Trigonometric Identities and Equations: Edexcel A-Level Pure Maths — 三角恒等式与方程：Edexcel A-Level 纯数学

Introduction | 引言

Trigonometric identities and equations form one of the most important topic areas in Edexcel A-Level Pure Mathematics. They appear consistently across Papers 1 and 2 of the Edexcel specification and are essential for success in topics ranging from calculus and coordinate geometry to vectors and complex numbers. Mastering trigonometric identities is not just about memorising formulas – it is about developing the algebraic fluency to recognise when and how to apply them in a variety of contexts.

三角恒等式与方程是 Edexcel A-Level 纯数学中最重要的专题之一。它们在 Edexcel 考试大纲的卷一和卷二中持续出现，对于从微积分和坐标几何到向量和复数等主题的成功至关重要。掌握三角恒等式不仅仅是记住公式 – 更重要的是培养代数流畅度，能够在各种情境中识别何时以及如何应用它们。

The Fundamental Trigonometric Identities | 基本三角恒等式

Before tackling complex equations, you must be completely comfortable with the foundational identities. The most basic identity, derived directly from the unit circle definition of sine and cosine, is the Pythagorean identity: sin squared theta plus cos squared theta equals one. This identity is so fundamental that it appears in virtually every trigonometric problem at A-Level, either explicitly or implicitly.

在攻克复杂方程之前，你必须完全熟悉基础恒等式。最基本的恒等式直接来源于单位圆对正弦和余弦的定义，即勾股恒等式：sin平方θ加cos平方θ等于1。这个恒等式如此基础，以至于它在A-Level的几乎每一个三角问题中都会出现，无论是显式的还是隐式的。

From this central identity, we can derive two additional forms by dividing through by cos squared theta or sin squared theta respectively. Dividing by cos squared theta yields: 1 plus tan squared theta equals sec squared theta. This form is particularly useful when an expression contains both tangent and secant functions, or when you need to convert between them. Dividing by sin squared theta gives: 1 plus cot squared theta equals cosec squared theta. While less commonly used, this third form is invaluable for problems involving cotangent and cosecant, which appear in the later stages of the Edexcel course and in Further Mathematics.

从这个核心恒等式出发，我们可以分别除以cos平方θ或sin平方θ来推导出另外两种形式。除以cos平方θ得到：1加tan平方θ等于sec平方θ。当表达式同时包含正切和正割函数，或者需要在它们之间转换时，这种形式特别有用。除以sin平方θ得到：1加cot平方θ等于cosec平方θ。虽然较少使用，但这第三种形式对于涉及余切和余割的问题非常宝贵，这些在Edexcel课程的后期阶段和进阶数学中会出现。

Compound Angle Formulae | 复合角公式

The compound angle formulae are the gateway to advanced trigonometry at A-Level. There are six key formulae to memorise, but they all follow logical patterns. For sine: sin of A plus B equals sin A cos B plus cos A sin B, and sin of A minus B equals sin A cos B minus cos A sin B. Notice how the sign between the terms matches the sign in the angle. For cosine: cos of A plus B equals cos A cos B minus sin A sin B, and cos of A minus B equals cos A cos B plus sin A sin B. Here, the sign between the terms is opposite to the sign in the angle – a common source of errors for students.

复合角公式是A-Level高级三角学的入口。有六个关键公式需要记忆，但它们都遵循逻辑模式。对于正弦：sin(A+B)等于sinAcosB加cosAsinB，sin(A-B)等于sinAcosB减cosAsinB。注意项之间的符号与角中的符号一致。对于余弦：cos(A+B)等于cosAcosB减sinAsinB，cos(A-B)等于cosAcosB加sinAsinB。这里项之间的符号与角中的符号相反 – 这是学生常犯错误的来源。

The tangent compound angle formula can be derived from the sine and cosine versions by dividing: tan of A plus B equals tan A plus tan B all over 1 minus tan A tan B. This formula is especially useful in coordinate geometry problems where you need to find the angle between two lines, or in problems involving the argument of a complex number in Further Mathematics. Edexcel examiners frequently test the application of compound angle formulae in unfamiliar contexts, such as proving that a given expression simplifies to a known value.

