IB Math: Product-to-Sum and Sum-to-Product Trigonometric Identities | IB数学：三角函数的积化和差与和差化积

📚 IB Math: Product-to-Sum and Sum-to-Product Trigonometric Identities | IB数学：三角函数的积化和差与和差化积

Trigonometric identities form a cornerstone of the IB Mathematics Analysis and Approaches (AA) and Applications and Interpretation (AI) syllabi. Among the most powerful yet frequently overlooked tools are the product-to-sum and sum-to-product identities, which convert products of sines and cosines into sums and vice versa. These identities simplify integrals, solve trigonometric equations, and appear in real-world modeling tasks.

三角函数恒等式是IB数学分析与方法（AA）以及应用与解释（AI）课程中的核心内容。积化和差与和差化积公式是最强大却常被忽视的工具之一，它们将正弦和余弦的乘积转化为和或差，反之亦然。这些公式能简化积分运算、求解三角方程，并应用于现实建模问题中。

In the IB Data Booklet, these formulas are printed for reference, but knowing how to apply them fluently and derive them quickly is what separates a strong candidate from an average one. This guide covers both sets of identities, their derivations, worked examples, and common pitfalls.

在IB数据手册中，这些公式以参考形式列出，但能否熟练应用并快速推导它们，正是区分优秀考生与一般考生的关键。本指南将涵盖两组恒等式、它们的推导过程、应用例题及常见误区。

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1. Why These Formulas Matter | 为什么这些公式很重要

In IB exams, you will meet expressions such as sin 75° cos 15° or sin 5θ + sin 3θ. Directly evaluating these using standard angle formulas is possible, but it is time-consuming and error-prone. The product-to-sum and sum-to-product identities collapse such expressions into single sine or cosine terms, making differentiation, integration, and equation solving far more efficient.

在IB考试中，你会遇到像 sin 75° cos 15° 或 sin 5θ + sin 3θ 这样的表达式。直接使用标准角公式计算固然可行，但既耗时又容易出错。积化和差与和差化积公式能将此类表达式压缩为单一的正弦或余弦项，从而大幅提高求导、积分和方程求解的效率。

Furthermore, these identities reveal the underlying relationship between trigonometric functions of different arguments, a theme that appears throughout the IB syllabus and past-paper questions. In particular, IB AA HL students will use them repeatedly in calculus questions involving integrals of products of trig functions. Mastery of these formulas is therefore not optional for top marks.

此外，这些公式揭示了不同角度三角函数之间的内在联系，这一主题贯穿IB教学大纲和历年真题。特别是IB AA HL学生在涉及三角函数乘积积分的微积分题目中会反复用到它们。因此，掌握这些公式并非可选项，而是获取高分的关键。

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2. Product-to-Sum Formulas | 积化和差公式

There are four product-to-sum identities. For any angles A and B:

积化和差公式共有四个。对于任意角 A 和 B：

sin A cos B = ½[sin(A + B) + sin(A − B)]

cos A sin B = ½[sin(A + B) − sin(A − B)]

cos A cos B = ½[cos(A + B) + cos(A − B)]

sin A sin B = ½[cos(A − B) − cos(A + B)]

Notice that the sine-sine formula has a minus sign and writes cos(A − B) first. This asymmetry is a common source of error, so pay special attention to it. The factor of ½ appears in every product-to-sum formula; forgetting it is one of the most frequent mistakes in IB exams.

请注意，sin A sin B 的公式带有负号，并且将 cos(A − B) 放在前面。这种不对称性常常导致错误，请特别留意。系数 ½ 出现在每一个积化和差公式中；遗漏它是IB考试中最常见的错误之一。

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3. Derivation of Product-to-Sum | 积化和差公式的推导

All four identities follow directly from the addition formulas for sine and cosine. Begin with:

这四条公式都可以直接由正弦和余弦的和角公式推导得出。首先写出：

sin(A + B) = sin A cos B + cos A sin B

sin(A − B) = sin A cos B − cos A sin B

Adding these two equations gives 2 sin A cos B = sin(A + B) + sin(A − B), which yields the first product-to-sum formula. Subtracting the second from the first gives 2 cos A sin B = sin(A + B) − sin(A − B).

将两式相加可得 2 sin A cos B = sin(A + B) + sin(A − B)，即第一个积化和差公式。用第一式减去第二式则得 2 cos A sin B = sin(A + B) − sin(A − B)。

Similarly, using the cosine addition formulas:

类似地，使用余弦的和角公式：

cos(A + B) = cos A cos B − sin A sin B

cos(A − B) = cos A cos B + sin A sin B

Adding yields the identity for cos A cos B; subtracting the first from the second and rearranging yields the identity for sin A sin B. Since these derivations are short, IB students are encouraged to reconstruct the formulas rather than memorize them blindly. In an exam, even if your memory fails, you can always return to the addition formulas.

相加即得 cos A cos B 的公式；用第二式减去第一式并整理则得 sin A sin B 的公式。由于推导过程简短，IB学生应当学会自行推导，而不是死记硬背。在考试中，即使记忆模糊，你总可以回到和角公式重新推出。

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4. Sum-to-Product Formulas | 和差化积公式

The sum-to-product identities express sums or differences of sine or cosine as products. Let x and y be any angles:

和差化积公式将正弦或余弦的和与差转化为乘积形式。设 x 和 y 为任意角：

sin x + sin y = 2 sin((x + y)/2) cos((x − y)/2)

sin x − sin y = 2 cos((x + y)/2) sin((x − y)/2)

cos x + cos y = 2 cos((x + y)/2) cos((x − y)/2)

cos x − cos y = −2 sin((x + y)/2) sin((x − y)/2)

The last formula is the most error-prone because of the leading negative sign. For example, cos 100° − cos 40° equals a negative quantity, which matches the sign of −2 sin 70° sin 30°. Note that all four formulas have a factor of 2 and use half-angles (x + y)/2 and (x − y)/2.

最后一个公式最易出错，因为其开头带有负号。例如，cos 100° − cos 40° 等于一个

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