Addition Formulae in Trigonometry | 三角函数加法公式

📚 Addition Formulae in Trigonometry | 三角函数加法公式

In Edexcel A-Level Mathematics, the addition formulae are essential trigonometric identities used to expand sin(A ± B), cos(A ± B) and tan(A ± B).

在 Edexcel A-Level 数学中，加法公式是核心的三角恒等式，用于展开 sin(A ± B)、cos(A ± B) 和 tan(A ± B)。

They appear in proofs, exact value calculations and equation solving, so a fluent command of them is required for high marks.

它们出现在证明、精确值计算和解方程中，因此熟练运用这些公式是取得高分的关键。

1. The Addition Formulae at a Glance | 加法公式一览

Below are the four forms you must memorise for the Edexcel specification.

以下是 Edexcel 考试大纲要求记住的四个公式。

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)

The ∓ notation means that when the left side has a plus sign, the right side has a minus sign, and vice versa.

符号 ∓ 表示当左边取加号时，右边对应取减号，反之亦然。

These identities hold for any real angles A and B, provided the tangent expressions are defined.

这些恒等式对任意实数角 A 和 B 都成立，但正切表达式需要有定义。

2. Geometric Derivation of cos(A − B) | cos(A − B) 的几何推导

To understand why the formulae work, start with cos(A − B).

为了理解公式的来源，先从 cos(A − B) 开始。

Take two points P and Q on the unit circle, where P has angle A and Q has angle B. Their coordinates are P = (cos A, sin A) and Q = (cos B, sin B).

在单位圆上取两点 P 和 Q，P 的角度为 A，Q 的角度为 B，则坐标为 P = (cos A, sin A)，Q = (cos B, sin B)。

The square of the distance PQ can be written in two equivalent ways. First, directly from coordinates:

距离 PQ 的平方可以用两种等价方式表示。首先，直接用坐标表示：

PQ² = (cos A − cos B)² + (sin A − sin B)²

Expanding the right side gives 2 − 2(cos A cos B + sin A sin B).

展开右边得到 2 − 2(cos A cos B + sin A sin B)。

After rotating the diagram by −B, the same distance is between the point at angle A − B and the point (1, 0), so:

将图形旋转 −B 后，同样的距离变为角度 A − B 的点与 (1, 0) 之间的距离，因此：

PQ² = (cos(A − B) − 1)² + (sin(A − B) − 0)²

Expanding this gives 2 − 2 cos(A − B).

展开后得到 2 − 2 cos(A

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