正切复合角公式可以通过除法从正弦和余弦版本推导出来：tan(A+B)等于(tanA加tanB)除以(1减tanAtanB)。这个公式在坐标几何问题中特别有用，当需要求两条直线之间的夹角时，或者在进阶数学中涉及复数辐角的问题中。Edexcel考官经常在不熟悉的语境中测试复合角公式的应用，例如证明给定表达式简化为已知值。

Double Angle Formulae | 倍角公式

The double angle formulae are special cases of the compound angle formulae where A equals B. The three most important forms for sin 2A, cos 2A, and tan 2A must be at your fingertips. For sine: sin 2A equals 2 sin A cos A. This is perhaps the most frequently used double angle identity across the entire A-Level syllabus. It appears in integration by substitution, in solving trigonometric equations, and in vector geometry problems.

倍角公式是复合角公式中A等于B的特殊情况。sin 2A、cos 2A和tan 2A的三种最重要形式必须烂熟于心。对于正弦：sin 2A等于2 sin A cos A。这可能是整个A-Level课程大纲中使用最频繁的倍角恒等式。它出现在换元积分法、解三角方程和向量几何问题中。

Cosine offers three equivalent forms of the double angle identity, and knowing when to use each one is a key problem-solving skill. The first form, cos 2A equals cos squared A minus sin squared A, is the most direct. The second, cos 2A equals 2 cos squared A minus 1, is used when you want to express everything in terms of cosine. The third, cos 2A equals 1 minus 2 sin squared A, is used when you want to express everything in terms of sine. Choosing the right form can transform a difficult integral or equation into something straightforward.

余弦有三种等价的倍角恒等式形式，知道何时使用每一种是一项关键的解题技能。第一种形式，cos 2A等于cos平方A减sin平方A，是最直接的。第二种，cos 2A等于2 cos平方A减1，当你想用余弦表示所有内容时使用。第三种，cos 2A等于1减2 sin平方A，当你想用正弦表示所有内容时使用。选择正确的形式可以将困难的积分或方程转化为简单的问题。

Solving Trigonometric Equations | 解三角方程

Solving trigonometric equations is a skill that Edexcel examines at every level, from basic equations in Year 12 through to the most challenging problems in Year 13. The general approach involves four steps: first, simplify the equation using identities so that it contains only one trigonometric function; second, solve the resulting algebraic equation for that function; third, find the principal values using inverse trigonometric functions; and fourth, find all solutions within the specified interval using the periodic properties of trigonometric functions.

解三角方程是Edexcel在每个层次都考察的技能，从12年级的基础方程到13年级最具挑战性的问题。一般方法包括四个步骤：首先，使用恒等式简化方程，使其只包含一个三角函数；其次，解出该函数的代数方程；第三，使用反三角函数求出主值；第四，利用三角函数的周期性质求出指定区间内的所有解。

Consider a typical Edexcel exam question: solve 3 cos 2x plus 5 sin x plus 1 equals 0 for x between 0 and 360 degrees. The key insight is to replace cos 2x using the identity cos 2x equals 1 minus 2 sin squared x. This transforms the equation into a quadratic in sin x: 3 times (1 minus 2 sin squared x) plus 5 sin x plus 1 equals 0, which simplifies to negative 6 sin squared x plus 5 sin x plus 4 equals 0. This is now a standard quadratic equation in the variable sin x, which can be solved by factorisation or the quadratic formula.

考虑一道典型的Edexcel考题：解方程 3 cos 2x 加 5 sin x 加 1 等于 0，其中 x 在0到360度之间。关键的思路是用恒等式 cos 2x 等于 1 减 2 sin平方x 来替换 cos 2x。这将方程转化为关于 sin x 的二次方程：3 乘以 (1 减 2 sin平方x) 加 5 sin x 加 1 等于 0，化简为 负6 sin平方x 加 5 sin x 加 4 等于 0。这现在是一个关于变量 sin x 的标准二次方程，可以通过因式分解或求根公式来解。

A critical skill that Edexcel examiners look for is the ability to find all solutions within a given range. After finding that sin x equals a certain value, you must use the CAST diagram or the graph of sine to identify every angle in the specified interval that satisfies the equation. The CAST diagram helps you remember which trigonometric ratios are positive in each quadrant: Cosine and its reciprocal are positive in the fourth quadrant, All are positive in the first, Sine in the second, and Tangent in the third – hence the name CAST, moving anticlockwise from the fourth quadrant.

