📚 Compound Angle Formulae | 复合角公式
In trigonometry, compound angle formulae allow us to express the sine, cosine, and tangent of sums or differences of two angles in terms of the trigonometric functions of the individual angles. These identities are fundamental for simplifying expressions, solving equations, and proving more complex relationships in mathematics and physics. Mastering them is essential for success in IB Mathematics: Analysis and Approaches and Applications and Interpretation.
在三角学中，复合角公式使我们能够将两个角之和或差的正弦、余弦和正切表示为单个角的三角函数的组合。这些恒等式是化简表达式、求解方程以及证明数学和物理学中更复杂关系的基础。掌握它们对于IB数学（分析与方法、应用与解释）的成功至关重要。
1. The Sine Addition and Subtraction Formulae | 正弦和差公式
For any angles A and B, the sine of the sum and the sine of the difference are given by: sin(A + B) = sin A cos B + cos A sin B sin(A - B) = sin A cos B - cos A sin B. Notice the sign pattern: the sign between the terms matches the sign inside the bracket.
对于任意角A和B，和角与差角的正弦公式为： sin(A + B) = sin A cos B + cos A sin B sin(A - B) = sin A cos B - cos A sin B。 注意符号规律：两项之间的符号与括号内的符号一致。
2. The Cosine Addition and Subtraction Formulae | 余弦和差公式
y different because the sign between the terms is opposite to the sign inside the bracket: cos(A + B) = cos A cos B - sin A sin B cos(A - B) = cos A cos B + sin A sin B. In words, 'cos cos, sin sin; opposite sign.'
余弦的复合角公式略有不同，因为两项之间的符号与括号内的符号相反： cos(A + B) = cos A cos B - sin A sin B cos(A - B) = cos A cos B + sin A sin B。 可记作：“cos cos, sin sin; 符号相反。”
3. The Tangent Addition and Subtraction Formulae | 正切和差公式
The tangent of a sum or difference can be expressed in terms of tan A and tan B: tan(A + B) = (tan A + tan B) / (1 - tan A tan B) tan(A - B) = (tan A - tan B) / (1 + tan A tan B). alid provided tan A, tan B, and tan(A ± B) are defined (i.e., no denominator equals zero).
和角与差角的正切可以用tan A和tan B表示： tan(A + B) = (tan A + tan B) / (1 - tan A tan B) tan(A - B) = (tan A - tan B) / (1 + tan A tan B)。 这些公式在tan A、tan B和tan(A ± B)均有定义（即分母不为零）时成立。
4. Derivation of sin(A ± B) Using a Geometric Approach | 用几何方法推导sin(A ± B)
(A - B), replace B with -B and use even/odd properties: sin(-B) = -sin B, cos(-B) = cos B.
对sin(A + B)的一种常见推导从两个共享一条边的直角三角形开始。设想角A和角B的放置使得组合角为A + B。通过用两种方式表示垂直高度并匹配分量，便得到sin(A + B) = sin A cos B + cos A sin B。对于sin(A - B)，将B替换为 -B，并利用奇偶性：sin(-B) = -sin B, cos(-B) = cos B。
5. Derivation of cos(A ± B) from Geometric Symmetry | 由几何对称性推导cos(A ± B)
o expressions gives cos(A - B) = cos A cos B + sin A sin B. For cos(A + B), substitute -B.
cos(A - B)的一个经典证明利用单位圆上的距离公式。考虑点P(cos A, sin A)和Q(cos B, sin B)。距离PQ的平方等于2 - 2 cos(A - B)。通过旋转坐标系使Q映射到(1,0)，同样的距离给出2 - 2 (cos A cos B + sin A sin B)。将两式等同便得到cos(A - B) = cos A cos B + sin A sin B。对于cos(A + B)，用 -B 代入。
6. Deriving tan(A ± B) from sin and cos Formulae | 从正弦余弦公式导出tan(A ± B)
an A + tan B) / (1 - tan A tan B). The same procedure with the difference leads to the corresponding formula.
由于tan(A + B) = sin(A + B) / cos(A + B)，我们可以代入复合表达式并将分子分母同除以cos A cos B（假设非零）。这样便得到tan(A + B) = (tan A + tan B) / (1 - tan A tan B)。对差角使用相同步骤即可得到相应公式。
7. Double Angle Formulae as Special Cases | 二倍角公式作为特例
When B = A, the compound angle formulae simplify to the double angle identities:
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)
These are extremely useful for integration, solving equations, and geometric proofs. In IB, you may need to choose the appropriate form of cos 2A depending on the context.当B = A时，复合角公式简化为二倍角恒等式。它们在积分、解方程以及几何证明中极其有用。在IB考试中，你可能需要根据上下文选择合适的cos 2A形式。
8. Half-Angle Formulae and Connections | 半角公式及相关联系
A) / sin A. These are not explicitly on the IB formula booklet but can be derived quickly.
通过重新整理余弦的二倍角公式，我们可以得到半角恒等式。例如，由cos 2θ = 1 - 2 sin² θ，令2θ = A，即得sin² (A/2) = (1 - cos A)/2。类似地，cos² (A/2) = (1 + cos A)/2。正切的半角公式为 tan(A/2) = sin A / (1 + cos A) = (1 - cos A) / sin A。这些虽未直接列于IB公式手册，但可以快速推导。
9. Simplifying Trigonometric Expressions | 化简三角表达式
° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°. Using exact values, this becomes (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4. Similar simplifications apply to expressions involving variables, where an expansion reveals a simpler form.
复合角公式能够将诸如 sin 75° 的表达式拆分为已知角：sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°。代入精确值得 (√2/2)(√3/2) + (√2/2)(1/2) = √6/4 + √2/4 = (√6 + √2)/4。类似的化简也适用于含有变量的表达式，展开后往往会呈现更简单的形式。
10. Solving Trigonometric Equations | 求解三角方程
ns within a specified interval. Compound and double angle identities often turn complex-looking equations into factorable forms.
诸如 sin 2x = cos x 的方程可通过将 sin 2x 写作 2 sin x cos x 来处理，然后整理得：2 sin x cos x - cos x = 0，cos x (2 sin x - 1) = 0。由此得出 cos x = 0 或 sin x = 1/2，进而在指定区间内求解。复合角与二倍角恒等式常能将看似复杂的方程转化为可因式分解的形式。
11. Proving More Advanced Identities | 证明更高级的恒等式
ddition formula, and then replacing sin 2A and cos 2A with their double angle forms. A methodical approach: rewrite the left side using compound formulae, simplify, and match the right side.
许多IB证明题需要处理角度和差。例如，证明 sin 3A = 3 sin A - 4 sin³ A，可通过将 sin 3A 写作 sin(2A + A)，应用正弦和角公式，然后用二倍角形式替代 sin 2A 和 cos 2A。有条理的方法是：利用复合角公式重写左边，化简，并匹配右边。
12. Common Mistakes and Tips for Exams | 常见错误与考试提示
Students often forget the sign reversal in cosine formulae – using the same sign for both terms. Another frequent error is misapplying the tangent formula: the middle sign in the denominator must be the opposite of the numerator's operation. When dealing with radians, always switch your calculator mode accordingly. In exact value questions, do not approximate: leave answers in surd form. Practise deriving one formula from another to reduce reliance on the formula booklet.
学生常忘记余弦公式中的符号反转——对两项使用相同的符号。另一个常见错误是误用正切公式：分母中间的符号必须与分子的运算符号相反。处理弧度时，务必相应切换计算器模式。在求精确值的题目中，不要取近似值：将答案保留为根式形式。练习从某个公式推导另一个公式，以减少对公式手册的依赖。
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