📚 SQA Higher Maths Formula & Theorem Quick Reference | SQA 高等数学公式定理速查手册
This quick reference guide is designed for Year 12 students preparing for SQA Higher Mathematics. It covers the essential formulae and theorems you will need across the syllabus, including algebra, trigonometry, calculus, vectors, and sequences. Each section pairs clear English explanations with Chinese translations to support bilingual learners, and every formula is presented without LaTeX — using simple Unicode symbols and standard mathematical notation.
本速查手册专为准备 SQA 高等数学的 Year 12 学生设计。它涵盖了考纲中必备的核心公式与定理，包含代数、三角、微积分、向量与数列等模块。每个要点先提供清晰的英文解释，再给出对应的中文翻译，帮助双语学习者快速掌握。所有公式均不依赖 LaTeX，采用 Unicode 符号和常规数学记法呈现。
1. Algebraic Operations and Surds | 代数运算与根式
Simplifying surds and applying index laws are fundamental skills. Rationalising the denominator removes a surd from the bottom of a fraction.
化简根式并运用指数法则是基础技能。分母有理化可以消去分母中的根号。
Key index rules: am × an = am+n, am ÷ an = am-n, (am)n = amn, a0 = 1, a-n = 1/an.
重要指数法则：am × an = am+n，am ÷ an = am-n，(am)n = amn，a0 = 1，a-n = 1/an。
For fractions with a surd denominator, multiply numerator and denominator by the conjugate. For example, for a/√b, use √b; for a/(b + √c), use (b – √c).
对于分母含根式的分式，分子分母同乘共轭根式。例如 a/√b 同乘 √b；a/(b + √c) 同乘 (b – √c)。
2. Quadratic Functions and the Discriminant | 二次函数与判别式
The general quadratic is ax² + bx + c = 0 (a ≠ 0). Its roots are given by the quadratic formula.
一般二次方程为 ax² + bx + c = 0（a ≠ 0）。其根由求根公式给出。
x = [-b ± √(b² – 4ac)] / (2a)
The discriminant D = b² – 4ac tells us the nature of the roots.
判别式 D = b² – 4ac 说明了根的性质。
- D > 0: two distinct real roots | 两个不等实根
- D = 0: one repeated real root (tangent to x-axis) | 一个重根（与 x 轴相切）
- D < 0: no real roots | 无实根
Completing the square rewrites the quadratic as y = a(x + p)² + q. The vertex (turning point) is at (-p, q).
配方法将二次式化为 y = a(x + p)² + q。顶点（转折点）坐标为 (-p, q)。
3. Polynomials and Division | 多项式与除法
Polynomial division, the remainder theorem and the factor theorem are central to algebraic manipulation in Higher Maths.
多项式除法、余式定理和因式定理是高等数学代数运算的核心。
Remainder theorem: When a polynomial f(x) is divided by (x – h), the remainder is f(h).
余式定理：多项式 f(x) 除以 (x – h) 时，余数为 f(h)。
Factor theorem: If f(h) = 0 then (x – h) is a factor of f(x).
因式定理：若 f(h) = 0，则 (x – h) 是 f(x) 的一个因式。
Use synthetic division for quick polynomial division, bringing down coefficients and repeating multiply-add steps.
使用综合除法可快速进行多项式除法：降下系数，重复乘加步骤。
4. Exponentials and Logarithms | 指数与对数
A logarithm answers the question: to what power must the base be raised to obtain a given number? The fundamental relationship is:
对数回答：底数需要多少次幂才能得到给定数字？基本关系为：
logₐ x = y ⇔ ay = x
Key log laws (valid for any admissible base):
重要对数法则（适用于任何可行底数）：
- logₐ (MN) = logₐ M + logₐ N
- logₐ (M/N) = logₐ M – logₐ N
- logₐ (Mk) = k logₐ M
Change of base formula: logₐ x = log_b x / log_b a. The natural logarithm ln x = logₑ x, where e ≈ 2.718.
换底公式：logₐ x = log_b x / log_b a。自然对数 ln x = logₑ x，其中 e ≈ 2.718。
The natural exponential ex and its inverse ln x appear frequently in growth and decay models.
