A-Level Mathematics: Essential Trigonometric Identities | A-Level 数学：常用三角恒等式归纳

📚 A-Level Mathematics: Essential Trigonometric Identities | A-Level 数学：常用三角恒等式归纳

Trigonometric identities are one of the most powerful tools in A-Level Mathematics. They allow us to simplify expressions, solve equations, prove further identities, and evaluate integrals and derivatives with ease. This guide consolidates the essential trigonometric identities you need for the Edexcel A-Level syllabus, organised by type and usage.

三角恒等式是 A-Level 数学中最强大的工具之一。它们帮助我们化简表达式、求解方程、证明更复杂的恒等式，并轻松处理积分与求导。本篇文章按照类型和用途，系统归纳 Edexcel A-Level 考纲中你必须掌握的常用三角恒等式。

1. Fundamental Pythagorean Identities | 基本毕达哥拉斯恒等式

The most fundamental identity in trigonometry is derived from the unit circle: for any angle θ, the square of sine plus the square of cosine equals one.

三角学中最基本的恒等式来源于单位圆：对于任意角 θ，正弦的平方加余弦的平方等于 1。

sin²θ + cos²θ ≡ 1

Dividing this identity by cos²θ gives a second useful form.

将该恒等式两边同时除以 cos²θ，得到第二个常用形式。

tan²θ + 1 ≡ sec²θ

Similarly, dividing by sin²θ yields the third Pythagorean identity.

同理，两边除以 sin²θ 得到第三个毕达哥拉斯恒等式。

1 + cot²θ ≡ csc²θ

These three identities are essential for simplifying expressions and solving trigonometric equations. In exam questions, recognising when to substitute sin²θ = 1 − cos²θ or cos²θ = 1 − sin²θ is a key skill.

这三个恒等式对于化简表达式和求解三角方程至关重要。在考试中，识别何时代入 sin²θ = 1 − cos²θ 或 cos²θ = 1 − sin²θ 是关键技能。

2. Compound Angle Formulas | 和角与差角公式

The compound angle formulas express trigonometric functions of sums and differences of angles in terms of functions of the individual angles.

和角与差角公式将两个角的和或差的三角函数，表示为各角三角函数的组合。

sin(A ± B) ≡ sinA cosB ± cosA sinB

cos(A ± B) ≡ cosA cosB ∓ sinA sinB

tan(A ± B) ≡ (tanA ± tanB) / (1 ∓ tanA tanB)

Pay close attention to the sign changes: for cosine, the sign inside the argument is opposite to the sign on the right-hand side. For example, cos(A + B) = cosA cosB − sinA sinB, while cos(A − B) = cosA cosB + sinA sinB.

特别注意符号变化：对于余弦，括号内的符号与右侧符号相反。例如，cos(A + B) = cosA cosB − sinA sinB，而 cos(A − B) = cosA cosB + sinA sinB。

These formulas are widely used to derive double-angle identities and to evaluate exact trigonometric values such as sin75° = sin(45° + 30°).

这些公式广泛用于推导倍角恒等式，以及计算精确三角值，例如 sin75° = sin(45° + 30°)。

3. Double Angle Identities | 二倍角公式

Setting A = B in the compound angle formulas produces the double-angle identities. These appear frequently in integration, differentiation, and equation solving.

在和角公式中令 A = B，即可得到二倍角公式。这些公式在积分、求导和方程求解中频繁出现。

sin2θ ≡ 2 sinθ cosθ

cos2θ ≡ cos²θ − sin²θ ≡ 2cos²θ − 1 ≡ 1 − 2sin²θ

tan2θ ≡ (2 tanθ) / (1 − tan²θ)

The three equivalent forms of cos2θ are especially important. Choosing the correct form often simplifies an integral or an equation significantly.

cos2θ 的三种等价形式尤其重要。选择正确的形式往往能大幅简化积分或方程。

For example, the identity cos2θ = 1 − 2sin²θ can be rearranged to express sin²θ in terms of cos2θ.

例如，恒等式 cos2θ = 1 − 2sin²θ 可以变形，用 cos2θ 表示 sin²θ。

sin²θ ≡ (1 − cos2θ) / 2

cos²θ ≡ (1 + cos2θ) / 2

These half-angle forms are essential for integrating powers of sine and cosine.

这两个半角形式对于对正弦、余弦的幂进行积分必不可少。

4. Harmonic Form: R cos(θ ± α) and R sin(θ ± α) | 辅助角公式：R cos(θ ± α) 与 R sin(θ ± α)

Expressions of the form a sinθ + b cosθ can be rewritten as a single trigonometric function. This transformation is known as the harmonic form or the R-formula.

