Formulae for Edexcel A-Level Mathematics | Edexcel A-Level 数学核心公式

📚 Formulae for Edexcel A-Level Mathematics | Edexcel A-Level 数学核心公式

Formulae are the backbone of A-Level Mathematics, enabling you to solve problems efficiently across pure, statistics, and mechanics. This article compiles the essential formulae you need to memorise and apply for the Edexcel specification.

公式是 A-Level 数学的支柱，可帮助你高效解决纯数学、统计和力学中的问题。本文整理了 Edexcel 考试大纲中必须熟记和运用的核心公式。

1. Quadratic Formula | 二次公式

ic equation is ax² + bx + c = 0 (a ≠ 0). Its solutions are given by the quadratic formula:

标准二次方程为 ax² + bx + c = 0（a ≠ 0），其解由二次公式给出：

x = (−b ± √b² − 4ac) / (2a)

nct real roots exist; if Δ = 0, one repeated real root; if Δ < 0, no real roots (complex).

判别式 Δ = b² − 4ac 决定了根的性质。若 Δ > 0，有两个相异实根；若 Δ = 0，有一个重实根；若 Δ < 0，无实根（复数根）。

elds the vertex of the parabola: (−b/(2a), −Δ/(4a)), and the axis of symmetry x = −b/(2a).

配方法也可得出抛物线的顶点：(−b/(2a), −Δ/(4a))，以及对称轴 x = −b/(2a)。

2. Laws of Indices and Surds | 指数与根式运算法则

× aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿ bⁿ, a⁰ = 1 (a ≠ 0), a⁻ⁿ = 1/aⁿ, and a^1/n = ⁿ√a.

对于实数 a, b 和有理指数 m, n 的关键指数法则：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，(ab)ⁿ = aⁿ bⁿ，a⁰ = 1（a ≠ 0），a⁻ⁿ = 1/aⁿ，a^1/n = ⁿ√a。

Surds follow: √ab = √a × √b, √a/b = √a / √b. Rationalising denominators, e.g., 1/√a = √a/a, is a common technique.

根式规则：√ab = √a × √b，√a/b = √a / √b。分母有理化（例如 1/√a = √a/a）是常用技巧。

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3. Logarithmic Rules | 对数运算法则

(x/y) = logₐx − logₐy, logₐ(xⁿ) = n logₐx. The change-of-base rule: logₐb = logₓb / logₓa.

对于底数 a（a > 0, a ≠ 1）：logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx − logₐy，logₐ(xⁿ) = n logₐx。换底公式：logₐb = logₓb / logₓa。

The natural logarithm, ln, uses base e: ln(eˣ) = x and e^{ln x} = x. Also logₐ1 = 0 and logₐa = 1.

自然对数 ln 的底为 e：ln(eˣ) = x，e^{ln x} = x。同时 logₐ1 = 0，logₐa = 1。

4. Trigonometric Identities | 三角恒等式

hagorean identity: sin²θ + cos²θ = 1. Derived forms: 1 + tan²θ = sec²θ, 1 + cot²θ = csc²θ.

勾股恒等式：sin²θ + cos²θ = 1。衍生形式：1 + tan²θ = sec²θ，1 + cot²θ = csc²θ。

Addition formulae: sin(A±B) = sinA cosB ± cosA sinB; cos(A±B) = cosA cosB ∓ sinA sinB; tan(A±B) = (tanA ± tanB) / (1 ∓ tanA tanB).

和角公式：sin(A±B) = sinA cosB ± cosA sinB；cos(A±B) = cosA cosB ∓ sinA sinB；tan(A±B) = (tanA ± tanB) / (1 ∓ tanA tanB)。

inθ cosθ; cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ; tan2θ = (2 tanθ) / (1 − tan²θ).

倍角公式：sin2θ =

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