📐 公式表

174/174 个公式

🔢基础算术(6)

整除规则高-3
\text{2: chẵn},\; \text{3: tổng CS}\vdots 3,\; \text{5: tận 0,5},\; \text{9: tổng CS}\vdots 9
GCD & LCM高-3
a \cdot b = \text{ƯCLN}(a,b) \cdot \text{BCNN}(a,b)
分数加法高-3
\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}
分数乘法高-3
\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}
分数除法高-3
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}
百分比高-3
p\% = \frac{p}{100}

📏六年级几何(5)

长方形面积高-3
S = a \times b
长方形周长高-3
P = 2(a + b)
正方形面积高-3
S = a^2
长方体体积高-3
V = a \times b \times c
正方体体积高-3
V = a^3

⚖️比例与比例式(4)

比例式高-2
\frac{a}{b} = \frac{c}{d} \Leftrightarrow ad = bc
等比数列高-2
\frac{a}{b} = \frac{c}{d} = \frac{a+c}{b+d} = \frac{a-c}{b-d}
正比例高-2
y = kx\;(k \ne 0)
反比例高-2
xy = k\;(k \ne 0)

△七年级几何(4)

三角形内角和高-2
\hat{A} + \hat{B} + \hat{C} = 180°
三角不等式高-2
|a - b| < c < a + b
勾股定理高-2
直角三角形，c为斜边
a^2 + b^2 = c^2
三角形全等高-2
+斜边-直角边
\text{c.c.c},\; \text{c.g.c},\; \text{g.c.g}

📝八年级代数(3)

因式分解高-1
ax^2 + bx + c = a(x - x_1)(x - x_2)
代数分式高-1
\frac{A}{B} + \frac{C}{D} = \frac{AD + BC}{BD}
一元一次方程高-1
ax + b = 0 \Rightarrow x = -\frac{b}{a}\;(a \ne 0)

▱八年级几何(4)

平行四边形面积高-1
S = a \times h
梯形面积高-1
S = \frac{1}{2}(a + b) \times h
菱形面积高-1
d₁, d₂: 对角线
S = \frac{1}{2}d_1 \times d_2
相似三角形高-1
面积比 = k², 体积比 = k³
\frac{a'}{a} = \frac{b'}{b} = \frac{c'}{c} = k

√九年级代数(6)

平方根高0
\sqrt{a^2} = |a|
积的根高0
\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}\;(a,b \ge 0)
商的根高0
\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\;(a \ge 0, b > 0)
二元一次方程组高0
\begin{cases} ax+by=e \ cx+dy=f \end{cases}
一次函数高0
a>0: 递增, a<0: 递减
y = ax + b\;(a \ne 0)
二次方程（介绍）高0
ax^2 + bx + c = 0\;(a \ne 0)

📐九年级几何(4)

三角比高0
\sin A = \frac{\text{đối}}{\text{huyền}},\; \cos A = \frac{\text{kề}}{\text{huyền}},\; \tan A = \frac{\text{đối}}{\text{kề}}
弧长高0
n = 弧度(°)
l = \frac{\pi R n}{180}
扇形面积高0
S = \frac{\pi R^2 n}{360}
圆周角高0
\hat{\text{nội tiếp}} = \frac{1}{2}\hat{\text{cung bị chắn}}

🔢重要恒等式(7)

和的平方高1
(a+b)^2 = a^2 + 2ab + b^2
差的平方高1
(a-b)^2 = a^2 - 2ab + b^2
平方差高1
a^2 - b^2 = (a-b)(a+b)
和的立方高1
(a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
差的立方高1
(a-b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
立方和高1
a^3 + b^3 = (a+b)(a^2 - ab + b^2)
立方差高1
a^3 - b^3 = (a-b)(a^2 + ab + b^2)

📐二次方程(6)

