📚 Addition Formulae | 和角公式
nd tangents of the individual angles A and B. Mastering these identities is essential for solving equations, proving other identities and tackling a wide range of Edexcel exam questions. In this article, we will explore each formula, derive them from first principles and illustrate their use with step-by-step examples.
和角公式是 A-Level 三角学的基石。它们将 sin(A ± B)、cos(A ± B) 和 tan(A ± B) 表示为单个角度 A 和 B 的正弦、余弦和正切的组合。掌握这些恒等式对于解方程、证明其他恒等式以及应对 Edexcel 考试中的各类题目至关重要。本文将逐一探讨每个公式，从基本原理出发进行推导，并通过逐步示例展示其应用。
1. The Sine Addition Formulae | 正弦和角公式
The sine addition formulae state:
正弦和角公式如下：
sin(A + B) = sin A cos B + cos A sin B
sin(A - B) = sin A cos B - cos A sin B
B) both terms are positive; for sin(A - B) the second term becomes negative. The structure 'sin cos ± cos sin' is easy to remember if you think of the sine function 'spreading out' to both angles.
请注意，两项之间的正负号与括号内的正负号一致。对于 sin(A + B) 两项均为正；对于 sin(A - B) 第二项变为负。'sin cos ± cos sin' 的结构很容易记住，只要想到正弦函数“扩展”到两个角度即可。
These formulae are valid for any angles measured in degrees or radians, and they form the basis for many other trigonometric identities.
这两个公式对任意角度（不论度数还是弧度）都成立，并且是许多其他三角恒等式的基础。
2. The Cosine Addition Formulae | 余弦和角公式
The cosine addition formulae are slightly different because the sign in the expression is opposite to the sign inside the argument.
余弦和角公式稍有不同，因为表达式中的正负号与括号内的正负号相反。
cos(A + B) = cos A cos B - sin A sin B
cos(A - B) = cos A cos B + sin A sin B
For cos(A + B) we subtract the product of sines, while for cos(A - B) we add that product. A common mnemonic is 'cos cos, sin sin, opposite sign'. Remembering this opposite nature helps avoid mixing up the formulas with the sine versions.
对于 cos(A + B)，我们要减去正弦的乘积，而对于 cos(A - B)，则加上该乘积。一个常见的记忆口诀是“cos cos, sin sin, 符号相反”。牢记这种相反的性质有助于避免将这些公式与正弦公式混淆。
These identities are particularly useful when we need to find exact values of cosines for angles like 15° or 75°, or when proving more complex results.
当我们需要求诸如 15° 或 75° 等角度的余弦精确值，或证明更复杂的结果时，这些恒等式特别有用。
3. The Tangent Addition Formulae | 正切和角公式
The tangent addition formulae can be derived by dividing the sine and cosine addition results. They are given by:
正切和角公式可以通过将正弦与余弦的和角结果相除推导得出。公式如下：
tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
tan(A - B) = (tan A - tan B) / (1 + tan A tan B)
hat these formulas are undefined whenever the denominator equals zero, i.e. when tan A tan B = 1 for tan(A + B) or tan A tan B = -1 for tan(A - B).
这里分子的正负号规律很明显，但分母的正负号再次与括号内的符号相反。必须牢记，当分母为零时公式无定义，即对 tan(A + B) 有 tan A tan B = 1，对 tan(A - B) 有 tan A tan B = -1。
Tangent addition formulae are heavily tested in solving equations where terms can be combined into a single tangent ratio.
正切和角公式在将各项合并为单一正切比的解方程题型中经常被考查。
4. Deriving the Formulae | 推导和角公式
A classic geometric proof starts with two points on the unit circle at angles A and B, and then uses the distance formula or rotation matrices. One of the most straightforward derivations is for cos(A - B). Consider points P(cos A, sin A) and Q(cos B, sin B) on the unit circle. The square of the distance PQ can be written in two ways: directly from coordinates and from the cosine rule in triangle POQ. Equating the two expressions gives:
一种经典的几何证明从单位圆上角度为 A 和 B 的两个点出发，然后利用距离公式或旋转矩阵。最直接的推导之一是针对 cos(A - B)。设单位圆上点 P(cos A, sin A) 和 Q(cos B, sin B)。PQ 间距离的平方可以用两种方式表示：直接从坐标计算，以及利用三角形 POQ 中的余弦定理。令这两个表达式相等可得：
cos(A - B) = cos A cos B + sin A sin B
ng the cosine expansion. The tangent formula follows from dividing the sine and cosine results.
由此，将 B 替换为 -B 便得到 cos(A + B)。然后利用 sin θ = cos(90° - θ) 并代入余弦展开式，即可得到正弦和角公式。正切公式则由正弦与余弦结果相除得出。
While you are not always required to reproduce the full proof in an exam, understanding the origin of these identities deepens your appreciation and helps you remember them correctly.
虽然考试中不经常要求完整写出证明过程，但理解这些恒等式的来源能加深理解，有助于正确记忆公式。
5. Finding Exact Values | 求精确值
5°, 60° and their radian equivalents. For example, to find sin 75° we write 75° = 45° + 30° and apply the sine addition formula:
最早的应用之一是计算那些可以表示为常见特殊角（如 30°、45°、60° 及其弧度等价）之和或差的三角比。例如，要求 sin 75°，我们将 75° 写为 45° + 30° 并应用正弦和角公式：
sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°
= (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4
We often express answers exactly using surds. Similar steps can be followed for cos 15°, tan 105°, and so on. This technique also works when one of the angles is given in terms of its sine or cosine, allowing us to find combined values without calculating the angles individually.