Edexcel考官看重的一个关键能力是找到给定范围内的所有解。在求出 sin x 等于某个值之后，你必须使用CAST图或正弦图像来找出指定区间内满足方程的每一个角度。CAST图帮助你记住每个象限中哪些三角比为正：余弦及其倒数在第四象限为正，所有比在第一象限为正，正弦在第二象限为正，正切在第三象限为正 – 因此得名CAST，从第四象限逆时针移动。

R cos (theta plus alpha) and Harmonic Form | R cos(θ+α)与调和形式

One of the most distinctive topics in Edexcel A-Level trigonometry is expressing a linear combination of sine and cosine as a single trigonometric function. The expression a sin theta plus b cos theta can be written as R sin of theta plus alpha, where R equals the square root of a squared plus b squared, and alpha satisfies tan alpha equals b over a. The equivalent form using cosine is R cos of theta minus alpha. This technique is sometimes called the harmonic form or the wave form, because it reveals the amplitude and phase shift of the combined wave.

Edexcel A-Level三角学中最具特色的专题之一是将正弦和余弦的线性组合表示为单个三角函数。表达式 a sin θ 加 b cos θ 可以写成 R sin(θ+α)，其中 R 等于 a 平方加 b 平方的平方根，α 满足 tan α 等于 b 除以 a。使用余弦的等价形式是 R cos(θ-α)。这种技巧有时被称为调和形式或波形形式，因为它揭示了组合波的振幅和相位偏移。

This technique is extensively tested because it bridges trigonometry with calculus. Once you have expressed a sin theta plus b cos theta in harmonic form, you can easily find its maximum and minimum values (R and negative R respectively) and the values of theta at which they occur. You can also differentiate and integrate the expression easily, since the derivative of R sin of theta plus alpha is simply R cos of theta plus alpha. Edexcel often sets questions that ask for the maximum value of a function involving both sine and cosine terms, and the harmonic form is almost always the most efficient approach.

这种技巧被广泛考察，因为它将三角学与微积分连接起来。一旦你将 a sin θ 加 b cos θ 表示为调和形式，你就能轻松找到它的最大值和最小值（分别为 R 和 负R）以及它们出现时的 θ 值。你还可以轻松地对表达式进行微分和积分，因为 R sin(θ+α) 的导数就是 R cos(θ+α)。Edexcel经常出题要求求同时包含正弦和余弦项的函数的最大值，而调和形式几乎总是最高效的方法。

Reciprocal Trigonometric Functions | 倒数三角函数

The reciprocal trigonometric functions – secant (sec), cosecant (cosec), and cotangent (cot) – are introduced in the second year of the Edexcel A-Level course. They are defined as: sec x equals 1 over cos x, cosec x equals 1 over sin x, and cot x equals 1 over tan x, which also equals cos x over sin x. These functions have their own graphs, domains, ranges, and asymptotic behaviour that you need to understand thoroughly.

倒数三角函数 – 正割(sec)、余割(cosec)和余切(cot) – 在Edexcel A-Level课程的第二年引入。它们的定义是：sec x 等于 1 除以 cos x，cosec x 等于 1 除以 sin x，cot x 等于 1 除以 tan x，也等于 cos x 除以 sin x。这些函数有自己的图像、定义域、值域和渐近行为，你需要彻底理解。

The derivatives of the reciprocal trigonometric functions are important results that you may be required to quote or derive. The derivative of sec x is sec x tan x. The derivative of cosec x is negative cosec x cot x. The derivative of cot x is negative cosec squared x. These can all be derived using the quotient rule from the definitions above, and Edexcel exam questions sometimes ask candidates to do exactly that as a proof exercise.

