# 式と証明

2次方程式の解と係数の関係

$$\alpha + \beta = -\frac{b}{a}, \quad \alpha\beta = \frac{c}{a}$$ $$\alpha + \beta = \frac{-b + \sqrt{D}}{2a} + \frac{-b - \sqrt{D}}{2a} = -\frac{b}{a}$$ $$\alpha \times \beta = \frac{-b + \sqrt{D}}{2a} \times \frac{-b - \sqrt{D}}{2a} = \frac{b^2 -D}{4a^2} = \frac{b^2 - (b^2 - 4ac)}{4a^2} = \frac{c}{a}$$

$$\text{整式}P(x)\text{を}x - \alpha\text{で割ったときの余りは}P(\alpha)$$

$$\text{整式}P(x)\text{を因数に持つ} \iff P(\alpha) = 0$$

$$\text{a > 0, b > 0のとき} \quad \frac{a + b}{2} \geq \sqrt{ab}$$ $$\text{等号が成り立つのは、a = bのとき}$$

# 図形と方程式

２点間の距離

$$\text{２点}A(x_1, y_1), B(x_2, y_2)\text{間の距離は}$$ $$AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$ $$\text{とくに, 原点}O\text{と}P(x, y)\text{の距離は}OP = \sqrt{x^2 + y^2}$$

$$\text{２点}A(x_1, y_1), B(x_2, y_2)\text{を結ぶ線分}AB\text{を}$$ $$m : n\text{に内分する点の座標は}(\frac{nx_1 + mx_2}{m + n}, \frac{ny_1 + ny_2}{m + n})$$ $$m : n\text{に外分する点の座標は}(\frac{-nx_1 + mx_2}{m - n}, \frac{-ny_1 + ny_2}{m - n})$$ $$\text{とくに, 線分}AB\text{の中点の座標は}(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})$$

$$\text{３点}A(x_1, y_1), B(x_2, y_2), C(x_3, y_3)\text{を頂点とする}\triangle ABC\text{の重心}G\text{の座標は}$$ $$(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3})$$

１点を通り, 傾きmの直線

$$\text{点}(x_1, y_1)\text{を通り, 傾き}m\text{の直線の方程式は}$$ $$y - y_1 = m(x - x_1)$$

２点を通る直線

$$\text{２点}(x_1, y_1), (x_2, y_2)\text{を通る直線の方程式は}$$ $$x_1 \ne x_2\text{のとき} \quad y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$$ $$x_1 = x_2\text{のとき} \quad x = x_1$$

２直線の平行条件・垂直条件

$$\text{２直線} y = mx + n, y = m'x + n'\text{について}$$ $$\text{平行条件は} \ m = m' \quad \text{垂直条件は} \ mm' = -1$$

$$\text{点}(x_1, y_1)\text{と直線}ax + by + c = 0\text{の距離}d\text{は}$$ $$d = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}}$$

$$\triangle ABC\text{の辺}BC\text{の中点を}M\text{とすれば}$$ $$AB^2 + AC^2 = 2(AM^2 + BM^2)$$

$$\text{点}(a, b)\text{を中心とする半径}r\text{の円の方程式は}$$ $$(x-a)^2 + (y-b)^2 = r^2$$ $$\text{とくに原点を中心とする半径}r\text{の円の方程式は}$$ $$x^2 + y^2 = r^2$$

$$D > 0 \iff \text{円と直線の共有点は２個}$$ $$D = 0 \iff \text{円と直線の共有点は１個}$$ $$D < 0 \iff \text{円と直線の共有点はない}$$

$$\text{円} \ x^2 + y^2 = r^2 \ \text{の周上の点}P(x_1, y_1)\text{における接線の方程式は}$$ $$x_1x + y_1y = r^2$$

$$\text{直線} \ y = mx + n \ \text{を}l\text{とすれば}$$ $$y > mx + n \ \text{の表す領域は} \quad \text{直線}l\text{の上側}$$ $$y < mx + n \ \text{の表す領域は} \quad \text{直線}l\text{の下側}$$

$$\text{円} \ (x - a)^2 + (y - b)^2 = r^2 \ \text{を}C\text{とすれば}$$ $$(x - a)^2 + (y - b)^2 < r^2 \ \text{の表す領域は} \quad \text{円}C\text{の内部}$$ $$(x - a)^2 + (y - b)^2 > r^2 \ \text{の表す領域は} \quad \text{円}C\text{の外部}$$

