📚 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。
inomial 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)。
c. 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。
nθ 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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