倒数三角函数的导数是重要的结果，你可能需要引用或推导它们。sec x 的导数是 sec x tan x。cosec x 的导数是 负cosec x cot x。cot x 的导数是 负cosec平方x。这些都可以使用商法则从上述定义中推导出来，Edexcel考题有时会要求考生将这一点作为证明练习来完成。

Proving Trigonometric Identities | 证明三角恒等式

Proof questions involving trigonometric identities are a staple of Edexcel A-Level examinations. The examiner is assessing your ability to manipulate algebraic expressions and to recognise which identity to apply at each step. The golden rule of identity proofs is to start from the more complicated side and simplify it until it matches the simpler side. Never start by assuming the identity is true and working on both sides simultaneously – that is logically circular.

涉及三角恒等式的证明题是Edexcel A-Level考试的主要内容。考官评估的是你操控代数表达式的能力，以及在每一步中识别应该应用哪个恒等式的能力。恒等式证明的黄金法则是从较复杂的一侧开始，简化它直到与较简单的一侧匹配。永远不要先假设恒等式成立然后同时处理两侧 – 这在逻辑上是循环论证。

A common strategy is to express everything in terms of sine and cosine. Since all six trigonometric functions can be expressed using just sine and cosine, this simplifies the problem to algebraic manipulation of these two basic functions. For example, to prove that tan x plus cot x equals sec x cosec x, write tan x as sin x over cos x and cot x as cos x over sin x. Combine the fractions to get sin squared x plus cos squared x all over sin x cos x. The numerator simplifies to 1 by the Pythagorean identity, and the denominator is exactly 1 over sec x cosec x, completing the proof.

一个常见的策略是用正弦和余弦表示所有内容。由于所有六个三角函数都可以仅用正弦和余弦表示，这将问题简化为对这两个基本函数的代数操作。例如，要证明 tan x 加 cot x 等于 sec x cosec x，将 tan x 写成 sin x 除以 cos x，cot x 写成 cos x 除以 sin x。合并分数得到 (sin平方x 加 cos平方x) 除以 (sin x cos x)。分子由勾股恒等式简化为1，分母正好是 1 除以 (sec x cosec x)，完成证明。

Trigonometric Integration | 三角积分

Integration of trigonometric functions is a major component of Edexcel A-Level Pure Mathematics, appearing in both Year 12 and Year 13 content. The basic integrals you must know are: the integral of sin x is negative cos x plus C, and the integral of cos x is sin x plus C. The integral of sec squared x is tan x plus C, and the integral of cosec squared x is negative cot x plus C. The integral of sec x tan x is sec x plus C, and the integral of cosec x cot x is negative cosec x plus C.

三角函数的积分是Edexcel A-Level纯数学的一个重要组成部分，出现在12年级和13年级的内容中。你必须知道的基本积分是：sin x 的积分是 负cos x 加 C，cos x 的积分是 sin x 加 C。sec平方x 的积分是 tan x 加 C，cosec平方x 的积分是 负cot x 加 C。sec x tan x 的积分是 sec x 加 C，cosec x cot x 的积分是 负cosec x 加 C。

More advanced integration techniques involving trigonometry include using the double angle formulae to integrate sin squared x or cos squared x. Since there is no direct antiderivative for sin squared x, you must use the identity sin squared x equals one half times (1 minus cos 2x) to rewrite it before integrating. Similarly, cos squared x equals one half times (1 plus cos 2x). These identities convert a squared trigonometric function into a linear combination of constants and double-angle cosines, both of which are straightforward to integrate.

涉及三角学的更高级积分技巧包括使用倍角公式来积分 sin平方x 或 cos平方x。由于 sin平方x 没有直接的原函数，你必须使用恒等式 sin平方x 等于 二分之一 乘以 (1 减 cos 2x) 来重写它然后再积分。类似地，cos平方x 等于 二分之一 乘以 (1 加 cos 2x)。这些恒等式将平方三角函数转化为常数和倍角余弦的线性组合，两者都很容易积分。

The substitution method is frequently tested with trigonometric integrands. The substitution u equals sin x is useful when the integrand contains cos x dx, since du equals cos x dx. Similarly, u equals cos x pairs with integrands containing sin x dx. For integrals involving expressions like the square root of a squared minus x squared, the trigonometric substitution x equals a sin theta (or x equals a tan theta for a squared plus x squared, or x equals a sec theta for x squared minus a squared) transforms the radical into a trigonometric expression that can be simplified using the Pythagorean identity.