判别式高1
\Delta = b^2 - 4ac
求根公式高1
x = \frac{-b \pm \sqrt{\Delta}}{2a}
判别式Δ'高1
当b为偶数时
\Delta' = b'^2 - ac \;(b=2b')
韦达：根的和高1
x_1 + x_2 = -\frac{b}{a}
韦达：根的积高1
x_1 \cdot x_2 = \frac{c}{a}
根的平方和高1
S = x₁+x₂, P = x₁·x₂
x_1^2 + x_2^2 = S^2 - 2P

⚖️不等式(3)

二次三项式的符号高1
f(x)=ax^2+bx+c,\; a>0:\; f(x)\ge 0 \;\forall x \Leftrightarrow \Delta \le 0
AM-GM不等式高1
\frac{a+b}{2} \ge \sqrt{ab} \;(a,b \ge 0)
绝对值不等式高1
|a+b| \le |a| + |b|

🔗方程组(2)

克莱姆法则高1
x = \frac{D_x}{D},\; y = \frac{D_y}{D},\; D \ne 0
消元法高1
\begin{cases} a_1x+b_1y=c_1 \ a_2x+b_2y=c_2 \end{cases}

➡️向量与坐标（平面）(4)

向量加法高1
\vec{a}+\vec{b} = (a_1+b_1,\; a_2+b_2)
点积高1
\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 = |\vec{a}||\vec{b}|\cos\theta
两点距离高1
AB = \sqrt{(x_B-x_A)^2 + (y_B-y_A)^2}
中点高1
M = \left(\frac{x_A+x_B}{2},\; \frac{y_A+y_B}{2}\right)

📏三角函数(16)

基本恒等式高2
\sin^2 x + \cos^2 x = 1
tan-cot关系高2
\tan x = \frac{\sin x}{\cos x},\; \cot x = \frac{\cos x}{\sin x}
1 + tan²x高2
1 + \tan^2 x = \frac{1}{\cos^2 x}
正弦和差角高2
\sin(a \pm b) = \sin a \cos b \pm \cos a \sin b
余弦和差角高2
\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b
正切和差角高2
\tan(a \pm b) = \frac{\tan a \pm \tan b}{1 \mp \tan a \tan b}
二倍角sin高2
\sin 2x = 2\sin x \cos x
二倍角cos高2
\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x
二倍角tan高2
\tan 2x = \frac{2\tan x}{1 - \tan^2 x}
降幂cos²高2
\cos^2 x = \frac{1 + \cos 2x}{2}
降幂sin²高2
\sin^2 x = \frac{1 - \cos 2x}{2}
积化和差高2
2\sin a \cos b = \sin(a+b) + \sin(a-b)
和差化积高2
\sin a + \sin b = 2\sin\frac{a+b}{2}\cos\frac{a-b}{2}
三角方程sin高2
\sin x = \sin \alpha \Leftrightarrow x = \alpha + k2\pi \;\text{hoặc}\; x = \pi - \alpha + k2\pi
三角方程cos高2
\cos x = \cos \alpha \Leftrightarrow x = \pm \alpha + k2\pi
三角方程tan高2
\tan x = \tan \alpha \Leftrightarrow x = \alpha + k\pi

🎲组合与概率(7)

排列高2
A_n^k = \frac{n!}{(n-k)!}
组合高2
C_n^k = \frac{n!}{k!(n-k)!}
二项式定理高2
(a+b)^n = \sum_{k=0}^{n} C_n^k a^{n-k} b^k
古典概率高2
P(A) = \frac{|A|}{|\Omega|}
并事件概率高2
P(A \cup B) = P(A) + P(B) - P(A \cap B)
对立事件高2
P(\bar{A}) = 1 - P(A)
伯努利试验高2
P(X=k) = C_n^k \cdot p^k \cdot (1-p)^{n-k}

📊数列(5)

等差数列通项高2
u_n = u_1 + (n-1)d
等差数列求和高2
S_n = \frac{n(u_1 + u_n)}{2} = \frac{n[2u_1 + (n-1)d]}{2}
等比数列通项高2
u_n = u_1 \cdot q^{n-1}
等比数列求和高2
S_n = u_1 \cdot \frac{1-q^n}{1-q}\;(q \ne 1)
无穷等比级数高2
S_\infty = \frac{u_1}{1-q}\;(|q|<1)