自然指数 ex 及其反函数 ln x 经常出现在增长与衰减模型中。
5. Trigonometric Identities and Exact Values | 三角恒等式与精确值
Mastery of trig identities allows you to simplify expressions and solve equations. The core identities are:
熟练掌握三角恒等式能够化简表达式并解方程。核心恒等式有：
sin² x + cos² x = 1
tan x = sin x / cos x
Double angle formulas:
倍角公式：
sin 2x = 2 sin x cos x
cos 2x = cos² x – sin² x = 2cos² x – 1 = 1 – 2sin² x
You must know exact values for key angles without a calculator:
你必须牢记以下关键角的精确值（无需计算器）：
| Angle θ | 角 θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
6. The Wave Function | 波的合成函数
Expressions of the form a sin x + b cos x can be combined into a single sine or cosine wave, making it easier to find maxima, minima and solve equations.
形如 a sin x + b cos x 的表达式可合成为一个正弦或余弦波，便于求最值和解方程。
a sin x + b cos x = k sin(x + α)
a sin x + b cos x = k cos(x – α)
where k = √(a² + b²) and the phase angle α satisfies tan α = b/a (for sin form) or tan α = a/b (for cos form). Check quadrant carefully.
其中 k = √(a² + b²)，相位角 α 满足 tan α = b/a（正弦形式）或 tan α = a/b（余弦形式）。请注意确定象限。
Maximum value is k, minimum is -k. This form appears in many SQA exam questions on trig equations.
最大值为 k，最小值为 -k。该形式常见于 SQA 三角方程考题。
7. Differentiation: Rules and Applications | 微分：法则与应用
Derivatives give the gradient of a curve and the rate of change. The basic derivative for a power is:
导数给出曲线的斜率与变化率。幂函数的基本导数为：
d/dx (xⁿ) = n xⁿ⁻¹
Standard derivatives you must know:
必须记住的标准导数：
- d/dx (sin x) = cos x
- d/dx (cos x) = -sin x
- d/dx (ex) = ex
- d/dx (ln x) = 1/x
Key differentiation rules:
核心微分法则：
- Chain rule: dy/dx = dy/du × du/dx, or for f(g(x)), derivative is f'(g(x))·g'(x).
- Product rule: If y = u v, then dy/dx = u’v + u v’.
- Quotient rule: If y = u/v, then dy/dx = (u’v – u v’) / v².
Applications include finding equations of tangents and normals, stationary points, and optimisation.
应用包括求切线和法线方程、驻点以及最优化问题。
8. Integration: Antiderivatives and Area | 积分：不定积分与面积
Integration reverses differentiation. For a power function (n ≠ -1):
积分是微分的逆运算。对于幂函数（n ≠ -1）：
∫ xⁿ dx = xⁿ⁺¹ / (n + 1) + C
Standard integrals to remember:
需记住的标准积分：
- ∫ cos x dx = sin x + C
- ∫ sin x dx = -cos x + C
- ∫ ex dx = ex + C
- ∫ 1/x dx = ln|x| + C
Definite integrals calculate the area under a curve between two limits a and b: Area = ∫ₐᵇ f(x) dx.
定积分计算曲线在上下限 a 与 b 之间的面积：Area = ∫ₐᵇ f(x) dx。
If the curve goes below the x‑axis, take absolute area by splitting the interval. Also solve simple differential equations like dy/dx = g(x) by direct integration.
如果曲线在 x 轴下方，需分割区间取绝对值求面积。通过直接积分也可解简单的微分方程，如 dy/dx = g(x)。
9. Straight Line and Circle | 直线与圆
Equations of a straight line can be expressed in several forms:
直线方程可以表示为多种形式：
- Gradient-intercept: y = m x + c
- Point-gradient: y – b = m (x – a)
Distance between two points (x₁, y₁) and (x₂, y₂): √[(x₂ – x₁)² + (y₂ – y₁)²]. Midpoint is ((x₁+x₂)/2, (y₁+y₂)/2).
两点间距离：√[(x₂ – x₁)² + (y₂ – y₁)²]。中点为 ((x₁+x₂)/2, (y₁+y₂)/2)。
The equation of a circle with centre (a, b) and radius r is:
圆心为 (a, b)、半径为 r 的圆方程为：
(x – a)² + (y – b)² = r²
The condition for a line y = mx + c to be tangent to a circle: substitute into the circle and set the discriminant of the resulting quadratic to zero.
直线 y = mx + c 与圆相切的条件：代入圆的方程，令所得二次方程的判别式为零。
10. Vectors in 3D | 三维向量
Vectors describe position and direction in space. A vector a = (a₁, a₂, a₃) has magnitude |a| = √(a₁² + a₂² + a₃²).
向量描述
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