换元法在三角被积函数中经常被考察。当被积函数包含 cos x dx 时，换元 u 等于 sin x 很有用，因为 du 等于 cos x dx。类似地，u 等于 cos x 与被积函数中的 sin x dx 配对。对于涉及如 根号下(a平方减x平方) 这样的表达式的积分，三角换元 x 等于 a sin θ（或者对于 a平方加x平方 用 x 等于 a tan θ，对于 x平方减a平方 用 x 等于 a sec θ）将根式转化为可以使用勾股恒等式简化的三角表达式。

Modelling with Trigonometric Functions | 三角函数的建模应用

Edexcel places strong emphasis on mathematical modelling, and trigonometric functions are ideally suited to model periodic phenomena. The general form of a periodic model is y equals a sin of b times (t minus c) plus d, or equivalently y equals a cos of b times (t minus c) plus d. The parameter a represents the amplitude (half the difference between maximum and minimum), the period is 2 pi over b (or 360 degrees over b if working in degrees), c represents the horizontal shift (phase), and d represents the vertical shift (the central value around which the oscillation occurs).

Edexcel非常强调数学建模，而三角函数非常适合模拟周期性现象。周期模型的一般形式是 y 等于 a sin(b(t减c)) 加 d，或等价地 y 等于 a cos(b(t减c)) 加 d。参数 a 表示振幅（最大值与最小值之差的一半），周期为 2π 除以 b（如果使用角度制则为 360度 除以 b），c 表示水平移动（相位），d 表示垂直移动（振荡围绕的中心值）。

Typical modelling questions on Edexcel papers involve tides, temperature variations over a day or year, the height of a point on a Ferris wheel, the depth of water in a harbour, or the voltage in an alternating current circuit. The question will provide real-world data and ask you to construct a trigonometric model, then use that model to make predictions. For instance, given that the depth of water in a harbour varies between 6 metres and 14 metres with a period of 12 hours, and that high tide occurs at 2 am, you can construct the model: depth equals 4 cos of (pi over 6 times (t minus 2)) plus 10, where t is measured in hours after midnight.

Edexcel试卷上典型的建模问题涉及潮汐、一天或一年中的温度变化、摩天轮上某点的高度、港口水深或交流电路中的电压。题目会提供真实世界的数据，要求你构建一个三角模型，然后用该模型进行预测。例如，给定港口水深在6米到14米之间变化，周期为12小时，高潮发生在凌晨2点，你可以构建模型：水深 等于 4 cos(π/6 × (t 减 2)) 加 10，其中 t 以午夜后的小时数计量。

Inverse Trigonometric Functions | 反三角函数

The inverse trigonometric functions – arcsin, arccos, and arctan – are essential for solving equations and appear frequently in Edexcel A-Level questions. The notation arcsin x means the angle whose sine is x. Since trigonometric functions are not one-to-one over their entire domains, we restrict their domains to define the inverse functions uniquely. For arcsin x, the range is from negative pi over 2 to pi over 2 inclusive. For arccos x, the range is from 0 to pi inclusive. For arctan x, the range is from negative pi over 2 to pi over 2 (excluding the endpoints).

反三角函数 – arcsin、arccos和arctan – 对于解方程至关重要，并频繁出现在Edexcel A-Level考题中。记号 arcsin x 表示正弦值为 x 的角。由于三角函数在其整个定义域上不是一一对应的，我们限制它们的定义域以唯一定义反函数。对于 arcsin x，值域是从负π/2到π/2（含端点）。对于 arccos x，值域是从0到π（含端点）。对于 arctan x，值域是从负π/2到π/2（不含端点）。

The derivatives of inverse trigonometric functions are standard results that you should know for Edexcel. The derivative of arcsin x is 1 over the square root of 1 minus x squared. The derivative of arccos x is negative 1 over the square root of 1 minus x squared. The derivative of arctan x is 1 over 1 plus x squared. These can be derived using implicit differentiation, and Edexcel may ask you to reproduce the derivation as part of a longer problem.

