Trigonometric Identities | 三角函数恒等式

📚 Trigonometric Identities | 三角函数恒等式

Trigonometric identities are equations involving trigonometric functions that hold true for every value of the variable for which both sides are defined. They are among the most powerful tools in A-Level mathematics, enabling us to simplify expressions, prove equations, and solve trigonometric equations that would otherwise be intractable.

三角函数恒等式是对定义域内所有变量取值均成立的三角方程，它们是 A-Level 数学中最强大的工具之一，帮助我们化简表达式、证明等式，并解决原本难以处理的三角方程。

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1. The Fundamental Identities | 基本恒等式

Every trigonometric identity system begins with the Pythagorean identity, derived directly from the unit circle definition of sine and cosine:

整个恒等式体系源于勾股恒等式，它直接由正弦和余弦的单位圆定义推导而来：

sin²θ + cos²θ ≡ 1

Dividing this equation by cos²θ yields the second Pythagorean identity; dividing by sin²θ yields the third:

将该式两边除以 cos²θ 得到第二个勾股恒等式；除以 sin²θ 得到第三个：

1 + tan²θ ≡ sec²θ,    cot²θ + 1 ≡ csc²θ

Alongside these, we define the quotient and reciprocal identities:

与此并列的是商数恒等式和倒数恒等式：

tan θ ≡ sin θ / cos θ,    cot θ ≡ cos θ / sin θ

sec θ ≡ 1 / cos θ,    csc θ ≡ 1 / sin θ,    cot θ ≡ 1 / tan θ

Note the use of the “identically equal” symbol ≡, which distinguishes an identity from a conditional equation. An identity holds for all θ, whereas a conditional equation such as sin θ = ½ only holds for specific values. In examination work, always ask yourself whether the statement you have written is an identity or an equation — the treatment of the two is fundamentally different.

注意这里使用了”恒等”符号 ≡，用于区分恒等式与条件方程。恒等式对一切 θ 成立，而条件方程（如 sin θ = ½）仅对特定取值成立。在考试作答时，务必问自己写下的语句是恒等式还是方程——两者处理方法截然不同。

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2. Compound Angle Formulas | 复角公式（和角公式）

The compound angle formulas express trigonometric functions of sums or differences of two angles. They are tested repeatedly in A-Level examinations and form the foundation of nearly every other identity in this article:

复角公式将两个角度之和或差的三角函数展开表示。它们在 A-Level 考试中反复出现，是本文几乎所有其他恒等式的基础：

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

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

cos(A + B) = cos A cos B − sin 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)

tan(A − B) = (tan A − tan B) / (1 + tan A tan B)

A useful memory aid is the mnemonic for sin(A + B): “sin cos plus cos sin” — the sign follows the order of the terms. For cosine, the middle sign reverses: “cos cos minus sin sin”.

记忆技巧：sin(A + B) 的口诀是”散扣加扣散”，即 sin cos + cos sin；而 cos(A + B) 中间符号相反，为”扣扣减散散”，即 cos cos − sin sin。

Worked example | 示例：Evaluate sin 75° exactly.

例题：精确求 sin 75° 的值。

sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°

= (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2) / 4

This technique of substituting standard angles is a frequent examination requirement and also appears in questions asking for exact values in surd form.

利用标准角度代入是一种常见的考试要求，也常出现在要求以根式形式给出精确值的问题中。

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3. Double Angle Formulas | 二倍角公式

Setting B = A in the compound angle formulas produces the double angle identities, which are indispensable for solving equations and integrating powers of trigonometric functions:

在复角公式中令 B = A，即得二倍角恒等式。它们在解方程和积分三角函数幂时不可或缺：

sin 2A = 2 sin A cos A

cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A

tan 2A = 2 tan A / (1 − tan²A)

The three equivalent forms of cos 2A are particularly important. Rearranging them yields the power-reduction identities, which convert squared functions into linear functions of twice the angle:

cos 2A 的三种等价形式尤为重要。移项整理即可得到降幂公式，将平方函数化为倍角的线性函数：

cos²A = (1 + cos 2A) / 2,    sin²A = (1 − cos 2A) / 2

These power-reduction identities are essential in A-Level calculus, particularly when integrating expressions such as sin²x or cos²x, which cannot be integrated directly by simple rules.

降幂公式在 A-Level 微积分中至关重要，尤其是在积分 sin²x 或 cos²x 这类无法直接用简单法则积分的表达式时。

Worked example | 示例：Given that sin A = 3/5 and A is acute, find the exact value of cos 2A.

例题：已知 sin A = 3/5，且 A 为锐角，求 cos 2A 的精确值。

cos 2A = 1 − 2 sin²A = 1 − 2(9/25) = 1 − 18/25 = 7/25

Choosing the most convenient form of cos 2A eliminates the need to find cos A first — a valuable shortcut under time pressure.

选择最方便的 cos 2A 形式可以免去先求 cos A 的步骤——这在时间紧张时是宝贵的捷径。

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4. The Harmonic Form | 辅助角形式（R sin 与 R cos）

Expressions of the form a sin θ + b cos θ can always be rewritten as a single sine (or cosine) function with a phase shift. This is known as the harmonic form:

形如 a sin θ + b cos θ 的表达式总可以改写为带相位平移的单一正弦（或余弦）函数，这称为辅助角形式：

a sin θ + b cos θ = R sin(θ + α)

= R cos(θ − β)

where R = √(a² + b²) and the phase angle α (or β) satisfies:

其中 R = √(a² + b²)，相位角 α（或 β）满足：

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