Double-Angle Formulae | 二倍角公式

📚 Double-Angle Formulae | 二倍角公式

In Edexcel A Level Mathematics, double-angle formulae express trigonometric functions of 2A in terms of functions of A. They are essential tools for solving trigonometric equations, proving identities, simplifying expressions and calculating exact values without finding the original angle.

在爱德思 A Level 数学中，二倍角公式将 2A 的三角函数用 A 的三角函数表示。它们是解三角方程、证明恒等式、化简表达式以及在不求原角的情况下计算精确值的重要工具。

1. The Three Key Formulae | 三个核心公式

For any angle A, the double-angle formulae are:

对于任意角 A，二倍角公式为：

• 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 identities are derived from the compound-angle formulae and are valid in both degrees and radians, provided the functions are defined.

这些恒等式由两角和公式推导而来，在角度制与弧度制下均成立，但需保证所涉及的三角函数有定义。

2. Deriving sin 2A from the Compound-Angle Formula | 由两角和公式推导 sin 2A

Start with the compound-angle formula sin(A + B) = sin A cos B + cos A sin B. Setting B = A gives:

从两角和公式 sin(A + B) = sin A cos B + cos A sin B 出发，令 B = A，得到：

sin 2A = sin A cos A + cos A sin A = 2 sin A cos A

This shows why the factor 2 appears and why both sine and cosine of A are needed. The derivation also reminds you that sin 2A is not the same as 2 sin A.

这解释了为什么会出现系数 2，以及为什么需要同时用到 A 的正弦和余弦。该推导也提醒你，sin 2A 并不等于 2 sin A。

3. Deriving cos 2A and Its Three Versions | 推导 cos 2A 及其三种形式

Using cos(A + B) = cos A cos B − sin A sin B and setting B = A gives:

利用 cos(A + B) = cos A cos B − sin A sin B，并令 B = A，得到：

cos 2A = cos² A − sin² A

Then substitute sin² A = 1 − cos² A or cos² A = 1 − sin² A to obtain two alternative forms:

再代入 sin² A = 1 − cos² A 或 cos² A = 1 − sin² A，可得到另外两种等价形式：

cos 2A = 2 cos² A − 1 = 1 − 2 sin² A

All three forms are equivalent; the best choice depends on the expression you are simplifying or the equation you are solving.

三种形式完全等价；选择哪一种取决于你要化简的表达式或要解的方程。

4. Deriving tan 2A | 推导 tan 2A

From the compound-angle formula tan(A + B) = (tan A + tan B) / (1 − tan A tan B), set B = A:

由两角和公式 tan(A + B) = (tan A + tan B) / (1 − tan A tan B)，令 B = A：

tan 2A = 2 tan A / (1 − tan² A)

Alternatively, you can derive this by dividing sin 2A by cos 2A. Note that tan 2A is undefined when the denominator is zero, that is when tan A = ±1, and the formula also requires tan A itself to be defined.

你也可以用 sin 2A 除以 cos 2A 得到该公式。注意当分母为零时 tan 2A 无定义，即 tan A = ±1 时；同时该公式要求 tan A 本身有定义。

5. Choosing the Right Cosine Form | 选择合适的余弦形式

The three forms of cos 2A are useful in different situations. Choosing the correct form is often the key to a quick solution.

cos 2A 的三种形式在不同情况下各有用途。选择正确的形式往往是快速解题的关键。