📚 Using the Angle Addition Formulae | 使用角度加法公式
The angle addition formulae are a cornerstone of A-Level trigonometry. They allow you to expand expressions like sin(A + B) or cos(A – B) and are vital for solving equations, proving identities, and finding exact values without a calculator. Mastery of these formulae opens the door to double angles, harmonic forms, and more advanced techniques.
角度加法公式是 A-Level 三角学的基础。它们可以将 sin(A+B) 或 cos(A–B) 等表达式展开，对于解方程、证明恒等式和不用计算器求精确值至关重要。掌握这些公式可以让你轻松应对二倍角、调和形式等更高级的技巧。
1. The Core Angle Addition Formulae | 核心角度加法公式
The three fundamental pairs of addition formulae are for sine, cosine, and tangent. Make sure you know the signs carefully – the cosine formula switches the sign between the sum and difference cases.
三组基本的加法公式分别针对正弦、余弦和正切。务必仔细区分符号——余弦公式在和角与差角的情况下符号是相反的。
sin (A ± B) = sin A cos B ± cos A sin B
cos (A ± B) = cos A cos B ∓ sin A sin B
tan (A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
For cosine, note the opposite signs: cos(A + B) uses a minus, cos(A – B) uses a plus. The tangent formula also uses opposite signs in the denominator.
对于余弦，注意符号相反：cos(A+B) 中用减号，cos(A–B) 中用加号。正切公式的分母符号也相反。
You can derive these from the unit circle or geometric proofs, but for the exam, memorising them accurately is essential. In Edexcel exams, these are provided in the formula booklet, but you must be able to use them fluently.
这些公式可以从单位圆或几何证明中推导出来，但在考试中准确记忆至关重要。在爱德思考试中，公式手册会提供这些公式，但你必须能够熟练运用。
2. Finding Exact Values | 求精确值
One of the easiest applications is to express angles like 15°, 75°, 105° as sums or differences of standard angles (30°, 45°, 60°). Then apply the addition formulae to obtain exact surd values.
最简单的应用之一是将 15°、75°、105° 等角度表示为标准角（30°, 45°, 60
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