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初中数学
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整式乘法公式
$$( a + b )( a - b ) = a^2 - b^2$$ $$(a \pm b)^2=a^2 \pm 2ab + b^2$$ $$(a \pm b)(a^2 \mp ab + b^2)=a^3 \pm b^3$$
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一元二次方程
一元二次方程 $ ax^2 + bx + c = 0 $的解为
$$ x = {- b \pm \sqrt{ b^2 - 4ac } \over {2a} } $$
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三角形
正弦定理
$$ { a \over \sin{A} } = { b \over \sin{B} } = { c \over \sin{C} } = 2R $$
余弦定理$$ \cos{A} = { b^2 + c^2 - a^2 \over 2bc } $$
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圆
圆的周长
$$ C = 2 \pi R $$
圆的面积$$ S = \pi R^2 $$
扇形面积$$ S = \frac 12 lR $$
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整式乘法公式
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高中数学
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代数
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函数
$$ \log_a(MN) = \log_a(M) + \log_a(N) $$
$$ \log_a(\frac{M}{N}) = \log_a(M) - \log_a(N) $$
$$ \log_a(M^n) = n\log_a(M) $$
$$ \log_b{M} = \frac{\log_a{M}}{\log_a{b}} $$
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数列
等差数列的和
$$ S_n = { n( a_1 + a_n ) \over 2 } $$
等比数列的和$$ S_n = \begin{cases} a_1(1 - q^n) \over { 1 - q} & q \neq 1 \\ na_1 & q = 1 \end{cases} $$
求和公式$$ \sum_{k=1}^n k ={ n(n + 1) \over 2 } $$ $$ \sum_{k=1}^n k^2 = { n(n + 1)(2n + 1) \over 6 } $$ $$ \sum_{k=1}^n k^3 = { \left[ { n(n + 1) \over 2 } \right] ^ 2 } $$
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排列组合与二项式定理
$$ A_n^m = \frac{n!}{(n - m)!} = n(n -1)(n - 2)\cdots(n -m + 1) $$
$$ C_n^m = \frac{A_n^m}{m!} = \frac{n!}{m!(n - m)!} $$
$$ C_{n+1}^m = C_n^m + C_n^{m - 1} $$
$$ C_n^m = C_n^{n - m} $$
$$ (a + b)^n = C_n^0a^n + C_n^1a^{n-1}b + C_n^2a^{n-2}b^2 + \cdots + C_n^nb^n $$
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函数
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三角函数
$$ \sin^2{\alpha} + \cos^2{\alpha} = 1 $$
$$ \sin(\alpha \pm \beta) = \sin{\alpha}\cos{\beta} \pm \cos{\alpha}\sin{\beta} $$
$$ \cos(\alpha \pm \beta) = \cos{\alpha}\cos{\beta} \mp \sin{\alpha}\sin{\beta} $$
$$ \tan(\alpha \pm \beta) = \frac{\tan{\alpha} \pm \tan{\beta}}{1 \mp \tan{\alpha}\tan{\beta}} $$
$$ \sin{2\alpha} = 2\sin{\alpha}\cos{\alpha} $$
$$ \cos{2\alpha} = \cos^2{\alpha} - \sin^2{\alpha} = 2\cos^2{\alpha} - 1 = 1 - 2\sin^2{\alpha} $$
$$ \tan{2\alpha} = \frac{2\tan{\alpha}}{1 - \tan^2{\alpha}} $$
$$ \sin{\frac{\alpha}{2}} = \pm\sqrt{\frac{1 - \cos{\alpha}}{2}} $$
$$ \cos{\frac{\alpha}{2}} = \pm\sqrt{\frac{1 + \cos{\alpha}}{2}} $$
$$ \tan{\frac{\alpha}{2}} = \pm\sqrt{\frac{1 - \cos{\alpha}}{1 + \cos{\alpha}}} = \frac{1 - \cos{\alpha}}{\sin{\alpha}} = \frac{\sin{\alpha}}{1 + \cos{\alpha}}$$