反三角函数的导数是Edexcel要求掌握的标准结果。arcsin x 的导数是 1 除以 根号下(1减x平方)。arccos x 的导数是 负1 除以 根号下(1减x平方)。arctan x 的导数是 1 除以 (1加x平方)。这些可以使用隐函数微分法推导，Edexcel可能会要求你在一个较长的题目中重现推导过程。

Exam Techniques and Common Pitfalls | 考试技巧与常见陷阱

Edexcel A-Level trigonometry questions carry significant weight – typically 8 to 15 marks each in Pure Mathematics papers. The most common mistake students make is forgetting to find all solutions within the specified interval. After using the inverse trigonometric function on your calculator, you get a principal value, but there may be other angles in the required range that produce the same trigonometric ratio. Always sketch the graph or use the CAST diagram to identify every solution.

Edexcel A-Level三角学题目分值很重 – 在纯数学试卷中通常每题8到15分。学生最常见的错误是忘记找到指定区间内的所有解。使用计算器上的反三角函数后，你会得到一个主值，但在所需范围内可能还有其他角度产生相同的三角比值。始终画出图像或使用CAST图来识别每一个解。

Another frequent pitfall is failing to check whether solutions are expressed in degrees or radians. Edexcel questions may switch between the two, and using the wrong mode on your calculator will produce entirely wrong answers. The question will specify the unit – look for the degree symbol or the absence of it (which implies radians in A-Level contexts). When the interval is given in terms of pi, it is always radians.

另一个常见的陷阱是没有检查解是以角度制还是弧度制表示的。Edexcel题目可能在两者之间切换，在计算器上使用错误的模式会产生完全错误的答案。题目会指定单位 – 寻找度数符号或缺少度数符号（在A-Level背景下暗示弧度制）。当区间以π的形式给出时，始终是弧度制。

Finally, many students lose marks by not simplifying their final answers. Edexcel mark schemes expect exact values where possible – for example, sin 60 degrees should be written as root 3 over 2, not as the decimal 0.866. Similarly, angles should be given as exact multiples of pi when working in radians, and surd forms should be simplified. Leaving an answer as sin theta equals 0.5 is acceptable only if the question explicitly asks for the value of sin theta rather than theta itself.

最后，许多学生因为不简化最终答案而失分。Edexcel评分标准期望尽可能使用精确值 – 例如，sin 60度应写为√3/2，而不是小数0.866。类似地，使用弧度制时角度应给出π的精确倍数，根式应简化。只有在题目明确要求求sinθ的值而非θ本身时，将答案留为sinθ等于0.5才是可接受的。

Summary | 总结

Trigonometric identities and equations represent a substantial and high-value topic within the Edexcel A-Level Pure Mathematics specification. Success in this area requires more than memorisation – it demands fluency in applying the Pythagorean identities, compound and double angle formulae, harmonic form, and reciprocal function properties across a wide range of problem types. The ability to move confidently between different trigonometric forms, to recognise which identity to apply in a given situation, and to systematically find all solutions within a specified interval are the hallmarks of a strong A-Level mathematics student. Regular practice with past paper questions, combined with a deep understanding of the underlying mathematical structures, is the most reliable path to mastering this essential topic.

三角恒等式与方程是Edexcel A-Level纯数学大纲中一个重要且分值高的专题。在这一领域的成功需要的不仅仅是记忆 – 它要求在多种问题类型中熟练应用勾股恒等式、复合角和倍角公式、调和形式以及倒数函数性质。自信地在不同三角形式之间转换、在给定情境中识别应使用哪个恒等式、以及系统地找到指定区间内的所有解的能力，是优秀A-Level数学学生的标志。定期练习历年真题，结合对底层数学结构的深刻理解，是掌握这一基本专题的最可靠途径。