---
# System prepended metadata

title: お勉強用
tags: [memo]

---

# お勉強用

###### tags: `memo`

## 解説

:::success
:memo: **解の公式**

$ax^2 + bx + c = 0$ のとき $x$ を求める時こうするよ！

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

:::

:::success
:memo: **ネイピア数の定義**

$$
e = \lim_{n \rightarrow \infty} \left( 1 + \cfrac{1}{n} \right)^n
$$

$$
e^x = \sum_{n=0}^{\infty} \cfrac{x^n}{n!}
$$
:::

:::success
:memo: **覚えておきたい三角関数の関係**

#### (1) よく使う
$$
\tan \alpha = \cfrac{\sin \alpha}{\cos \alpha}
$$

#### (2) よく使う
$$
\sin^2 \alpha + \cos^2 \alpha = 1
$$

#### (3) あんま使わん
$$
1 + \tan^2 \alpha = \cfrac{1}{\cos^2 \alpha}
$$

:::

:::success
:memo: **正弦定理**
$$
\cfrac{a}{\sin A} = \cfrac{b}{\sin B} = \cfrac{c}{\sin C} = 2R
$$

:memo: **余弦定理**
$$
\left\{
\begin{array}{l}
a^2 = b^2 + c^2 -2bc \cos A \\
b^2 = c^2 + a^2 -2ca \cos B \\
c^2 = a^2 + b^2 -2ab \cos C
\end{array}
\right.
$$

![正弦定理説明用画像](https://i.imgur.com/aUUgZ1H.png)
:::

### (5)
:::spoiler
$$
\begin{align}
    2R &= \cfrac{3}{\sin 60^{\circ}} \\
    &= \cfrac{3}{\cfrac{\sqrt{3}}{2}} \\
    &= 3 \div \cfrac{\sqrt{3}}{2} \\
    &= 3 \times \cfrac{2}{\sqrt{3}} \\
    &= 3 \times \cfrac{2}{\sqrt{3}} \times \cfrac{\sqrt{3}}{\sqrt{3}} \\
    &= 3 \times \cfrac{2 \sqrt{3}}{3} \\
    &= 2 \sqrt{3}
\end{align}
$$

よって両辺を $2$ で割ってあげると $R = \sqrt{3}$
:::

### 7-(1)
:::spoiler
$\sin^2 \alpha + \cos^2 \alpha = 1$ の関係を用いて解く

$$
\begin{align}
    \cos^2 A &= 1 - \sin^2 A \\
    &= 1 - \left( \cfrac{4}{5} \right)^2 \\
    &= 1 - \cfrac{16}{25} \\
    &= \cfrac{9}{25}
\end{align}
$$

$$
\cos A = \sqrt{\cfrac{9}{25}} = \pm \cfrac{3}{5}
$$

ここで、鈍角なので $\cos A = -\cfrac{3}{5}$
:::

### 7-(2)
:::spoiler
$$
\begin{align}
    \tan A &= \cfrac{\sin A}{\cos A} \\
    &= \cfrac{4}{5} \div \left( -\cfrac{3}{5} \right) \\
    &= \cfrac{4}{5} \times \left( -\cfrac{5}{3} \right) \\
    &= -\cfrac{4}{3}
\end{align}
$$
:::

$$
\begin{align}
\int e^{-x^2}\mathrm{d}x \\
\int \frac{\sin x}{x}\mathrm{d}x \\
\int \frac{1}{\ln x}\mathrm{d}x \\
\int \frac{1}{1+x^4}\mathrm{d}x \\
\int \sqrt{1+x^4}\mathrm{d}x \\
\end{align}
$$