# 三角関数

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

$$\sin(\theta + 2n \pi) = \sin \theta \quad \quad \sin(-\theta) = -\sin \theta$$ $$\cos(\theta + 2n \pi) = \cos \theta \quad \quad \cos(-\theta) = \cos \theta$$ $$\tan(\theta + 2n \pi) = \tan \theta \quad \quad \tan(-\theta) = -\tan \theta$$ $$\sin(\theta + \pi) = -\sin \quad \quad \sin(\pi - \theta) = \sin \theta$$ $$\cos(\theta + \pi) = -\cos \quad \quad \cos(\pi - \theta) = -\cos \theta$$ $$\tan(\theta + \pi) = \tan \quad \quad \tan(\pi - \theta) = -\tan \theta$$ $$\sin(\theta + \frac{\pi}{2}) = \cos \theta \quad \quad \sin(\frac{\pi}{2} - \theta) = \cos \theta$$ $$\cos(\theta + \frac{\pi}{2}) = -\sin \theta \quad \quad \cos(\frac{\pi}{2} - \theta) = \sin \theta$$ $$\tan(\theta + \frac{\pi}{2}) = -\frac{1}{\tan \theta} \quad \quad \tan(\frac{\pi}{2} - \theta) = \frac{1}{\tan \theta}$$

# 正弦定理

$$\sin (\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$$ $$\sin (\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$$

$$\cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$$ $$\cos (\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$$

$$\tan (\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$$ $$\tan (\alpha - \beta) = \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$$
2倍角の公式
$$\sin2 \alpha = 2\sin \alpha \cos \alpha$$ $$\cos2 \alpha = \cos^2 \alpha - \sin^2 \alpha = 1 - 2\sin^2 \alpha = 2\cos^2 \alpha - 1$$ $$\tan2 \alpha = \frac{2\tan \alpha}{1 - tan^2 \alpha}$$

$$\sin^2\frac{\alpha}{2} = \frac{1 - \cos2\alpha}{2}, \quad \cos^2\frac{\alpha}{2} = \frac{1 + \cos2\alpha}{2}$$
3倍角の公式
$$\sin3\alpha = 3\sin\alpha - 4\sin^3\alpha$$ $$\cos3\alpha = 4\cos^3\alpha - 3\cos\alpha$$ $$\tan3\alpha = \frac{3\tan\alpha - \tan^3\alpha}{1 - 3tan^2\alpha}$$

# 和と積の変換公式

$$\sin\alpha\cos\beta = \frac{1}{2}\{\sin(\alpha + \beta) + \sin(\alpha - \beta)\}$$ $$\cos\alpha\sin\beta = \frac{1}{2}\{\sin(\alpha + \beta) - \sin(\alpha - \beta)\}$$ $$\cos\alpha\cos\beta = \frac{1}{2}\{\cos(\alpha + \beta) + \cos(\alpha - \beta)\}$$ $$\sin\alpha\sin\beta = -\frac{1}{2}\{\cos(\alpha + \beta) - \cos(\alpha - \beta)\}$$

$$\sin A + \sin B = 2\sin\frac{A + B}{2}\cos\frac{A - B}{2}$$ $$\sin A - \sin B = 2\cos\frac{A + B}{2}\sin\frac{A - B}{2}$$ $$\cos A + \cos B = 2\cos\frac{A + B}{2}\cos\frac{A - B}{2}$$ $$\cos A - \cos B = -2\sin\frac{A + B}{2}\sin\frac{A - B}{2}$$

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

ただし

$$\cos \alpha = \frac{a}{\sqrt{a^2 + b^2}}$$ $$\sin \alpha = \frac{b}{\sqrt{a^2 + b^2}}$$

