📚 Trigonometric Identities for Edexcel A-Level Maths | 爱德思A-Level数学：三角恒等式
In Edexcel A-Level Mathematics, trigonometric identities are essential tools for simplifying expressions, proving results and solving equations. This article covers the main identities you need to know, from the Pythagorean identity to compound-angle and double-angle formulae, with worked examples and exam strategies. Understanding how to choose and apply the right identity is a key skill for both AS and A-Level Pure Mathematics papers.
在爱德思A-Level数学中，三角恒等式是化简表达式、证明结论和求解方程的核心工具。本文涵盖你需要掌握的主要恒等式，从毕达哥拉斯恒等式到复合角公式和倍角公式，并提供例题和考试策略。理解如何选择并应用正确的恒等式是AS和A-Level纯数学考试中的关键技能。
1. The Pythagorean Identity | 毕达哥拉斯恒等式
The most important trigonometric identity is sin² θ + cos² θ = 1. It holds for every real value of θ and comes directly from the unit circle definition of sine and cosine. This identity is the foundation for many other results and is used constantly in proofs and equation solving.
最重要的三角恒等式是 sin² θ + cos² θ = 1。它对任意实数 θ 都成立，并且直接来源于单位圆上正弦和余弦的定义。这个恒等式是许多其他结论的基础，在证明和方程求解中被频繁使用。
sin² θ + cos² θ = 1
You can rearrange this identity to express sine in terms of cosine, or cosine in terms of sine. These rearrangements are useful when an equation contains both sin θ and cos θ, or when a proof requires a single trigonometric function.
你可以重新整理这个恒等式，用余弦表示正弦，或者用正弦表示余弦。当方程同时包含 sin θ 和 cos θ，或者证明中需要只出现一个三角函数时，这些变形非常有用。
sin² θ = 1 – cos² θ and cos² θ = 1 – sin² θ
In Edexcel exams, you may be asked to prove an identity such as (1 – sin θ)(1 + sin θ) = cos² θ. Expanding the left-hand side gives 1 – sin² θ, which is exactly cos² θ by the Pythagorean identity. Recognising this structure saves time and avoids unnecessary algebraic mistakes.
在爱德思考试中，你可能需要证明诸如 (1 – sin θ)(1 + sin θ) = cos² θ 的恒等式。展开左边得到 1 – sin² θ，根据毕达哥拉斯恒等式，它正好等于 cos² θ。识别这种结构可以节省时间，并避免不必要的代数错误。
2. Tangent and Quotient Identities | 正切与商数恒等式
The tangent function is defined as the ratio of sine to cosine. This quotient identity is extremely useful when simplifying expressions that contain both tangent and sine or cosine. It also allows you to rewrite tan θ in terms of sin θ and cos θ when solving equations.
正切函数定义为正弦与余弦的比值。这个商数恒等式在化简同时包含正切和正弦或余弦的表达式时非常有用。它也允许你在解方程时将 tan θ 改写为 sin θ 和 cos θ。
tan θ = sin θ / cos θ, cos θ ≠ 0
The related cotangent quotient identity is cot θ = cos θ / sin θ, provided sin θ ≠ 0. In Edexcel A-Level work, cot θ is often introduced alongside sec θ and cosec θ, so it is worth becoming fluent in moving between tan θ and cot θ.
相关的余切商数恒等式是 cot θ = cos θ / sin θ，其中 sin θ ≠ 0。在爱德思A-Level课程中，cot θ 通常与 sec θ 和 cosec θ 一起引入，因此熟练地在 tan θ 和 cot θ 之间转换是值得的。
For example, if you need to simplify sin θ / tan θ, replace tan θ with sin θ / cos θ to get sin θ ÷ (sin θ / cos θ) = sin θ × (cos θ / sin θ) = cos θ. This kind of rewriting is common in proof questions.
例如，如果要化简 sin θ / tan θ，将 tan θ 替换为 sin θ / cos θ，得到 sin θ ÷ (sin θ / cos θ) = sin θ × (cos θ / sin θ) = cos θ。这种改写方法在证明题中很常见。
3. Reciprocal Trigonometric Functions | 倒数三角函数
Edexcel A-Level Mathematics introduces three reciprocal trigonometric functions: secant, cosecant and cotangent. They are defined as follows and appear in identities, differentiation and integration.
爱德思A-Level数学引入了三个倒数三角函数：正割、余割和余切。它们的定义如下，并出现在恒等式、微分和积分中。
sec θ = 1 / cos θ, cosec θ = 1 / sin θ, cot θ = 1 / tan θ = cos θ / sin θ
Two further Pythagorean-type identities follow from dividing the main Pythagorean identity by cos² θ or sin² θ. These are very important in calculus and in solving equations involving sec, cosec or cot.
