📚 Year 12 SQA Mathematics: Formula & Theorem Quick Reference | 12年级 SQA 数学：公式定理速查手册
This handbook collects the essential formulas and theorems needed for the SQA Higher Mathematics course. It is designed for quick revision, with each key result presented in a clear, exam-ready format followed by concise explanations in both English and Chinese.
本手册汇集了 SQA Higher 数学课程所需的核心公式与定理，便于快速复习。每个关键结果以清晰、贴合考试要求的形式呈现，并配有中英双语简练解释。
1. Algebraic Operations & Quadratic Theory | 代数运算与二次方程理论
x = [-b ± √b² - 4ac] / (2a)
utions of ax² + bx + c = 0, where a ≠ 0. It is used when factoring is not straightforward.
二次求根公式给出方程 ax² + bx + c = 0 (a ≠ 0) 的解，当因式分解不直接时使用该公式。
Δ = b² - 4ac
ct real roots if Δ > 0, one repeated real root if Δ = 0, and no real roots if Δ < 0.
判别式决定根的性质：若 Δ > 0 有两个不等实根；Δ = 0 有一个重根；Δ < 0 没有实根。
Sum of roots = –b/a ; Product of roots = c/a
For ax² + bx + c = 0, the sum of the roots α + β is –/a, and their product αβ is c/a. These relationships help form equations from given roots.
对于 ax² + bx + c = 0，两根之和 α + β = –b/a，两根之积 αβ = c/a。这些关系可用于由给定根构造方程。
(x + p)² + q form
Completing the square rewrites x² + bx + c as (x + b/2)² – (b/2)² + c. The vertex of the parabola is (–b/2, c – (b/2)²).
配方法将 x² + bx + c 改写为 (x + b/2)² – (b/2)² + c。抛物线的顶点为 (–b/2, c – (b/2)²)。
2. Polynomials & Synthetic Division | 多项式与综合除法
Remainder Theorem: f(a) = remainder when f(x) ÷ (x – a)
Substituting x = a into a polynomial f(x) gives the remainder upon division by (x – his is a quick way to evaluate polynomial division without carrying out the full division.
将 x = a 代入多项式 f(x) 得到除以 (x – a) 的余数。这是快速求多项式除法余数的方法，无需完整执行除法。
Factor Theorem: (x – a) is a factor if f(a) = 0
A linear expression (x – exactly if and only if f(a) = 0. This helps factorise cubics and higher-order polynomials.
当且仅当 f(a) = 0 时，线性表达式 (x – a) 能整除多项式。该定理用于三次及更高次多项式的因式分解。
Synthetic division
Synthetic division is a streamlined method for dividing a polynomial by a linear factor (x – a). Coefficients are written in a row, and the process involves multiplying by a and adding.
综合除法是除以线性因式 (x – a) 的简化方法。将系数排成一行，通过乘 a 再相加的过程来完成除法。
Polynomial equation from roots
If roots r₁, r₂, ..., rₙ are known, the monic polynomial is (x – r₁)(x – r₂)...(x – rₙ). Expanding gives the standard form.
若已知根 r₁, r₂, ..., rₙ，则首一多项式为 (x – r₁)(x – r₂)...(x – rₙ)。展开后得到标准形式。
3. Trigonometric Functions & Identities | 三角函数与恒等式
Radian measure: π rad = 180°
he conversion is θ (rad) = θ (°) × π/180. The arc length s = rθ and sector area A = ½ r²θ.
在 Higher 数学中角度使用弧度制。换算关系为 θ (弧度) = θ (°) × π/180。弧长公式 s = rθ，扇形面积 A = ½ r²θ。
sin²θ + cos²θ = 1
²θ and 1 + cot²θ = cosec²θ, though sec, cosec, and cot are used less frequently at Higher.
基本恒等式。可变形为 tan²θ + 1 = sec²θ 及 1 + cot²θ = cosec²θ，不过 sec、cosec、cot 在 Higher 中使用较少。
Addition formulas: sin(A ± B) = sin A cos B ± cos A sin B ; cos(A ± B) = cos A cos B ∓ sin A sin B
These expand the sine and cosine of a sum or difference. They are essential for solving trigonometric equations and proving identities.
这些公式将和差角的正弦、余弦展开，是解三角方程和证明恒等式的基础。
Double-angle formulas: sin 2θ = 2 sin θ cos θ ; cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ
Derived from the addition formulas, the double-angle identities are heavily used in integration and solving equations.
由和角公式推导出的倍角恒等式，在积分和解方程中大量使用。
Exact values for 0, π/6, π/4, π/3, π/2
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| π/6 | 1/2 | √3/2 | √3/3 |
| π/4 | √2/2 | √2/2 | 1 |
| π/3 | √3/2 | 1/2 | √3 |
| π/2 | 1 | 0 | undefined |
Memorising these exact values is mandatory; they appear frequently on the non‑calculator paper.
熟记这些精确值必不可少，它们经常出现在非计算器试卷中。
4. Differentiation Rules & Applications | 微分法则与应用
Power rule: d/dx (xⁿ) = n xⁿ⁻¹
The derivative of x to the power n is n times x to the power n−1. This is valid for any real constant n.
x 的 n 次幂的导数为 n 乘以 x 的 n−1 次幂。此式对任意实常数 n 成立。
Sum/difference rule: d/dx [f(x) ± g(x)] = f '(x) ± g '(x)
Differentiation is linear; the derivative of a sum is the sum of the derivatives.
微分是线性的；和的导数等于导数的和。
Chain rule: dy/dx = dy/du × du/dx
When y is a function of u and u is a function of x, multiply the derivative of the outer function by the derivative of the inner function.
