数学II

二項定理

(a+b)n=nC0an+nC1an−1b+nC2an−2b2+⋯+nCn−1abn−1+nCnbn(a+b)^n = {}_n\mathrm{C}_0 a^n + {}_n\mathrm{C}_1 a^{n-1}b + {}_n\mathrm{C}_2 a^{n-2}b^2 + \cdots + {}_n\mathrm{C}_{n-1} ab^{n-1} + {}_n\mathrm{C}_n b^n

$$
(a+b)^n = 
{}_n\mathrm{C}_0 a^n + 
{}_n\mathrm{C}_1 a^{n-1}b + 
{}_n\mathrm{C}_2 a^{n-2}b^2 + 
\cdots + 
{}_n\mathrm{C}_{n-1} ab^{n-1} + 
{}_n\mathrm{C}_n b^n
$$

虚数単位

i=−1i = \sqrt{-1}

$$
i = \sqrt{-1}
$$

複素数

α=a+bi\alpha = a + bi

$$
\alpha = a + bi
$$

共役複素数

α‾=a−bi\overline{\alpha} = a - bi

$$
\overline{\alpha} = a - bi
$$

α‾‾=α\overline{\overline{\alpha}} = \alpha

$$
\overline{\overline{\alpha}} = \alpha
$$

α+α‾=2a\alpha + \overline{\alpha} = 2a

$$
\alpha + \overline{\alpha} = 2a
$$

αα‾=a2+b2\alpha \overline{\alpha} = a^2 + b^2

$$
\alpha \overline{\alpha} = a^2 + b^2
$$

N乗根

x3=a  ⟺  x=a3x^3 = a \iff x = \sqrt[3]{a}

$$
x^3 = a \iff x = \sqrt[3]{a}
$$

xn=a  ⟺  x=anx^n = a \iff x = \sqrt[n]{a}

$$
x^n = a \iff x = \sqrt[n]{a}
$$

点と直線の距離

d=∣ax1+by1+c∣a2+b2d = \dfrac{| ax_1 + by_1 + c |}{\sqrt{a^2 + b^2}}

$$
d = \dfrac{| ax_1 + by_1 + c |}{\sqrt{a^2 + b^2}}
$$

円

(x−a)2+(y−b)2=r2(x - a)^2 + (y - b)^2 = r^2

$$
(x - a)^2 + (y - b)^2 = r^2
$$

円の接線の方程式

x1x+y1y=r2x_1 x + y_1 y = r^2

$$
x_1 x + y_1 y = r^2
$$

加法定理

sin⁡(α±β)=sin⁡αcos⁡β±cos⁡αsin⁡β\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta

$$
\sin(\alpha \pm \beta) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta
$$

cos⁡(α±β)=cos⁡αcos⁡β∓sin⁡αcos⁡β\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \cos \beta

$$
\cos(\alpha \pm \beta) = \cos \alpha \cos \beta \mp \sin \alpha \cos \beta
$$

tan⁡(α±β)=tan⁡α±tan⁡β1∓tan⁡αtan⁡β\tan(\alpha \pm \beta) = \dfrac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}

$$
\tan(\alpha \pm \beta) = \dfrac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}
$$

2倍角の公式

sin⁡2α=2sin⁡αcos⁡α\sin2 \alpha = 2 \sin \alpha \cos \alpha

$$
\sin2 \alpha = 2 \sin \alpha \cos \alpha
$$

cos⁡2α=cos⁡2α−sin⁡2α=2cos⁡2α−1=1−2sin⁡2α\begin{aligned} \cos2 \alpha &= \cos^2 \alpha - \sin^2 \alpha \\ &= 2 \cos^2 \alpha - 1 \\ &= 1 - 2 \sin^2 \alpha \end{aligned}

$$
\begin{aligned} 
\cos2 \alpha &= \cos^2 \alpha - \sin^2 \alpha \\ 
&= 2 \cos^2 \alpha - 1 \\ 
&= 1 - 2 \sin^2 \alpha 
\end{aligned}
$$

tan⁡2α=2tan⁡α1−tan⁡2α\tan2 \alpha = \dfrac{2 \tan \alpha}{1 - \tan^2 \alpha}

$$
\tan2 \alpha = \dfrac{2 \tan \alpha}{1 - \tan^2 \alpha}
$$

半角の公式

sin⁡2α2=1−cos⁡α2\sin^2 \dfrac{\alpha}{2} = \dfrac{1 - \cos \alpha}{2}

$$
\sin^2 \dfrac{\alpha}{2} = \dfrac{1 - \cos \alpha}{2}
$$

cos⁡2α2=1+cos⁡α2\cos^2 \dfrac{\alpha}{2} = \dfrac{1 + \cos \alpha}{2}

$$
\cos^2 \dfrac{\alpha}{2} = \dfrac{1 + \cos \alpha}{2}
$$

tan⁡2α2=1−cos⁡α1+cos⁡α\tan^2 \dfrac{\alpha}{2} = \dfrac{1 - \cos \alpha}{1 + \cos \alpha}

