📚 Year 11 SQA Advanced Mathematics: Formula & Theorem Quick Reference Handbook | Year 11 SQA 进阶数学：公式定理速查手册
This quick reference handbook compiles the essential formulas and theorems required for the SQA Higher Mathematics course (Year 11). Use it as a rapid revision tool to reinforce your understanding of algebra, trigonometry, calculus, vectors, sequences, and more. Each section presents key results with concise explanations, followed by Chinese translations for bilingual learners.
本速查手册汇编了 SQA 进阶数学（Year 11 / Higher）所需的核心公式与定理。可作为快速复习工具，巩固代数、三角、微积分、向量、数列等领域的知识。每个小节先提供英文要点，再附中文解释，便于双语学习者掌握。
1. Algebraic Operations & Factorisation | 代数运算与因式分解
Expanding brackets: multiply each term inside by the factor outside. For example, 3(x + 2y) = 3x + 6y and (x + 4)(x – 3) = x² + x – 12.
展开括号：将括号外的因式乘以括号内的每一项。例如 3(x + 2y) = 3x + 6y 及 (x+4)(x-3) = x² + x – 12。
Factorising is the reverse process. Always look for a common factor first: 6x² – 9x = 3x(2x – 3). For a quadratic trinomial x² + bx + c, find two numbers with sum b and product c. For x² + 5x + 6, the numbers 2 and 3 give (x+2)(x+3).
因式分解是逆向过程。首先提取公因子：6x² – 9x = 3x(2x – 3)。对于二次三项式 x² + bx + c，寻找两数之和为 b、积为 c。例如 x² + 5x + 6，2 和 3 给出 (x+2)(x+3)。
Difference of two squares: a² – b² = (a – b)(a + b).
平方差公式：a² – b² = (a – b)(a + b)。
Completing the square: rewrite x² + 6x + 2 as (x+3)² – 7. The expression a(x + p)² + q has vertex (-p, q).
完成平方：将 x² + 6x + 2 写成 (x+3)² – 7。形如 a(x + p)² + q 的表达式其顶点为 (-p, q)。
2. Quadratic Functions & the Discriminant | 二次函数与判别式
The quadratic formula for ax² + bx + c = 0 is x = [-b ± √(b² – 4ac)] / (2a). The discriminant Δ = b² – 4ac determines the nature of the roots: if Δ > 0, two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, no real roots.
二次方程 ax² + bx + c = 0 的求根公式为 x = [-b ± √(b² – 4ac)] / (2a)。判别式 Δ = b² – 4ac 决定了根的性质：若 Δ > 0，有两个不等实根；Δ = 0，有一个重实根；Δ < 0，无实根。
The parabola y = a(x – p)² + q has vertex (p, q) and axis of symmetry x = p. The graph crosses the x-axis at the roots, and the y-intercept is (0, c).
抛物线 y = a(x – p)² + q 的顶点为 (p, q)，对称轴为 x = p。图像与 x 轴相交于根处，y 轴截距为 (0, c)。
For a quadratic function, the sum of the roots = -b/a and the product = c/a.
二次函数的两根之和 = -b/a，两根之积 = c/a。
3. Polynomials, Remainder & Factor Theorems | 多项式、余式与因式定理
Remainder theorem: when a polynomial f(x) is divided by (x – h), the remainder is f(h). Factor theorem: (x – h) is a factor of f(x) if and only if f(h) = 0.
余式定理：多项式 f(x) 除以 (x – h) 时，余式为 f(h)。因式定理：(x – h) 是 f(x) 的因式当且仅当 f(h) = 0。
To factorise a cubic, use synthetic division or long division after identifying a factor via the factor theorem. For x³ – 4x² + x + 6, test factors of the constant term (±1, ±2, ±3, ±6) to find a root.
分解三次多项式时，先利用因式定理试验可能的根（常数项因子），再通过综合除法或长除法进行因式分解。例如 x³ – 4x² + x + 6，可试 ±1, ±2, ±3, ±6。
A polynomial of degree n has at most n real roots, and its graph crosses the x-axis at most n times.
n 次多项式最多有 n 个实根，其图像最多与 x 轴相交 n 次。
4. Exponentials & Logarithms | 指数与对数函数
The natural logarithm ln x is the inverse of eˣ: ln(eˣ) = x and e^(ln x) = x. The base-10 log is log₁₀ x; base a log is logₐ x. Changes of base: logₐ b = log_c b / log_c a.