♾️极限(4)

sinx/x极限高2
\lim_{x \to 0} \frac{\sin x}{x} = 1
欧拉数高2
\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e
(eˣ-1)/x极限高2
\lim_{x \to 0} \frac{e^x - 1}{x} = 1
ln(1+x)/x极限高2
\lim_{x \to 0} \frac{\ln(1+x)}{x} = 1

📈导数(12)

幂函数导数高3
(x^n)' = nx^{n-1}
sin导数高3
(\sin x)' = \cos x
cos导数高3
(\cos x)' = -\sin x
tan导数高3
(\tan x)' = \frac{1}{\cos^2 x}
cot导数高3
(\cot x)' = -\frac{1}{\sin^2 x}
eˣ导数高3
(e^x)' = e^x
aˣ导数高3
(a^x)' = a^x \ln a
ln x导数高3
(\ln x)' = \frac{1}{x}
logₐx导数高3
(\log_a x)' = \frac{1}{x \ln a}
乘法法则高3
(uv)' = u'v + uv'
除法法则高3
\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}
链式法则高3
[f(u)]' = f'(u) \cdot u'

📐指数与对数(7)

同底乘法高3
a^m \cdot a^n = a^{m+n}
同底除法高3
a^m \div a^n = a^{m-n}
幂的幂高3
(a^m)^n = a^{mn}
对数乘法高3
\log_a(xy) = \log_a x + \log_a y
对数除法高3
\log_a\frac{x}{y} = \log_a x - \log_a y
对数幂高3
\log_a x^n = n \log_a x
换底公式高3
\log_a b = \frac{\ln b}{\ln a} = \frac{\log_c b}{\log_c a}

∫积分(8)

幂函数积分高3
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \;(n \ne -1)
1/x积分高3
\int \frac{1}{x} \, dx = \ln|x| + C
eˣ积分高3
\int e^x \, dx = e^x + C
sin积分高3
\int \sin x \, dx = -\cos x + C
cos积分高3
\int \cos x \, dx = \sin x + C
曲线下面积高3
S = \int_a^b |f(x)| \, dx
旋转体体积高3
V = \pi \int_a^b [f(x)]^2 \, dx
分部积分高3
\int u \, dv = uv - \int v \, du

🔮复数(3)

代数形式高3
z = a + bi \;(a,b \in \mathbb{R},\; i^2 = -1)
模高3
|z| = \sqrt{a^2 + b^2}
共轭高3
\bar{z} = a - bi,\; z \cdot \bar{z} = |z|^2

△平面几何(8)

勾股定理高1
a^2 + b^2 = c^2
三角形面积高1
S = \frac{1}{2}ah
三角形面积(sin)高1
S = \frac{1}{2}ab\sin C
海伦公式高1
p = (a+b+c)/2
S = \sqrt{p(p-a)(p-b)(p-c)}
正弦定理高1
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R
余弦定理高1
a^2 = b^2 + c^2 - 2bc\cos A
圆面积高1
S = \pi r^2
圆周长高1
C = 2\pi r

🧊立体几何(8)

棱柱体积高3
V = S_{đáy} \cdot h
棱锥体积高3
V = \frac{1}{3}S_{đáy} \cdot h
圆柱体积高3
V = \pi r^2 h
圆柱侧面积高3
S_{xq} = 2\pi r h
圆锥体积高3
V = \frac{1}{3}\pi r^2 h
圆锥侧面积高3
l = 母线
S_{xq} = \pi r l
球体积高3
V = \frac{4}{3}\pi r^3
球表面积高3
S = 4\pi r^2

📍空间坐标几何(9)