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

3倍角の公式

sin⁡3α=3sin⁡α−4sin⁡3α\sin3 \alpha = 3 \sin \alpha - 4 \sin^3 \alpha

$$
\sin3 \alpha = 3 \sin \alpha - 4 \sin^3 \alpha
$$

cos⁡3α=−3cos⁡α+4cos⁡3α\cos3 \alpha = -3 \cos \alpha + 4 \cos^3 \alpha

$$
\cos3 \alpha = -3 \cos \alpha + 4 \cos^3 \alpha
$$

三角関数の合成

asin⁡θ+bcos⁡θ=a2+b2sin⁡(θ+α)a \sin \theta + b \cos \theta = \sqrt{a^2+b^2} \sin(\theta + \alpha)

$$
a \sin \theta + b \cos \theta = \sqrt{a^2+b^2} \sin(\theta + \alpha)
$$

対数

log⁡aM=p  ⟺  M=ap\log_a M = p \iff M = a^p 

$$
\log_a M = p \iff M = a^p 
$$

対数の性質

log⁡a1=0\log_a 1 = 0

$$
\log_a 1 = 0
$$

log⁡aa=1\log_a a = 1

$$
\log_a a = 1
$$

log⁡aMN=log⁡aM+log⁡aN\log_a MN = \log_a M + \log_a N

$$
\log_a MN = \log_a M + \log_a N
$$

log⁡aMN=log⁡aM−log⁡aN\log_a \dfrac{M}{N} = \log_a M - \log_a N

$$
\log_a \dfrac{M}{N} = \log_a M - \log_a N
$$

log⁡aMr=rlog⁡aM\log_a M^r = r \log_a M

$$
\log_a M^r = r \log_a M
$$

底の変換公式

log⁡ab=log⁡cblog⁡ca\log_a b = \dfrac{\log_c b}{\log_c a}

$$
\log_a b = \dfrac{\log_c b}{\log_c a}
$$

導関数

f′(x)=lim⁡Δx→0f(x+Δx)−f(x)Δxf'(x) = \lim_{\varDelta x \to 0} \dfrac{ f(x+\varDelta x) - f(x) }{\varDelta x}

$$
f'(x) = \lim_{\varDelta x \to 0} \dfrac{ f(x+\varDelta x) - f(x) }{\varDelta x}
$$

接線の方程式

y−f(a)=f′(a)(x−a)y - f(a) = f'(a)(x - a)

$$
y - f(a) = f'(a)(x - a)
$$

増減表

x⋯−a⋯a⋯f′(x)+0−0+f(x)↗2a3↘−2a3↗\begin{array}{c|ccccc} x & \cdots & -a & \cdots & a & \cdots \\ \hline f'(x) & + & 0 & - & 0 & + \\ \hline f(x) & \nearrow & 2a^3 & \searrow & -2a^3 & \nearrow \end{array} 

$$
\begin{array}{c|ccccc} 
x & \cdots & -a & \cdots & a & \cdots \\ 
\hline 
f'(x) & + & 0 & - & 0 & + \\ 
\hline 
f(x) & \nearrow & 2a^3 & \searrow & -2a^3 & \nearrow 
\end{array} 
$$

不定積分

∫f(x) dx=F(x)+C\int f(x) \ dx = F(x) + C

$$
\int f(x) \ dx = F(x) + C
$$

定積分

∫abf(x) dx=[F(x)]ab=F(b)−F(a)\int_{a}^{b} f(x) \ dx = \left[ F(x) \right]^{b}_{a} = F(b) - F(a)

$$
\int_{a}^{b} f(x) \ dx = \left[ F(x) \right]^{b}_{a} = F(b) - F(a)
$$

微分と積分の関係

ddx∫axf(t) dt=ddx[F(t)]ax=ddx{F(x)−F(a)}=F′(x)\begin{aligned} & \dfrac{d}{dx} \int_{a}^{x} f(t) \ dt \\ &= \dfrac{d}{dx} \left[ F(t) \right]^{x}_{a} \\ &= \dfrac{d}{dx} \{ F(x) - F(a) \} \\ &= F'(x) \end{aligned} 

$$
\begin{aligned} 
& \dfrac{d}{dx} \int_{a}^{x} f(t) \ dt \\ 
&= \dfrac{d}{dx} \left[ F(t) \right]^{x}_{a} \\ 
&= \dfrac{d}{dx} \{ F(x) - F(a) \} \\ 
&= F'(x) 
\end{aligned} 
$$