散
s_{xy} = \frac{1}{n}\{(x_1 - \overline{x})(y_1 - \overline{y}) + (x_2 - \overline{x})(y_2 - \overline{y}) \cdots + (x_n - \overline{x})(y_n - \overline{y})\}

s_{xy} = \frac{1}{n}\{(x_1 - \overline{x})(y_1 - \overline{y}) + (x_2 - \overline{x})(y_2 - \overline{y}) \cdots + (x_n - \overline{x})(y_n - \overline{y})\}