两点距离(3D)高3
AB = \sqrt{(x_B-x_A)^2 + (y_B-y_A)^2 + (z_B-z_A)^2}
中点(3D)高3
M = \left(\frac{x_A+x_B}{2}, \frac{y_A+y_B}{2}, \frac{z_A+z_B}{2}\right)
平面方程高3
法向量: n⃗(a,b,c)
ax + by + cz + d = 0
点到平面距离高3
d(M, \alpha) = \frac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2+b^2+c^2}}
直线方程(3D)高3
方向向量: u⃗(a,b,c)
\frac{x-x_0}{a} = \frac{y-y_0}{b} = \frac{z-z_0}{c}
球面方程高3
(x-a)^2 + (y-b)^2 + (z-c)^2 = R^2
叉积高3
\vec{a} \times \vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \ a_1 & a_2 & a_3 \ b_1 & b_2 & b_3 \end{vmatrix}
两平面夹角高3
\cos(\alpha, \beta) = \frac{|a_1 a_2 + b_1 b_2 + c_1 c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\sqrt{a_2^2+b_2^2+c_2^2}}
异面直线距离高3
d = \frac{|[\vec{u_1}, \vec{u_2}] \cdot \vec{M_1 M_2}|}{|\vec{u_1} \times \vec{u_2}|}

📏直线(Oxy)(7)

一般式高1
ax + by + c = 0
斜截式高1
k = tan α
y = kx + m
两点式高1
\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1}
点到直线距离高1
d(M_0, \Delta) = \frac{|ax_0 + by_0 + c|}{\sqrt{a^2 + b^2}}
两直线夹角高1
\tan \alpha = \left|\frac{k_1 - k_2}{1 + k_1 k_2}\right|
平行条件高1
d_1 \parallel d_2 \Leftrightarrow \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}
垂直条件高1
d_1 \perp d_2 \Leftrightarrow a_1 a_2 + b_1 b_2 = 0

⭕圆与椭圆(4)

圆方程高1
(x-a)^2 + (y-b)^2 = R^2
圆的一般方程高1
圆心I(a,b), R² = a²+b²-c
x^2 + y^2 - 2ax - 2by + c = 0
椭圆标准方程高1
c² = a² - b², e = c/a
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\;(a > b > 0)
圆的切线高1
在(C)上M(x₀,y₀)处
(x_0 - a)(x - a) + (y_0 - b)(y - b) = R^2

📊统计学(10)

算术平均高1
\bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n}
加权平均高1
\bar{x} = \frac{\sum n_i x_i}{\sum n_i}
方差高1
S^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2
标准差高1
S = \sqrt{S^2} = \sqrt{\frac{1}{n}\sum(x_i - \bar{x})^2}
中位数高1
n偶: Mₑ = (x_{n/2} + x_{n/2+1})/2
M_e = x_{(n+1)/2}\;\text{(n lẻ)}
众数高1
M_o = \text{giá trị xuất hiện nhiều nhất}
极差高1
R = x_{max} - x_{min}
四分位数高1
IQR = Q₃ - Q₁
Q_1,\; Q_2 (= M_e),\; Q_3
相关系数高1
r = \frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum(x_i-\bar{x})^2 \cdot \sum(y_i-\bar{y})^2}}
线性回归高1
a = r·Sy/Sx, b = ȳ - aẍ
\hat{y} = ax + b

📉函数与分析(8)

单调递增条件高3
f'(x) > 0 \;\forall x \in (a,b) \Rightarrow f \nearrow
极值（导数变号）高3
f'(x_0) = 0,\; f'\text{ đổi dấu qua } x_0
拐点高3
f''(x_0) = 0,\; f''\text{ đổi dấu}
水平渐近线高3
\lim_{x \to \pm\infty} f(x) = L \Rightarrow y = L
垂直渐近线高3
\lim_{x \to a} f(x) = \pm\infty \Rightarrow x = a
三次函数高3
总有1个拐点
y = ax^3 + bx^2 + cx + d\;(a \ne 0)
分式函数高3
VA: x=-d/c, HA: y=a/c
y = \frac{ax+b}{cx+d}\;(c \ne 0, ad-bc \ne 0)
切线方程高3
y - y_0 = f'(x_0)(x - x_0)