📚 IB Mathematics: Compound Angle Identities Explained | IB数学：复合角恒等式解析
Compound angle identities are a cornerstone of trigonometry in the IB Mathematics curriculum. They allow us to express trigonometric functions of sums or differences of angles in terms of functions of the individual angles, enabling simplifications, exact value calculations, and proofs of many other identities.
复合角恒等式是 IB 数学课程中三角学的基石。它们让我们能够将和角或差角的三角函数用单个角的三角函数表达，从而简化表达式、计算精确值并证明许多其他恒等式。
1. What Are Compound Angle Identities? | 复合角恒等式简介
Compound angle identities apply to expressions such as sin(A+B), cos(A−B) and tan(A+B). They break these into products and sums of sines and cosines of A and B. These identities are valid for all angles measured in radians or degrees, provided the expressions are defined.
复合角恒等式适用于诸如 sin(A+B)、cos(A−B) 和 tan(A+B) 的表达式。它们将这些表达式分解为 A 和 B 的正弦、余弦的乘积与和。这些恒等式对所有以弧度或角度度量的角都成立，只要表达式有意义。
The six fundamental compound angle identities are summarised below.
下面总结了六个基本的复合角恒等式。
| Identity | Formula |
|---|---|
| sin(A+B) | sinA cosB + cosA sinB |
| sin(A−B) | sinA cosB − cosA sinB |
| cos(A+B) | cosA cosB − sinA sinB |
| cos(A−B) | cosA cosB + sinA sinB |
| tan(A+B) | (tanA + tanB) / (1 − tanA tanB) |
| tan(A−B) | (tanA − tanB) / (1 + tanA tanB) |
2. Deriving sin(A+B) | 推导 sin(A+B)
One elegant way to derive the formula for sin(A+B) is to compare the entries of the rotation matrix product R
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