将主要的毕达哥拉斯恒等式除以 cos² θ 或 sin² θ，可以得到另外两个毕达哥拉斯型恒等式。它们在微积分以及求解包含 sec、cosec 或 cot 的方程时非常重要。
1 + tan² θ = sec² θ
1 + cot² θ = cosec² θ
| Identity | Common rearrangements |
|---|---|
| sin² θ + cos² θ = 1 | sin² θ = 1 – cos² θ; cos² θ = 1 – sin² θ |
| 1 + tan² θ = sec² θ | tan² θ = sec² θ – 1; sec² θ – tan² θ = 1 |
| 1 + cot² θ = cosec² θ | cot² θ = cosec² θ – 1; cosec² θ – cot² θ = 1 |
| tan θ = sin θ / cos θ | sin θ = tan θ cos θ; cos θ = sin θ / tan θ |
When proving identities with sec θ or cosec θ, it is often helpful to convert everything into sin θ and cos θ first. For example, to prove sec² θ – 1 = tan² θ, write sec² θ = 1 / cos² θ and tan² θ = sin² θ / cos² θ, then use the Pythagorean identity.
在证明包含 sec θ 或 cosec θ 的恒等式时，通常先把所有表达式都转换为 sin θ 和 cos θ。例如，要证明 sec² θ – 1 = tan² θ，可以写成 sec² θ = 1 / cos² θ 和 tan² θ = sin² θ / cos² θ，然后使用毕达哥拉斯恒等式。
4. Compound-Angle Formulae | 复合角公式
Compound-angle formulae allow you to find the sine, cosine or tangent of a sum or difference of two angles. They are used to derive exact values for angles that are not in the standard triangles, and they also appear in differentiation and integration.
复合角公式允许你求两个角的和或差的正弦、余弦或正切。它们用于推导非标准三角形中角度的精确值，同时也出现在微分和积分中。
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)
For example, sin 75° can be found by writing 75° = 45° + 30°. Using the addition formula gives sin 75° = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.
例如，可以通过将 75° 写成 45° + 30° 来求 sin 75°。使用加法公式得到 sin 75° = sin 45° cos 30° + cos 45° sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4。
When using the tangent addition formula, remember to check that the denominator is not zero. If tan A tan B = 1, the expression is undefined, which corresponds to tan(A + B) being undefined at an odd multiple of π/2.
使用正切加法公式时，记得检查分母是否为零。如果 tan A tan B = 1，表达式就没有定义，这对应于 tan(A + B) 在 π/2 的奇数倍处无定义。
5. Double-Angle Formulae | 倍角公式
The double-angle formulae are obtained by setting A = B in the compound-angle formulae. They are particularly useful for solving equations such as sin 2θ = sin θ, and for integrating powers of sine and cosine.
倍角公式是通过在复合角公式中令 A = B 得到的。它们在求解诸如 sin 2θ = sin θ 的方程以及对正弦和余弦的幂进行积分时特别有用。
sin 2θ = 2 sin θ cos θ
cos 2θ = cos² θ – sin² θ = 2 cos² θ – 1 = 1 – 2 sin² θ
tan 2θ = 2 tan θ / (1 – tan² θ)
There are three equivalent forms for cos 2θ. Choosing the correct form can make an equation much easier to solve. For instance, if an equation contains sin θ and cos 2θ, use cos 2θ = 1 – 2 sin² θ so that the whole equation can be written in terms of sin θ only.
cos 2θ 有三种等价形式。选择正确的形式可以使方程更容易求解。例如，如果方程同时含有 sin θ 和 cos 2θ，使用 cos 2θ = 1 – 2 sin² θ，这样整个方程就可以只用 sin θ 来表示。
Double-angle identities are also used in integration. For example, cos² θ can be rewritten as (1 + cos 2θ)/2, and sin² θ can be rewritten as (1 – cos 2θ)/2, which allows integration of even powers of sine and cosine.
倍角恒等式也用于积分。例如，cos² θ 可以改写为 (1 + cos 2θ)/2，sin² θ 可以改写为 (1 – cos 2θ)/2，这样就可以对正弦和余弦的偶次幂进行积分。
6. R-Formulae, Harmonic Form | R公式与辅助角形式
The R-formulae, also called harmonic form, let you rewrite an expression of the form a sin θ ± b cos θ as a single sine or cosine function. This is extremely useful for finding maximum and minimum values and for solving equations of the form a sin θ + b cos θ = c.
R公式，也称为辅助角形式，允许你将 a sin θ ± b cos θ 这类表达式改写为单个正弦或余弦函数。这在求最大值和最小值以及求解 a sin θ + b cos θ = c 形式的方程时非常有用。
a sin θ + b cos θ = R sin(θ + α), R = √(a² + b²), tan α = b / a
a sin θ – b cos θ = R sin(θ – α), R = √(a² + b²), tan α = b / a
a cos θ + b sin θ = R cos(θ – α), R = √(a² + b²), tan α = b / a
a cos θ – b sin θ = R cos(θ + α), R = √(a² + b²), tan α = b / aPublished by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
屏轩国际教育cambridge primary/secondary checkpoint, cat4, ukiset,ukcat,igcse,alevel,PAT,STEP,MAT, ibdp,ap,ssat,sat,sat2课程辅导，国外大学本科硕士研究生博士课程论文辅导