Formula Summary Handbook | 公式汇总手册

📚 Formula Summary Handbook | 公式汇总手册

This handbook collects essential formulas required for the Edexcel A-Level Mathematics specification, covering Pure Mathematics, Mechanics, and Statistics. Use it as a quick reference for revision and exam preparation.

本手册汇集了Edexcel A-Level数学课程所需的必备公式，涵盖纯数、力学和统计，可作为复习备考的快速参考。

1. Algebraic Identities and Laws | 代数恒等式与运算法则

Laws of indices: aᵐ × aⁿ = aᵐ⁺ⁿ, (aᵐ)ⁿ = aᵐⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.

指数定律：aᵐ × aⁿ = aᵐ⁺ⁿ，(aᵐ)ⁿ = aᵐⁿ，aᵐ / aⁿ = aᵐ⁻ⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ。

Binomial expansion for positive integer n: (a+b)ⁿ = Σ (k=0 to n) ⁿCₖ aⁿ⁻ᵏ bᵏ, where ⁿCₖ = n!/(k!(n-k)!).

正整数n的二项式展开：(a+b)ⁿ = Σ (k=0 到 n) ⁿCₖ aⁿ⁻ᵏ bᵏ，其中 ⁿCₖ = n!/(k!(n-k)!)。

Special case: (1+x)ⁿ ≈ 1 + nx + n(n-1)/2! x² + … for |x| < 1 (valid for any real n).

特例：(1+x)ⁿ ≈ 1 + nx + n(n-1)/2! x² + …（对任意实数n，|x|<1 时成立）。

2. Quadratic Functions and Discriminant | 二次函数与判别式

Quadratic formula: roots of ax² + bx + c = 0 are x = (-b ± √(b² – 4ac)) / (2a).

二次公式：方程 ax² + bx + c = 0 的根为 x = (-b ± √(b² – 4ac)) / (2a)。

Discriminant Δ = b² – 4ac. If Δ > 0: two distinct real roots; Δ = 0: one repeated real root; Δ < 0: no real roots.

判别式 Δ = b² – 4ac。若 Δ > 0：两个不等实根；Δ = 0：一个重根；Δ < 0：无实根。

Axis of symmetry: x = -b/(2a). Vertex coordinates: (-b/(2a), (4ac – b²)/(4a)).

对称轴：x = -b/(2a)；顶点坐标：(-b/(2a), (4ac – b²)/(4a))。

Sum and product of roots: sum = -b/a, product = c/a.

根的和与积：和 = -b/a，积 = c/a。

3. Trigonometry and Identities | 三角学与恒等式

Pythagorean identity: sin²θ + cos²θ = 1. Also tanθ = sinθ / cosθ.

毕达哥拉斯恒等式：sin²θ + cos²θ = 1。以及 tanθ = sinθ / cosθ。

Double-angle formulas: sin(2θ) = 2 sinθ cosθ; cos(2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ; tan(2θ) = 2tanθ / (1 – tan²θ).

二倍角公式：sin(2θ) = 2 sinθ cosθ；cos(2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ；tan(2θ) = 2tanθ / (1 – tan²θ)。

Sine rule: a / sin A = b / sin B = c / sin C. Cosine rule: a² = b² + c² – 2bc cos A.

正弦定理：a / sin A = b / sin B = c / sin C。余弦定理：a² = b² + c² – 2bc cos A。

Area of triangle: ½ ab sin C.

三角形面积：½ ab sin C。

4. Exponentials and Logarithms | 指数与对数

Logarithm laws: logₐ(xy) = logₐx + logₐy; logₐ(x/y) = logₐx – logₐy; logₐ(xⁿ) = n logₐx.

对数定律：logₐ(xy) = logₐx + logₐy；logₐ(x/y) = logₐx – logₐy；logₐ(xⁿ) = n logₐx。

Change of base: logₓy = logₐy / logₐx. Special case: ln x = logₑx.

换底公式：logₓy = logₐy / logₐx。特例：自然对数 ln x = logₑx。

Derivative of e

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