自然对数 ln x 是 eˣ 的反函数：ln(eˣ) = x 且 e^(ln x) = x。常用对数记作 log₁₀ x；一般底数 a 的对数为 logₐ x。换底公式：logₐ b = log_c b / log_c a。
Laws of logs: log(xy) = log x + log y; log(x/y) = log x – log y; log(xⁿ) = n log x.
对数运算法则：log(xy) = log x + log y；log(x/y) = log x – log y；log(xⁿ) = n log x。
Exponential growth/decay model: N(t) = N₀ e^(kt), where N₀ is initial quantity, k the growth (k>0) or decay (k<0) constant.
指数增长/衰减模型：N(t) = N₀ e^(kt)，N₀ 为初始量，k 为增长 (k>0) 或衰减 (k<0) 常数。
5. Trigonometric Identities & Exact Values | 三角恒等式与精确值
Fundamental identities: sin²θ + cos²θ = 1; tanθ = sinθ / cosθ. Derived: sin²θ = 1 – cos²θ; cos²θ = 1 – sin²θ.
基本恒等式：sin²θ + cos²θ = 1；tanθ = sinθ / cosθ。由此得 sin²θ = 1 – cos²θ；cos²θ = 1 – sin²θ。
Double-angle formulas: sin(2θ) = 2 sinθ cosθ; cos(2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ.
倍角公式：sin(2θ) = 2 sinθ cosθ；cos(2θ) = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ。
Know exact values for 0°, 30°, 45°, 60°, 90° in degrees and 0, π/6, π/4, π/3, π/2 in radians. Example: sin 30° = ½, cos 45° = √2/2, tan 60° = √3.
熟记特殊角（0°, 30°, 45°, 60°, 90° 及对应弧度）的精确值。例如 sin 30° = ½，cos 45° = √2/2，tan 60° = √3。
- sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3
- sin 30° = ½, cos 30° = √3/2, tan 30° = 1/√3
- sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1
- sin 45° = √2/2, cos 45° = √2/2, tan 45° = 1
- sin 60° = √3/2, cos 60° = ½, tan 60° = √3
- sin 60° = √3/2, cos 60° = ½, tan 60° = √3
6. Solving Trigonometric Equations | 解三角方程
To solve a trigonometric equation, use identities to rewrite in terms of one function, then find all solutions in the given interval. The CAST diagram helps determine the quadrant(s) where the function is positive or negative.
解三角方程时，先用恒等式化为单一三角函数的方程，再在所给区间内求出所有解。CAST 图有助于确定函数在各象限的正负。
For example, 2 sin²θ – sinθ = 0 can be factored as sinθ(2 sinθ – 1) = 0, giving sinθ = 0 or sinθ = ½. Then use CAST or reference angles to find all solutions in [0, 360°].
例如，2 sin²θ – sinθ = 0 可因式分解为 sinθ(2 sinθ – 1) = 0，得到 sinθ = 0 或 sinθ = ½，再借助 CAST 图或参考角求出 [0°, 360°] 内的所有解。
Remember the periodic nature: sinθ and cosθ have period 360° (2π), tanθ has period 180° (π). Always check for extraneous solutions.
牢记周期性：sinθ 和 cosθ 的周期为 360° (2π)，tanθ 的周期为 180° (π)。务必检查增根。
7. Differentiation: Rules & Techniques | 微分：法则与技巧
The derivative of f(x) is f'(x) or dy/dx. Basic rule: d/dx (xⁿ) = n xⁿ⁻¹. Constant multiple: d/dx [c·f(x)] = c·f'(x).
函数 f(x) 的导数记为 f'(x) 或 dy/dx。基本法则：d/dx (xⁿ) = n xⁿ⁻¹。常数倍法则：d/dx [c·f(x)] = c·f'(x)。
Sum rule: d/dx [f(x) ± g(x)] = f'(x) ± g'(x). Chain rule: if y = f(g(x)), then dy/dx = f'(g(x))·g'(x).
和差法则：d/dx [f(x) ± g(x)] = f'(x) ± g'(x)。链式法则：若 y = f(g(x))，则 dy/dx = f'(g(x))·g'(x)。
Product rule: d/dx (uv) = u’v + uv’. Quotient rule: d/dx (u/v) = (u’v – uv’) / v².
乘积法则：d/dx (uv) = u’v + uv’。商法则：d/dx (u/v) = (u’v – uv’) / v²。
Derivatives of special functions: d/dx (sin x) = cos x; d/dx (cos x) = -sin x; d/dx (tan x) = sec² x = 1/cos² x; d/dx (eˣ) = eˣ; d/dx (ln x) = 1/x.