# 指数関数

$$a \neq 0, b \neq 0\text{で}, m, n \ \text{が整数のとき}$$ $$a^0 = 1 \quad \quad \quad \quad \quad \quad a^{-1} = \frac{1}{a^n}$$ $$a^ma^n = a^{m+n} \quad \quad \quad a^m \div a^n = a^{m - n}$$ $$(a^m)^n = a^{mn}$$ $$(ab)^n = a^nb^n \quad \quad \quad (\frac{a}{b})^n = \frac{a^n}{b^n}$$

$$a \gt 0, b \lt 0 \ \text{で}, m, n, p \ \text{が正の整数のとき}$$ $$\sqrt[n]{a} \sqrt[n]{b} = \sqrt[n]{ab} \quad \quad \frac{\sqrt[n]{a}}{\sqrt[n]{b}} = \sqrt[n]{\frac{a}{b}}$$ $$(\sqrt[n]{a})^m = \sqrt[n]{a^m} \quad \quad \sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}$$ $$\sqrt[np]{a^{mp}} = \sqrt[n]{a^m}$$

$$a \gt 0\ \text{で}, m \ \text{が整数,} \ n \ \text{が正の整数のとき}$$ $$a^{\frac{m}{n}} = \sqrt[n]{a^m}$$

$$a > 0, \ b > 0 \ \text{で,}\ p, q \ \text{が有理数のとき}$$ $$a^pa^q = a^{p+q} \quad \quad a^p \div a^q = a^{p-q}$$ $$(a^p)^q = a^{pq}$$ $$(ab)^p = a^pb^p \quad \quad (\frac{a}{b})^p = \frac{a^p}{b^p}$$

# 対数関数

$$a > 0, a \ne 1, M > 0 \ \text{のとき}$$ $$\log_a M = p \iff a^p = M$$

$$a > 0, a \ne 1, M > 0, N > 0 \ \text{のとき}$$ $$\log_a MN = \log_a M + \log_a N$$ $$\log_a \frac{M}{N} = \log_a M - \log_a N$$ $$\log_a M^r = r\log_a M$$

$$\log_a \frac{1}{N} = - \log_a N$$ $$\log_a \sqrt[n]{M} = \frac{1}{M} \log_a M$$

$$a, b, c \ \text{が正の数で,} \ a \ne 1, c \ne 1 \ \text{のとき}$$ $$\log_a b = \frac{log_c b}{log_c a}$$

$$\text{定義域は正の実数全体, 値域は実数全体である。}$$ $$\text{グラフは点(1, 0)を通り,} \ y \text{軸がグラフの漸近線になる。}$$ $$a > 1 \ \text{のとき,} \ y = \log_a X \ \text{は増加関数である。}$$ $$\text{すなわち} \quad 0 < p < q \iff \log_a p < \log_a q$$ $$0 < a < 1 \ \text{のとき,} \ y = \log_a X \ \text{は減少関数である。}$$ $$\text{すなわち} \quad 0 < p < q \iff \log_a p > \log_a q$$

# 微分と積分

$$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(a)}{h}$$ $$f'(x) = \lim_{\varDelta x \to 0} \frac{\varDelta y}{\varDelta x} = \lim_{\varDelta x \to 0} \frac{f(x + \varDelta x) - f(x)}{\varDelta x}$$

x^nの導関数

$$(x^n)' = nx^{n-1}$$

$$c\text{が定数で,} \quad y = c \ \text{ならば} \quad y' = 0$$ $$k\text{が定数で,} \quad y = kf(x) \ \text{ならば} \quad y' = kf'(x)$$ $$y = f(x) + g(x) \ \text{ならば} \quad y' = f'(x) + g'(x)$$ $$y = f(x) - g(x) \ \text{ならば} \quad y' = f'(x) - g'(x)$$

x^nの不定積分

$$\int x^ndx = \frac{1}{n+1}x^{n+1} + C$$

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

２曲線間の面積