形如 a sinθ + b cosθ 的表达式可以改写为单个三角函数。这种变换称为辅助角公式或 R 公式。

a sinθ + b cosθ ≡ R sin(θ + α)

a sinθ − b cosθ ≡ R sin(θ − α)

a cosθ + b sinθ ≡ R cos(θ − α)

a cosθ − b sinθ ≡ R cos(θ + α)

where R = √(a² + b²) and α is found from tanα = b/a, with the quadrant determined by the signs of a and b.

其中 R = √(a² + b²)，α 由 tanα = b/a 确定，具体象限由 a 和 b 的符号决定。

This form is particularly useful for finding maximum and minimum values, solving equations of the form a sinθ + b cosθ = c, and sketching graphs.

这种形式特别适用于求最大值和最小值、求解 a sinθ + b cosθ = c 形式的方程，以及画函数图像。

5. Product-to-Sum and Sum-to-Product Identities | 积化和差与和差化积公式

These identities convert products of trigonometric functions into sums or differences, and vice versa. They are useful in solving certain equations and in integration.

这类恒等式将三角函数的乘积转换为和或差，反之亦然。它们在求解特定方程和积分中非常有用。虽然 Edexcel 考纲不强制要求记忆，但在进阶题目中可能用作潜在工具。

2 sinA cosB ≡ sin(A + B) + sin(A − B)

2 cosA sinB ≡ sin(A + B) − sin(A − B)

2 cosA cosB ≡ cos(A + B) + cos(A − B)

2 sinA sinB ≡ cos(A − B) − cos(A + B)

The corresponding sum-to-product forms are obtained by substitution:

对应的和差化积形式通过代入得到：

sinX + sinY ≡ 2 sin((X + Y)/2) cos((X − Y)/2)

sinX − sinY ≡ 2 cos((X + Y)/2) sin((X − Y)/2)

cosX + cosY ≡ 2 cos((X + Y)/2) cos((X − Y)/2)

cosX − cosY ≡ −2 sin((X + Y)/2) sin((X − Y)/2)

6. Small Angle Approximations | 小角近似

For small angles measured in radians, trigonometric functions can be approximated by simple polynomials. These approximations are valid when θ is close to zero.

对于以弧度为单位的小角，三角函数可以用简单的多项式近似。当 θ 接近 0 时，这些近似成立。

sinθ ≈ θ − θ³/6

cosθ ≈ 1 − θ²/2

tanθ ≈ θ + θ³/3

In Edexcel A-Level, you are expected to know the first-order approximations.

在 Edexcel A-Level 中，主要要求掌握一阶近似。

sinθ ≈ θ, tanθ ≈ θ, cosθ ≈ 1 − θ²/2

These are derived from Maclaurin series and are essential in mechanics, particularly in the simple pendulum model and small oscillation problems.

这些近似由麦克劳林级数推导而来，在力学中尤其重要，例如单摆模型和小振动问题。

7. Inverse Trigonometric Functions | 反三角函数

The inverse functions arcsin x, arccos x, and arctan x are used to solve trigonometric equations for specific angle ranges.

反三角函数 arcsin x、arccos x 和 arctan x 用于在特定角度范围内求解三角方程。

y = arcsin x ⇒ x = sin y, −π/2 ≤ y ≤ π/2

y = arccos x ⇒ x = cos y, 0 ≤ y ≤ π

y = arctan x ⇒ x = tan y, −π/2 < y < π/2

These restricted domains ensure that each inverse function is single-valued. A useful identity for integration is the derivative of arctan x.

这些限制区间确保每个反函数都是单值的。积分中一个有用的恒等式是 arctan x 的导数。

d/dx (arctan x) = 1 / (1 + x²)

d/dx (arcsin x) = 1 / √(1 − x²)

d/dx (arccos x) = −1 / √(1 − x²)

These derivatives are essential for integrating rational functions of the form 1/(a² + x²) and 1/√(a² − x²).

这些导数对于积分形如 1/(a² + x²) 和 1/√(a² − x²) 的有理函数至关重要。

8. Secant, Cosecant and Cotangent | 正割、余割与余切

These three reciprocal trigonometric functions are defined as follows:

这三个倒数三角函数定义如下：

secθ ≡ 1/cosθ, cscθ ≡ 1/sinθ, cotθ ≡ 1/tanθ = cosθ/sinθ

Their derivatives are also essential in calculus:

它们的导数在微积分中同样重要：

d/dx (sec x) = sec x tan x

d/dx (csc x) = −csc x cot x

d/dx (cot x) = −csc² x

These derivatives are used extensively when integrating trigonometric expressions, especially those involving tan x and sec x.

