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30 Verified Equations

CSCA Mathematics Formula Catalog

CSCA 数学核心公式与定理库

Complete reference for all 24 CSCA Math formulas: quadratic inequalities, logarithms, trigonometry, progressions, differential calculus, and analytic conics.

Sets & Inequalities

6 Formulas

Quadratic Formula

一元二次方程求根公式

Solves quadratic equations of the form ax² + bx + c = 0. Here, 'x' represents the roots, 'a', 'b', and 'c' are coefficients, and the term under the square root (b² - 4ac) is the discriminant.

$\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]$
Proof

Union of Sets

集合的并集

The union of two sets A and B contains all elements that are in A, in B, or in both. Denoted by 'A ∪ B'.

$\[ A \cup B = \{ x \mid x \in A \text{ or } x \in B \} \]$
Proof

Intersection of Sets

集合的交集

The intersection of two sets A and B contains all elements that belong to both A and B. Denoted by 'A ∩ B'.

$\[ A \cap B = \{ x \mid x \in A \text{ and } x \in B \} \]$
Proof

Complement of a Set

集合的补集

The complement of a subset A with respect to a universal set U contains all elements in U that do not belong to A. Denoted by ∁_U A.

$\[ \complement_{U} A = \{ x \in U \mid x \notin A \} \]$
Proof

Rational Inequality Equivalence

分式不等式的等价转化

Solves fractional inequalities by converting the division into multiplication. It maintains sign properties while ensuring the denominator 'cx + d' is non-zero.

$\[ \frac{ax + b}{cx + d} \le 0 \iff (ax + b)(cx + d) \le 0 \quad \text{and} \quad cx + d \ne 0 \]$
Proof

AM-GM Inequality

基本均值不等式

The Arithmetic Mean-Geometric Mean inequality states that for any non-negative real numbers 'a' and 'b', their arithmetic mean is greater than or equal to their geometric mean. Equality holds if and only if a = b.

$\[ \frac{a + b}{2} \ge \sqrt{ab} \quad (a, b \ge 0) \]$
Proof

Functions

6 Formulas

Change of Base Formula

对数的换底公式

Allows logarithm evaluation in any desired base 'c' (usually base e or base 10) by expressing it as the ratio of two logarithms in that base.

$\[ \log_a b = \frac{\log_c b}{\log_c a} \quad (a, b, c > 0, \, a, c \ne 1) \]$
Proof

Logarithmic Properties

对数的基本运算性质

Core operational laws of logarithmic functions: the log of a product is the sum of the logs, and the log of a quotient is the difference of the logs.

$\[ \log_a (xy) = \log_a x + \log_a y \quad \text{and} \quad \log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y \]$
Proof

Trigonometric Pythagorean Identity

同角三角关系（平方关系）

The fundamental relation expressing the Pythagorean theorem in terms of trigonometric functions. Useful for finding sine if cosine is known, or vice versa.

$\[ \sin^2 \theta + \cos^2 \theta = 1 \]$
Proof

Double-Angle Formula (Sine)

正弦的二倍角公式

Expresses the sine of double an angle in terms of the sine and cosine of the single angle.

$\[ \sin 2\theta = 2 \sin \theta \cos \theta \]$
Proof

Double-Angle Formula (Cosine)

余弦的二倍角公式

Expresses the cosine of double an angle in terms of the single angle's cosine and sine.

$\[ \cos 2\theta = \cos^2 \theta - \sin^2 \theta \]$
Proof

Function Parity (Odd & Even)

函数的奇偶性

An even function is symmetric about the y-axis, satisfying f(-x) = f(x). An odd function is symmetric about the origin, satisfying f(-x) = -f(x). Both require a domain symmetric about x = 0.

$\[ \text{Even: } f(-x) = f(x), \quad \text{Odd: } f(-x) = -f(x) \]$
Proof

Sequences

4 Formulas

Arithmetic Sequence Nth Term

等差数列通项公式

Calculates the general term of an arithmetic sequence. Here, 'a₁' is the first term, 'd' is the common difference, and 'n' is the index.

$\[ a_n = a_1 + (n - 1)d \]$
Proof

Arithmetic Sequence Sum

等差数列前n项和公式

Finds the sum of the first n terms of an arithmetic sequence. Represents the sum of terms multiplied by their average.

$\[ S_n = \frac{n(a_1 + a_n)}{2} \]$
Proof

Geometric Sequence Nth Term

等比数列通项公式

Calculates the general term of a geometric sequence. Here, 'a₁' is the first term, 'q' is the common ratio, and 'n' is the term index.