特殊函数的导数：d/dx (sin x) = cos x；d/dx (cos x) = -sin x；d/dx (tan x) = sec² x = 1/cos² x；d/dx (eˣ) = eˣ；d/dx (ln x) = 1/x。
8. Integration: Anti-differentiation & Area | 积分：反导与面积
Integration is the reverse of differentiation. The indefinite integral: ∫ f(x) dx = F(x) + C, where F'(x) = f(x). Basic rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1).
积分是微分的逆运算。不定积分：∫ f(x) dx = F(x) + C，其中 F'(x) = f(x)。基本积分公式：∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1)。
Standard integrals: ∫ sin x dx = -cos x + C; ∫ cos x dx = sin x + C; ∫ eˣ dx = eˣ + C; ∫ 1/x dx = ln|x| + C.
标准积分表：∫ sin x dx = -cos x + C；∫ cos x dx = sin x + C；∫ eˣ dx = eˣ + C；∫ 1/x dx = ln|x| + C。
For ∫ f(ax+b) dx, use the ‘reverse chain rule’: if ∫ f(x) dx = F(x) + C, then ∫ f(ax+b) dx = (1/a) F(ax+b) + C.
对于 ∫ f(ax+b) dx，使用“反向链式法则”：若 ∫ f(x) dx = F(x) + C，则 ∫ f(ax+b) dx = (1/a) F(ax+b) + C。
Definite integral: ∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) – F(a). It gives the signed area between the curve and the x-axis from a to b.
定积分：∫ₐᵇ f(x) dx = [F(x)]ₐᵇ = F(b) – F(a)。它表示曲线与 x 轴之间从 a 到 b 的有向面积。
9. Arithmetic & Geometric Sequences | 等差与等比数列
Arithmetic sequence: each term differs by a common difference d. The nth term uₙ = a + (n-1)d. Sum to n terms: Sₙ = n/2 [2a + (n-1)d] = n/2 (a + l), where l is the last term.
等差数列：相邻两项之差为常数 d。第 n 项 uₙ = a + (n-1)d。前 n 项和：Sₙ = n/2 [2a + (n-1)d] = n/2 (a + l)，其中 l 为末项。
Geometric sequence: each term is multiplied by a common ratio r. The nth term uₙ = a rⁿ⁻¹. Sum to n terms: Sₙ = a(1 – rⁿ)/(1 – r) for r ≠ 1. If |r| < 1, the sum to infinity S∞ = a / (1 - r).
等比数列：相邻两项之比为常数 r。第 n 项 uₙ = a rⁿ⁻¹。前 n 项和：Sₙ = a(1 – rⁿ)/(1 – r) (r ≠ 1)。若 |r| < 1，无穷级数和 S∞ = a / (1 - r)。
For both types, a represents the first term. Use these formulas to find unknown terms or sums in applied problems.
两种数列中，a 均表示首项。利用这些公式可在应用题中求未知项或求和。
10. Vectors in the Plane & Space | 平面与空间向量
A vector can be written in component form as a = (a₁, a₂) in 2D or a = (a₁, a₂, a₃) in 3D. The zero vector is (0,0). Magnitude: |a| = √(a₁² + a₂²) (2D) or √(a₁² + a₂² + a₃²) (3D).
向量可用分量形式表示，如二维 a = (a₁, a₂)，三维 a = (a₁, a₂, a₃)。零向量为 (0,0)。模长：|a| = √(a₁² + a₂²)（二维）或 √(a₁² + a₂² + a₃²)（三维）。
Scalar (dot) product: a·b = a₁b₁ + a₂b₂ (+ a₃b₃). Geometrically, a·b = |a||b| cos θ, where θ is the angle between the vectors. If a·b = 0, the vectors are perpendicular.
数量积（点积）：a·b = a₁b₁ + a₂b₂ (+ a₃b₃)。几何意义：a·b = |a||b| cos θ，θ 为两向量夹角。若 a·b = 0，则向量垂直。
Unit vector: a unit vector in direction a is â = a / |a|. Common unit vectors in 3D are i, j, k.
单位向量：沿 a 方向的单位向量为 â = a / |a|。三维空间中常用单位向量 i, j, k。
To find the angle between two vectors, use cos θ = (a·b) / (|a||b|). For 2D vectors, the gradient of a line in direction v = (v₁, v₂) is v₂/v₁ (v₁ ≠ 0).
求两向量夹角：cos θ = (a·b) / (|a||b|)。对于二维向量，方向向量 v = (v₁, v₂) 决定的直线斜率为 v₂/v₁ (v₁ ≠ 0)。
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