📚 Formulae & Theorems Quick-Reference Handbook | 公式定理速查手册
This handbook provides a concise collection of essential formulae and theorems for the Year 10 Eduqas Level 2 Certificate in Further Mathematics. It is designed as a quick revision aid to reinforce algebraic manipulation, coordinate geometry, trigonometry, calculus, and other key topics.
本手册为10年级Eduqas二级进阶数学证书课程提供必备公式与定理的简明汇编。它作为快速复习工具，旨在强化代数运算、坐标几何、三角学、微积分及其他关键主题。
1. Algebraic Identities & Expansion | 代数恒等式与展开
Mastering algebraic identities allows efficient expansion and factorisation. The perfect square forms and the difference of two squares are fundamental building blocks.
掌握代数恒等式可高效展开和因式分解。完全平方形式与平方差公式是基础构件。
(a + b)² = a² + 2ab + b²
(a − b)² = a² − 2ab + b²
(a + b)(a − b) = a² − b²
ll expansions, and the concept extends to trinomials: (x + a)(x + b) = x² + (a + b)x + ab.
分配律 a(b + c) = ab + ac 支撑所有展开，这一概念可推广至三项式：(x + a)(x + b) = x² + (a + b)x + ab。
2. Factor Theorem & Polynomial Division | 因式定理与多项式除法
(x − a) is a factor. This is the Factor Theorem, a crucial tool for solving higher-degree equations and simplifying rational expressions.
对于多项式 f(x)，若 f(a) = 0 则 (x − a) 是一个因式。这就是因式定理，是求解高次方程和化简有理式的关键工具。
Polynomial long division or synthetic division is used to divide a polynomial by a linear factor, yielding a quotient and remainder. The Remainder Theorem states that when f(x) is divided by (x − a), the remainder is f(a).
多项式长除法或综合除法用于将多项式除以一次因式，得到商式和余数。余数定理指出，当 f(x) 除以 (x − a) 时，余数为 f(a)。
3. Binomial Expansion | 二项式展开
The binomial expansion expresses (a + b)ⁿ as a sum of terms involving binomial coefficients. For a positive integer n, the expansion is governed by Pascal's triangle or the combination formula.
二项式展开将 (a + b)ⁿ 表示为包含二项式系数的各项之和。对于正整数 n，展开由帕斯卡三角形或组合公式控制。
(a + b)ⁿ = nC0 aⁿ + nC1 aⁿ⁻¹ b + nC2 aⁿ⁻² b² + … + nCr aⁿ⁻ʳ bʳ + … + nCn bⁿ
When expanding (1 + x)ⁿ, the general term is nCr xʳ. For rational and negative n where |x| < 1, the expansion is infinite and uses the generalised binomial theorem.
在展开 (1 + x)ⁿ 时，通项为 nCr xʳ。对于满足 |x| < 1 的有理数或负数 n，展开是无穷的，并使用广义二项式定理。
4. Coordinate Geometry – Straight Lines | 坐标几何—直线
d (x₂, y₂) is m = (y₂ − y₁)/(x₂ − x₁). The equation of a line can be expressed in gradient–intercept form y = mx + c or point–gradient form y − y₁ = m(x − x₁).
过点 (x₁, y₁) 和 (x₂, y₂) 的直线斜率 m = (y₂ − y₁)/(x₂ − x₁)。直线方程可表示为斜截式 y = mx + c 或点斜式 y − y₁ = m(x − x₁)。
((x₁ + x₂)/2, (y₁ + y₂)/2), and the distance between points is √[(x₂ − x₁)² + (y₂ − y₁)²].
平行直线斜率相同。垂直直线满足 m₁ × m₂ = −1。线段中点为 ((x₁ + x₂)/2, (y₁ + y₂)/2)，两点间距离为 √[(x₂ − x₁)² + (y₂ − y₁)²]。
5. Coordinate Geometry – Circles | 坐标几何—圆
− a)² + (y − b)² = r². When the centre is at the origin, this simplifies to x² + y² = r².
对于圆心为 (a, b)、半径为 r 的圆，标准方程为 (x − a)² + (y − b)² = r²。当圆心在原点时，方程简化为 x² + y² = r²。
from x² + y² + 2gx + 2fy + c = 0, complete the square to obtain centre (−g, −f) and radius √g² + f² − c. A tangent to a circle touches at exactly one point and is perpendicular to the radius at that point.
要从 x² + y² + 2gx + 2fy + c = 0 求圆心和半径，通过配方法得到圆心 (−g, −f) 和半径 √g² + f² − c。圆的切线仅在一点接触圆，且在该点垂直于半径。
6. Trigonometry – Right-Angled Triangles | 三角学—直角三角形
The SOH CAH TOA mnemonic summarises the three primary trigonometric ratios in a right-angled triangle. These relationships link an acute angle to the side lengths.
SOH CAH TOA 助记口诀总结了直角三角形中三个基本的三角比。这些关系将锐角与边长联系起来。
sin θ = opposite / hypotenuse
cos θ = adjacent / hypotenuse
tan θ = opposite / adjacent
es Exact trigonometric values for 0°, 30°, 45°, 60° and 90° must be memorised. For example, sin 30° = ½, cos 45° = 1/√2, tan 60° = √3.
必须熟记 0°、30°、45°、60° 和 90° 的精确三角函数值。例如 sin 30° = ½，cos 45° = 1/√2，tan 60° = √3。
7. Sine & Cosine Rules | 正弦与余弦定理
The sine rule is used for non-right-angled triangles when either two angles and one side are known, or two sides and a non-included angle are given. Its formula relates the side lengths to the sine of opposite angles.