当 y 是 u 的函数且 u 是 x 的函数时，外函数的导数乘以内函数的导数。
Product rule: d/dx (u v) = u' v + u v'
For the product of two functions, differentiate the first and multiply by the second, then add the first multiplied by derivative of the second.
对于两个函数的乘积，第一项求导乘第二项，加上第一项乘第二项求导。
Quotient rule: d/dx (u/v) = (u' v – u v') / v²
For a division of two functions, the denominator becomes v² and the numerator is 'lo‑dhi minus hi‑dlo'.
对于分式函数，分母为 v²，分子为“低乘高导减去高乘低导”。
Special derivatives: d/dx (sin x) = cos x; d/dx (cos x) = –sin x; d/dx (eˣ) = eˣ; d/dx (ln x) = 1/x
These fundamental derivatives are used repeatedly in chain, product, and quotient rule applications.
这些基本导数在链式法则、乘积法则及商法则的应用中反复出现。
Stationary points: f '(x) = 0
Solve f '(x) = 0 to find x‑ he nature table or second derivative to classify maxima, minima, and points of inflection.
解方程 f '(x) = 0 求得驻点的 x 坐标。利用导数符号表或二阶导数判别极大值、极小值和拐点。
5. Integration Rules & Area | 积分法则与面积
Power rule: ∫ xⁿ dx = xⁿ⁺¹/(n+1) + C, n ≠ –1
The reverse of the power rule for differentiation. Add one to the exponent and divide by the new exponent. The constant of integration C is essential.
微分的幂法则逆运算。指数加 1 再除以新指数。积分常数 C 必不可少。
Linearity: ∫ [a f(x) + b g(x)] dx = a ∫ f(x)dx + b ∫ g(x)dx
Integration preserves linear combinations, allowing term‑by‑term integration.
积分保持线性组合，可以逐项积分。
Special integrals: ∫ sin x dx = –cos x + C; ∫ cos x dx = sin x + C; ∫ eˣ dx = eˣ + C; ∫ 1/x dx = ln|x| + C
These standard integrals are derived directly from differentiation and must be memorised.
这些标准积分由微分直接得出，必须牢记。
Definite integral and area: ∫ₐᵇ f(x) dx
Evaluate the antiderivative at b and subtract the value at a. If f(x) is negative, the integral gives a signed area; the physical area is found by splitting the interval where the function changes sign.
计算原函数在 b 和 a 的值并相减。若 f(x) 为负，积分给出带符号的面积；实际面积需对函数变号处分割区间。
Area between curves
The area enclosed between y = f(x) and y = g(x) from a to b is ∫ₐᵇ |f(x) – g(x)| dx. In practice, identify which graph is above and integrate (upper – lower).
两曲线 y = f(x) 与 y = g(x) 在 a 到 b 之间所围面积是 ∫ₐᵇ |f(x) – g(x)| dx。实际求解时先确定上方图形，积分（上方减下方）。
6. Exponential & Logarithmic Functions | 指数与对数函数
Index laws: aˣ aʸ = aˣ⁺ʸ; (aˣ)ʸ = aˣʸ; aˣ / aʸ = aˣ⁻ʸ; a⁻ˣ = 1/aˣ
These laws hold for any positive real base a. They are the foundation for manipulating exponential equations.
这些法则对任意正实数底 a 成立，是处理指数方程的基础。
Logarithm laws: logₐ (xy) = logₐ x + logₐ y; logₐ (x/y) = logₐ x – logₐ y; logₐ (xᵇ) = b logₐ x
Logarithms turn multiplication into addition and division into subtraction. The power rule brings exponents in front.
对数将乘法化为加法，除法化为减法；幂法则将指数前置。
Natural logarithm: ln x = logₑ x
base e. It satisfies ln e = 1 and ln 1 = 0. The function y = ln x is defined for x > 0.
自然对数是底数为 e 的对数。满足 ln e = 1 和 ln 1 = 0。函数 y = ln x 的定义域为 x > 0。
Change of base: logₐ b = logₓ b / logₓ a
Any logarithm can be expressed in terms of logs of another base. Commonly used to compute values on a calculator with only ln or log₁₀.
任何对数可用另一底的对数表示，常用于计算器上只有 ln 或 log₁₀ 时的换算。
Exponential equations: aˣ = b ⇔ x = logₐ b
Take logs of both sides to bring the exponent down. For eˣ, natural logs are most convenient.
两边取对数将指数降下来。对于 eˣ，使用自然对数最方便。
7. Vectors & Scalar Product | 向量与标量积
Vector components: a = a₁i + a₂j + a₃k
A vector in three dimensions is represented by its components along the unit vectors i, j, and k.
三维空间中的向量由其沿单位向量 i、j、k 的分量表示。
Magnitude: |a| = √a₁² + a₂² + a₃²
The magnitude (length) of a vector is found by the square root of the sum of squares of its components.
向量的模（长度）为其各分量平方和的平方根。
Scalar (dot) product: a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos θ
The dot product is a scalar. The component form and the geometric form are equivalent. It is used to find angles and to test perpendicularity.
点积是一个标量。代数分量形式和几何形式等价，用于求夹角和判断垂直。
Perpendicular vectors: a·b = 0 ⇔ a ⟂ b
Two non‑zero vectors are perpendicular if and only if their scalar product is zero.
两个非零向量垂直当且仅当它们的点积为零。
Vector equation of a line: r = a + t d
A straight line in 3D passes through point A (a) and is parallel to direction vector d. Parameter t scalars the direction.
三维空间中的直线过点 A（向量 a）且平行于方向向量 d。参数 t 按比例沿方向延伸。
8. Straight Lines & Circles | 直线与圆
Distance between two points: d
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