$$ \sin{\alpha} = \frac{2\tan{\frac{\alpha}{2}}}{1 + \tan^2{\frac{\alpha}{2}}} $$
$$ \cos{\alpha} = \frac{1 - \tan^2{\frac{\alpha}{2}}}{1 + \tan^2{\frac{\alpha}{2}}} $$
$$ \tan{\alpha} = \frac{2\tan{\frac{\alpha}{2}}}{1 - \tan^2{\frac{\alpha}{2}}} $$
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代数
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解析几何
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直线方程
$$ y - y_1 = k(x - x_1) $$
$$ y = kx + b $$
$$ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} $$
$$ \frac{x}{a} = \frac{y}{b} = 1 $$
$$ Ax + By + C = 0 $$
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两直线关系
平行
$$ l_1 \parallel l_2 \Leftrightarrow \frac{A_1}{A_2} = \frac{B_1}{B_2} \neq \frac{C_1}{C_2} $$
$$ l_1 \parallel l_2 \Leftrightarrow k_1 = k_2 且 b_1 \neq b_2 $$
重合$$ l_1 与 l_2 重合 \Leftrightarrow \frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2} $$
$$ l_1 与 l_2 重合 \Leftrightarrow k_1 = k_2 且 b_1 = b_2 $$
相交$$ l_1 与 l_2 相交 \Leftrightarrow \frac{A_1}{A_2} \neq \frac{B_1}{B_2} $$
$$ l_1 与 l_2 相交 \Leftrightarrow k_1 \neq k_2 $$
垂直$$ l_1 \perp l_2 \Leftrightarrow A_1A_2 + B_1B_2 = 0 $$
$$ l_1 \perp l_2 \Leftrightarrow k_1k_2 = -1 $$
$$ l_1到l_2的角$$$$ \tan{\theta} = \frac{k_2 - k_1}{1 + k_1k_2}(1 + k_1k_2 \neq 0) $$
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点到直线的距离
$$ d = \frac{\vert Ax_0 + By_0 + C \vert}{\sqrt{A^2 + B^2}} $$
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圆锥曲线
$$圆 (x - a)^2 + (y - b)^2 = R^2 $$
$$椭圆 \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $$
$$双曲线 \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 $$
$$抛物线 y^2 = 2px(p \gt 0) $$
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直线方程
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立体几何
$$ S_{圆柱侧} = 2 \pi rh $$
$$ S_{圆锥侧} = \pi rl $$
$$ S_{球} = 4 \pi r^2 $$
$$ V_{圆柱} = \pi r^2h $$
$$ V_{圆锥} = \frac{1}{3} \pi r^2h $$
$$ V_{球} = \frac{4}{3} \pi r^3 $$
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导数
$$ c' = 0 $$
$$ (x^n)' = nx^{n - 1} $$
$$ (\ln{x})' = \frac{1}{x} $$
$$ (\log_a{x})' = \frac{1}{x} \log_a{e} $$
$$ (e^x)' = e^x $$
$$ (a^x)' = a^x \ln{a} $$
$$ (\sin{x})' = \cos{x} $$
$$ (\cos{x})' = - \sin{x} $$
$$ (u \pm v)' = u' \pm v' $$
$$ (uv)' = u'v + uv' $$
$$ (\frac{u}{v})' = \frac{u'v - uv'}{v^2} $$
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积分
$$ \int{0}dx = c $$
$$ \int{x^m}dx = \frac{1}{m + 1}x^{m+1} + c $$
$$ \int{\frac{1}{x}}dx = \ln{\vert x \vert} + c $$
$$ \int{e^x}dx = e^x + c $$
$$ \int{a^x}dx = \frac{a^x}{\ln{a}} + c $$
$$ \int{\cos{x}}dx = \sin{x} + c $$
$$ \int{\sin{x}}dx = - \cos{x} + c $$