$\[ a_n = a_1 q^{n-1} \]$
Proof

Geometric Sequence Sum

等比数列前n项和公式

Computes the sum of the first n terms of a geometric sequence when the common ratio 'q' is not equal to 1. If q = 1, Sn = n * a₁.

$\[ S_n = \frac{a_1(1 - q^n)}{1 - q} \quad (q \ne 1) \]$
Proof

Derivatives

2 Formulas

Power Rule for Derivatives

导数的幂函数求导法则

The fundamental derivative rule for power functions. It states that the derivative of x^n with respect to x is n multiplied by x^(n-1).

$\[ \frac{d}{dx} x^n = n x^{n-1} \]$
Proof

Tangent Line Equation

曲线切线方程公式

Defines the equation of the line tangent to the curve y = f(x) at point (x₀, y₀). The derivative value f'(x₀) represents the slope of the tangent line.

$\[ y - y_0 = f'(x_0)(x - x_0) \]$
Proof

Plane Analytic Geometry

6 Formulas

Distance Formula

两点间距离公式

Calculates the Euclidean distance 'd' between two coordinates (x₁, y₁) and (x₂, y₂) in a two-dimensional Cartesian coordinate system.

$\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]$
Proof

Equation of a Circle

圆的标准方程

The standard equation of a circle centered at (a, b) with a radius of 'R'. Here, 'R' represents the radius and (a, b) is the center.

$\[ (x - a)^2 + (y - b)^2 = R^2 \]$
Proof

Equation of an Ellipse

椭圆的标准方程

The standard equation of an ellipse centered at the origin with foci on the x-axis. Satisfies the focal relation c² = a² - b², where 'c' is the focal coordinate and 'e = c/a' is the eccentricity.

$\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a > b > 0) \]$
Proof

Equation of a Hyperbola

双曲线的标准方程

The standard equation of a hyperbola centered at the origin with foci on the x-axis. Satisfies the relation c² = a² + b² and has eccentricity e = c/a (where e > 1).

$\[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad (a > 0, \, b > 0) \]$
Proof

Equation of a Parabola

抛物线的标准方程

The standard equation of a parabola opening to the right. It has a focus at (p/2, 0) and a directrix equation x = -p/2.

$\[ y^2 = 2px \quad (p > 0) \]$
Proof

Equation of a Line (Slope-Intercept & General Form)

直线的方程（斜截式与一般式）

Slope-intercept form (y = kx + b) uses slope 'k' and y-intercept 'b'. General form (Ax + By + C = 0) can represent any straight line in a 2D plane.

$\[ y = kx + b \quad \text{and} \quad Ax + By + C = 0 \]$
Proof

Vectors & Complex Numbers

2 Formulas

Vector Dot Product

平面向量的数量积（点积）

Calculates the algebraic dot product of two 2D vectors. It relates the geometric lengths of vectors and the cosine of the angle between them.

$\[ \mathbf{a} \cdot \mathbf{b} = a_x b_x + a_y b_y = |\mathbf{a}| |\mathbf{b}| \cos \theta \]$
Proof

Modulus of a Complex Number

复数的模

Calculates the modulus or absolute value of a complex number z = a + bi, representing its geometric distance from the origin in the complex plane.

$\[ |z| = \sqrt{a^2 + b^2} \quad (z = a + bi) \]$
Proof

Solid Geometry

1 Formula

Volume and Surface Area of a Sphere

球的体积与表面积公式

Volume 'V' and Surface Area 'S' of a sphere with radius 'R' in 3D space.

$\[ V = \frac{4}{3}\pi R^3 \quad \text{and} \quad S = 4\pi R^2 \]$
Proof

Probability & Statistics

3 Formulas

Classical Probability Formula

古典概型概率计算公式

Calculates the probability of an event 'A' in a classical probability model. 'N(A)' is the number of outcomes favorable to event 'A', and 'N' is the total number of equally likely outcomes in the sample space.

$\[ P(A) = \frac{N(A)}{N} \]$
Proof

Variance and Standard Deviation

样本方差与标准差公式

Variance (s²) measures the dispersion of a data set. It is the average of squared differences from the mean. Standard deviation (s) is the square root of variance.

$\[ s^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]$
Proof

Normal Distribution (3-Sigma Rule)

正态分布（3σ原则 / 68-95-99.7准则）

The 3-Sigma (68-95-99.7) rule for a normal distribution with mean 'μ' and standard deviation 'σ'. It describes the percentage of data falling within one, two, and three standard deviations from the mean.

$\[ P(\mu - \sigma \le X \le \mu + \sigma) \approx 0.6826 \quad \text{and} \quad P(\mu - 2\sigma \le X \le \mu + 2\sigma) \approx 0.9544 \]$
Proof