我们通常用根式精确表示答案。对于 cos 15°、tan 105° 等也可以采用类似步骤。当其中一个角度是通过其正弦或余弦条件给出时，这种方法依然有效，我们无需分别求出各个角度就能算出组合值。
6. Simplifying Expressions and Proving Identities | 化简表达式与证明恒等式
Using the sine subtraction formula backwards, this collapses to sin((A + B) - B) = sin A.
在化简三角表达式时，和角公式必不可少。考虑表达式 sin(A + B) cos B - cos(A + B) sin B。反向运用正弦减法公式，它可简化为 sin((A + B) - B) = sin A。
th the left-hand side, expand sin(A+B) using the addition formula and then separate the fraction. This yields:
要证明恒等式，例如 (sin(A+B))/(cos A cos B) = tan A + tan B，可以从左边开始，用和角公式展开 sin(A+B)，然后拆分分式，可得：
(sin A cos B + cos A sin B) / (cos A cos B) = tan A + tan B
These types of algebraic manipulations are common in Edexcel papers, so practising a variety of identities will build your fluency.
这类代数操作在 Edexcel 试卷中很常见，因此多做各种恒等式的练习可以提升熟练度。
7. Solving Trigonometric Equations | 解三角方程
eed to solve sin(x + 30°) = 2 cos x for 0° ≤ x ≤ 360°. Expanding the left-hand side gives:
许多方程需要使用和角公式改写为可解的形式。假设我们要在 0° ≤ x ≤ 360° 范围内解方程 sin(x + 30°) = 2 cos x。左边展开后得：
sin x cos 30° + cos x sin 30° = 2 cos x
(√3/2) sin x + (1/2) cos x = 2 cos x
Rearranging yields a relationship between sin x and cos x, which can be turned into a standard tan x equation. Finally, all solutions in the given interval can be found using the CAST diagram or by graphing.
整理后可得到 sin x 与 cos x 的关系，并将其转化为标准的 tan x 方程。最后，利用 CAST 图或图像求出给定区间内的所有解。
, be careful to check that the solutions do not make the denominator zero, as those must be excluded.
当方程涉及 tan(A ± B) 时，要仔细检查所求得的解是否会使分母为零，那些解需要被舍去。
8. Double-Angle Formulae as a Special Case | 特例：倍角公式
rmulae yields the double-angle identities, which are themselves essential A-Level content.
在和角公式中令 B = A 即可得到倍角恒等式，它们本身也是 A-Level 的核心内容。
sin 2A = 2 sin A cos A
cos 2A = cos² A - sin² A = 2 cos² A - 1 = 1 - 2 sin² A
tan 2A = 2 tan A / (1 - tan² A)
Note that for cosine we have three alternative forms, which are selected depending on whether the equation requires expressing everything in terms of sin A or cos A alone. The double-angle formulae are heavily used in integration, differentiation and solving more advanced trigonometric equations.
注意，余弦的倍角公式有三种等价形式，取决于方程是需要全部用 sin A 还是全部用 cos A 表示。倍角公式在积分、微分以及更复杂的三角方程求解中应用非常广泛。
9. 'Rsin(x+α)' and Harmonic Form | Rsin(x+α) 与辅助角形式
the addition formulae. By expanding R sin(θ + α) and comparing coefficients, we obtain R = √a² + b² and α = arctan(b/a) (with quadrant adjustments).
虽然这是一个独立主题，但将 a sin θ + b cos θ 写为 R sin(θ + α) 或 R cos(θ - α) 的方法直接依赖于和角公式。通过展开 R sin(θ + α) 并比较系数，我们得到 R = √a² + b² 和 α = arctan(b/a)（需根据象限调整）。
e a sin θ + b cos θ = c, as well as for finding maximum and minimum values of expressions.
例如，3 sin θ + 4 cos θ 可表示为 5 sin(θ + 53.1°)。这种转化对于求解 a sin θ + b cos θ = c 类方程，以及求表达式的最大值和最小值都非常有效。
Edexcel often combines harmonic form with addition formulae in a single question, so being comfortable with both is vital.
Edexcel 常在一道题中综合考查辅助角形式与和角公式，因此熟练掌握两者至关重要。
10. Common Mistakes and Tips | 常见错误与提示
Even confident students can slip up with addition formulae. Here are some typical pitfalls:
即使自信的学生也可能在和角公式上出错。以下是一些常见陷阱：
- Confusing the sign in cosine addition: remember 'cos(A + B) = cos A cos B - sin A sin B', sign opposite.
t formulae or misplacing the minus sign: tan(A + B) = (tan A + tan B) / (1 - tan A tan B).
- Using degrees and radians inconsistently when evaluating special angles.
- Applying the formula when one of the angles is not known explicitly – have enough information (e.g., sin A = 3/5 and cos B = 12/13) to find the missing ratios.
- 混淆余弦和角公式的符号：牢记 'cos(A + B) = cos A cos B - sin A sin B'，符号相反。
- 忘记正切公式的分母，或弄错减号位置：tan(A + B) = (tan A + tan B) / (1 - tan A tan B)。
- 在计算特殊角时，度数与弧度混用。
- 当其中一个角度不明确时仍生搬硬套公式——务必先检查是否有足够信息（如 sin A = 3/5 且 cos B = 12/13）求出缺失的三角比。
A solid tip is to write out the full formula before substituting numbers; this reduces sign errors. Also, double-check the quadrant of the resulting angle when using inverse functions.
一个可靠的技巧是，先完整写出公式再代入数字，这样可以减少符号错误。另外，在运用反函数时，一定要复查结果角所在的象限。
Practice regularly with past paper questions – the more you use the addition formulae, the more second nature they become.
定期练习往年真题——和角公式用得越多，它们就越会成为你的第二天性。
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