# 数学I

## 指数法則

$a^m \times a^n = a^{m+n}$

$$a^m \times a^n = a^{m+n}$$

$(a^m)^n = a^{mn}$

$$(a^m)^n = a^{mn}$$

$(ab)^n = a^{n}b^{n}$

$$(ab)^n = a^{n}b^{n}$$

## 展開公式

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

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

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

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

## 循環小数

$0.\dot{5}\dot{6} = 0.56565656 \ldots$

$$0.\dot{5}\dot{6} = 0.56565656 \ldots$$

$0.2\dot{1} = 0.21111111\ldots$

$$0.2\dot{1} = 0.21111111\ldots$$

$0.\dot{4}5\dot{6} = 0.456456456\ldots$

$$0.\dot{4}5\dot{6} = 0.456456456\ldots$$

## 有理化

$\dfrac{b}{\sqrt{a}} = \dfrac{b \times \sqrt{a}}{\sqrt{a} \times \sqrt{a}} \\ = \dfrac{b \sqrt{a}}{a}$

$$\dfrac{b}{\sqrt{a}} = \dfrac{b \times \sqrt{a}}{\sqrt{a} \times \sqrt{a}} \\ = \dfrac{b \sqrt{a}}{a}$$

## 根号

$\sqrt{(a+b) + 2\sqrt{ab}} = \sqrt{a} + \sqrt{b}$

$$\sqrt{(a+b) + 2\sqrt{ab}} = \sqrt{a} + \sqrt{b}$$

$\sqrt{(a+b) - 2\sqrt{ab}} = \sqrt{a} - \sqrt{b}$

$$\sqrt{(a+b) - 2\sqrt{ab}} = \sqrt{a} - \sqrt{b}$$

## 一次不等式

$|x| = a \iff x = \pm a$

$$|x| = a \iff x = \pm a$$

$|x| < a \iff -a < x < a$

$$|x| < a \iff -a < x < a$$

$|x| > a \iff x < -a, x > a$

$$|x| > a \iff x < -a, x > a$$

## 平方完成

$y = ax^2 + bx + c = a \left( x + \dfrac{b}{2a} \right)^2 - \dfrac{b^2 - 4ac}{4a}$

$$y = ax^2 + bx + c = a \left( x + \dfrac{b}{2a} \right)^2 - \dfrac{b^2 - 4ac}{4a}$$

## 判別式

$D^2 = b^2 - 4ac$

$$D^2 = b^2 - 4ac$$

$\begin{cases} D>0 \iff \text{異なる2つの実数解を持つ} \\ D=0 \iff \text{実数解を1つもつ（重解）} \\ D<0 \iff \text{実数解を持たない（異なる2つの虚数解を持つ）} \end{cases}$

$$\begin{cases} D>0 \iff \text{異なる2つの実数解を持つ} \\ D=0 \iff \text{実数解を1つもつ（重解）} \\ D<0 \iff \text{実数解を持たない（異なる2つの虚数解を持つ）} \end{cases}$$

## 三角比の変換公式

$\tan \theta = \dfrac{\sin \theta}{\cos \theta}$

$$\tan \theta = \dfrac{\sin \theta}{\cos \theta}$$

$\sin^{2} \theta + \cos^{2} \theta = 1$

$$\sin^{2} \theta + \cos^{2} \theta = 1$$

$1 + \tan^{2} \theta = \dfrac{1}{\cos^{2} \theta}$

$$1 + \tan^{2} \theta = \dfrac{1}{\cos^{2} \theta}$$

## 正弦定理

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

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

## 余弦定理

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

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

$b^2 = c^2 + a^2 - 2ca \cos B$

$$b^2 = c^2 + a^2 - 2ca \cos B$$

$c^2 = a^2 + b^2 - 2ab \cos C$

$$c^2 = a^2 + b^2 - 2ab \cos C$$

## 三角形の面積

$S = \dfrac{1}{2} ab \sin \theta$

$$S = \dfrac{1}{2} ab \sin \theta$$

## 集合の演算子

$a \in A, A \ni a$

$$a \in A, A \ni a$$

$a \notin A$

$$a \notin A$$

$A \subset B, B \supset A$

$$A \subset B, B \supset A$$

$A \not \subset B$

$$A \not \subset B$$

$A \cap B$

$$A \cap B$$

$A \cup B$

$$A \cup B$$

$\emptyset$

$$\emptyset$$

$\overline{A}$

$$\overline{A}$$

## ドモルガンの法則

$\overline{A \cup B} = \overline{A} \cap \overline{B}$

$$\overline{A \cup B} = \overline{A} \cap \overline{B}$$

$\overline{A \cap B} = \overline{A} \cup \overline{B}$

$$\overline{A \cap B} = \overline{A} \cup \overline{B}$$

## 和集合の要素の数

$n(A \cup B) = n(A) + n(B) - n(A \cap B \cap C)$

$$n(A \cup B) = n(A) + n(B) - n(A \cap B \cap C)$$

$n(\overline{A}) = n(U) - n(A)$

$$n(\overline{A}) = n(U) - n(A)$$

\begin{aligned} n(A \cup B \cup C) &= n(A) + n(B) + n(C) \\ &- n(A \cap B) -n(B \cap C) - n(C \cap A) \\ &+ n(A \cap B \cap C) \end{aligned}

\begin{aligned} n(A \cup B \cup C) &= n(A) + n(B) + n(C) \\ &- n(A \cap B) -n(B \cap C) - n(C \cap A) \\ &+ n(A \cap B \cap C) \end{aligned}

## 命題

$A \Rightarrow B$

$$A \Rightarrow B$$

$A \Leftarrow B$

$$A \Leftarrow B$$

$A \iff B$

$$A \iff B$$

## 平均値

\begin{aligned} \bar{x} &= \dfrac{\sum_{k=1}^{n} x_k}{n} \\ &= \dfrac{x_1 + x_2 + x_3 + \cdots + x_n}{n} \end{aligned}

\begin{aligned} \bar{x} &= \dfrac{\sum_{k=1}^{n} x_k}{n} \\ &= \dfrac{x_1 + x_2 + x_3 + \cdots + x_n}{n} \end{aligned}

## 中央値

$\tilde{x} = \begin{cases} x_\frac{n+1}{2} \quad \text{(nが奇数のとき)} \\ \dfrac{x_\frac{n}{2} + x_{\frac{n}{2}+1}}{2} \quad \text{(nが偶数のとき)} \end{cases}$

$$\tilde{x} = \begin{cases} x_\frac{n+1}{2} \quad \text{(nが奇数のとき)} \\ \dfrac{x_\frac{n}{2} + x_{\frac{n}{2}+1}}{2} \quad \text{(nが偶数のとき)} \end{cases}$$

## 偏差

$x_k-\bar{x}$

$$x_k-\bar{x}$$

## 分散

$S^2 = \dfrac{1}{n} \sum_{k=1}^{n} (x_k - \bar{x})^{2}$

$$S^2 = \dfrac{1}{n} \sum_{k=1}^{n} (x_k - \bar{x})^{2}$$

## 共分散

$S_{xy} = \dfrac{1}{n} \sum_{k=1}^n (x_k - \bar{x})(y_k - \bar{y})$

$$S_{xy} = \dfrac{1}{n} \sum_{k=1}^n (x_k - \bar{x})(y_k - \bar{y})$$