这些导数在积分三角表达式时被广泛使用，尤其是涉及 tan x 和 sec x 的表达式。

Moreover, the integral of tan x is expressed using sec x:

此外，tan x 的积分用 sec x 表示：

∫ tan x dx = ln|sec x| + C

∫ sec x dx = ln|sec x + tan x| + C

9. Key Identities for Integration | 积分常用恒等式

Several trigonometric identities are specifically useful for evaluating integrals. The following conversions are frequently required in Edexcel exam questions.

一些三角恒等式在计算积分时特别有用。以下转换在 Edexcel 考试题目中经常出现。

sin²x = (1 − cos2x)/2 replaces a squared sine function with a linear combination of cos doubling the angle.

sin²x = (1 − cos2x)/2 将正弦平方转化为含二倍角余弦的线性组合
cos²x = (1 + cos2x)/2 similarly simplifies integrals of cos²x.

cos²x = (1 + cos2x)/2 同样简化 cos²x 的积分
sin³x = (3sinx − sin3x)/4 is used for higher powers.

sin³x = (3sinx − sin3x)/4 用于更高次幂

For integrals of products like sinmx cosnx, use the product-to-sum identities to rewrite them as sums of sines or cosines.

对于 sinmx cosnx 这类乘积的积分，使用积化和差公式将其转换为正弦或余弦之和。

Another common technique is the t-substitution, where t = tan(x/2), which transforms any rational trigonometric expression into a rational algebraic expression.

另一个常用技巧是 t 代换，令 t = tan(x/2)，将任意有理三角表达式转化为有理代数表达式。

sin x = 2t / (1 + t²), cos x = (1 − t²) / (1 + t²), dx = 2 dt / (1 + t²)

10. Common Exam Pitfalls and Tips | 常见考试陷阱与技巧

Trigonometric identities require careful attention to signs, domains, and the correct form of an identity for the given context. Below are common pitfalls and tips.

三角恒等式需要特别注意符号、定义域以及针对具体问题选择正确的形式。以下是常见陷阱与技巧。

Always check the range of the angle when solving equations. Extra solutions may appear when squaring both sides.

求解方程时始终检查角度范围。两边平方可能会产生增根。
When using tan(A + B), remember that the formula is undefined when tanA tanB = 1.

使用 tan(A + B) 时，注意当 tanA tanB = 1 时公式无定义。
For harmonic form, always verify the quadrant of α; a common error is choosing the wrong angle.

对于辅助角公式，务必验证 α 所在的象限；常见的错误是选错角度。

In proof questions, start with the more complicated side and manipulate it until it matches the simpler side. Never move terms across the equals sign when proving an identity.

在证明题中，从更复杂的一边开始变形，直到与较简单的一边相同。证明恒等式时，绝不要将项移过等号。

11. Summary Table: Essential Identities | 核心恒等式总结表

The table below summarises the essential identities presented in this article for quick revision.

下表总结了本文介绍的核心恒等式，便于快速复习。

Identity Type	Formula
Pythagorean	sin²θ + cos²θ = 1, tan²θ + 1 = sec²θ, 1 + cot²θ = csc²θ
Compound Angle	sin(A±B) = sinAcosB ± cosAsinB, cos(A±B) = cosAcosB ∓ sinAsinB
Double Angle	sin2θ = 2sinθcosθ, cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
Harmonic Form	asinθ + bcosθ = Rsin(θ + α), R = √(a² + b²)
Product-to-Sum	2sinAcosB = sin(A+B) + sin(A−B)
Small Angle	sinθ ≈ θ, tanθ ≈ θ, cosθ ≈ 1 − θ²/2 (radians)

12. Conclusion | 结语

Mastering trigonometric identities is not about memorising every formula blindly. Understanding the relationships between the identities allows you to derive what you forget and to apply the most efficient method in each situation. Regular practice with past exam questions is the best way to become fluent in using these identities.

掌握三角恒等式不是盲目地记住每一个公式。理解恒等式之间的关系，能让你推导出暂时遗忘的公式，并在每种情况下选择最有效的方法。定期练习历年真题是熟练掌握这些恒等式的最佳途径。

For more revision resources, visit aleveler.com where you will find topic-wise past papers, worked solutions, and comprehensive study notes.

更多复习资源请访问 aleveler.com，那里有分类真题、详细解答和全套学习笔记。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

A-Level Mathematics: Essential Trigonometric Identities | A-Level 数学：常用三角恒等式归纳