正弦定理用于非直角三角形，当已知两角一边或两边及非夹角时。其公式将边长与其对角的正弦相联系。
a / sin A = b / sin B = c / sin C = 2R
The cosine rule extends Pythagoras' theorem to any triangle, linking the three sides and one angle. It is especially useful when two sides and the included angle are known, or all three sides are given.
余弦定理将勾股定理推广到任意三角形，将三边与一角联系起来。当已知两边及其夹角或三边时特别有用。
a² = b² + c² − 2bc cos A
lculated using Area = ½ ab sin C, where a and b are two sides and C is the included angle.
三角形面积可用 Area = ½ ab sin C 计算，其中 a 和 b 为两边，C 为夹角。
8. Trigonometric Identities & Equations | 三角恒等式与方程
viding by cos²θ yields tan²θ + 1 = sec²θ, and dividing by sin²θ gives 1 + cot²θ = cosec²θ.
平方恒等式 sin²θ + cos²θ = 1 是化简表达式和证明其他恒等式的基石。除以 cos²θ 得到 tan²θ + 1 = sec²θ，除以 sin²θ 得到 1 + cot²θ = cosec²θ。
angle formulae sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ.
其他有用的恒等式包括 tan θ = sin θ / cos θ，以及倍角公式 sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ − sin²θ = 2 cos²θ − 1 = 1 − 2 sin²θ。
θ = k or cos(θ + 30°) = 0.5, paying attention to the given interval to find all solutions.
解三角方程通常需要利用恒等式将方程化为形如 sin θ = k 或 cos(θ + 30°) = 0.5 的形式，并注意给定区间以找出所有解。
9. Radian Measure, Arc Length & Sector Area | 弧度制、弧长与扇形面积
entre of a circle by an arc equal in length to the radius. The conversion is π rad = 180°.
弧度提供了一种自然的角的度量方式，1 弧度是弧长等于半径所对的圆心角。换算关系为 π 弧度 = 180°。
Arc length s = rθ
Sector area A = ½ r²θ
When θ is in radians, the segment area can be found by subtracting the triangular area (½ r² sin θ) from the sector area. Radians are essential in calculus when differentiating or integrating trigonometric functions.
当 θ 以弧度为单位时，弓形面积可由扇形面积减去三角形面积 (½ r² sin θ) 求得。在涉及三角函数的微积分运算中，弧度是必不可少的。
10. Introduction to Differentiation | 微分导论
e derivative of a constant is zero, and the derivative of a sum is the sum of derivatives.
微分用于求曲线上任意点的斜率。对于 y = xⁿ，导数为 dy/dx = n xⁿ⁻¹。常数的导数为零，和的导数等于导数之和。
d/dx (xⁿ) = n xⁿ⁻¹
d/dx (f(x) + g(x)) = f'(x) + g'(x)
or point of inflection) is determined by the second derivative d²y/dx² or by examining the sign change of the gradient. If d²y/dx² > 0, the point is a local minimum; if d²y/dx² < 0, it is a local maximum.
驻点出现在 dy/dx = 0 处。其性质（极大值、极小值或拐点）由二阶导数 d²y/dx² 或通过考察斜率符号变化来确定。若 d²y/dx² > 0，该点为局部极小值；若 d²y/dx² < 0，则为局部极大值。
11. Introduction to Integration | 积分导论
Integration is the reverse process of differentiation, used to find the area under a curve or to recover a function from its derivative. The indefinite integral of xⁿ is (xⁿ⁺¹)/(n+1) + C, where n ≠ −1.
积分是微分的逆过程，用于求曲线下的面积或根据导数还原函数。xⁿ 的不定积分为 (xⁿ⁺¹)/(n+1) + C，其中 n ≠ −1。
∫ xⁿ dx = (xⁿ⁺¹)/(n+1) + C, n ≠ −1
signed area between the curve y = f(x) and the x-axis: ∫ₐᵇ f(x) dx = F(b) − F(a), where F'(x) = f(x). Areas below the axis are counted as negative unless absolute values are taken.
在上下限 a 和 b 之间的定积分给出曲线 y = f(x) 与 x 轴之间的带符号面积：∫ₐᵇ f(x) dx = F(b) − F(a)，其中 F'(x) = f(x)。x 轴下方的面积视为负值，除非取绝对值。
The constant of integration C must be included in indefinite integrals, and can often be evaluated using a boundary condition or point through which the curve passes.
不定积分必须包含积分常数 C，通常可利用曲线经过的边界条件或点来求得其数值。
12. Exponentials & Logarithms | 指数与对数
The exponential function y = eˣ has the remarkable property that its derivative is itself, dy/dx = eˣ. Similarly, the natural logarithm function ln x is the inverse of eˣ, and its derivative is 1/x.
指数函数 y = eˣ 具有一个非凡的性质：其导数等于自身，即 dy/dx = eˣ。类似地，自然对数函数 ln x 是 eˣ 的反函数，其导数为 1/x。
d/dx (eˣ) = eˣ
∫ eˣ dx = eˣ + C
d/dx (ln x) = 1/x
∫ 1/x dx = ln|x| + C
− logₐ y, and logₐ (xⁿ) = n logₐ x. The change of base rule is logₐ b = log_c b / log_c a.
对数定律在求解指数方程和对数方程时至关重要：logₐ (xy) = logₐ x + logₐ y，logₐ (x/y) = logₐ x − logₐ y，以及 logₐ (xⁿ) = n logₐ x。换底公式为 logₐ b = log_c b / log_c a。
Published by TutorHao | Further Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图