📚 Trigonometric Identities | 三角恒等式
Trigonometric identities are equations involving trigonometric functions that hold true for every value of the variable for which both sides are defined. In Edexcel A-Level Mathematics, a strong command of these identities is required to simplify complex expressions, prove equivalence between trigonometric statements, and solve equations efficiently. This article brings together all the key identities you need to know, illustrates their use through worked examples, and highlights common pitfalls you should avoid in the exam.
三角恒等式是包含三角函数的等式，对于使两边均有定义的变量取值恒成立。在 Edexcel A-Level 数学中，熟练掌握这些恒等式是化简复杂表达式、证明三角命题以及高效解方程的基础。本文汇总了所有你需要掌握的关键恒等式，通过实例展示它们的用法，并指出考试中应当避免的常见错误。
1. Basic Reciprocal and Quotient Identities | 基本倒数与商数恒等式
Every trigonometric function can be expressed in terms of sine and cosine. The three reciprocal functions — secant, cosecant and cotangent — are defined as 1/cosθ, 1/sinθ and cosθ/sinθ respectively. The quotient identity for tangent is central to many simplifications.
每个三角函数都可以用正弦和余弦表示。三个倒数函数——正割、余割和余切——分别定义为 1/cosθ、1/sinθ 和 cosθ/sinθ。正切的商数恒等式是许多化简工作的核心。
| tan θ = sin θ / cos θ | 正切由正弦除以余弦给出 |
| cot θ = cos θ / sin θ = 1 / tan θ | 余切是余弦除以正弦，即正切的倒数 |
| sec θ = 1 / cos θ | 正割是余弦的倒数 |
| cosec θ = 1 / sin θ | 余割是正弦的倒数 |
These definitions allow you to rewrite any trigonometric expression entirely in terms of sine and cosine, a useful first step when proving identities or solving equations.
这些定义允许你将任何三角表达式完全改写为正弦和余弦的形式，这在证明恒等式或解方程时是一个很有用的第一步。
2. Pythagorean Identities | 毕达哥拉斯恒等式
g this identity by cos²θ or sin²θ, we obtain two further forms that involve tan²θ, sec²θ, cot²θ and cosec²θ.
最基本的毕达哥拉斯恒等式是 sin²θ + cos²θ = 1。将这个恒等式除以 cos²θ 或 sin²θ，可以得到另外两种涉及 tan²θ、sec²θ、cot²θ 和 cosec²θ 的形式。
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
These identities are invaluable for replacing squared terms during simplification. For example, if an expression contains sec²θ - 1, you can immediately replace it with tan²θ.
这些恒等式在化简过程中替代平方项时非常宝贵。例如，若表达式含有 sec²θ - 1，你可以立刻将其替换为 tan²θ。
3. Sum and Difference Formulae | 和差公式
) in terms of the sines and cosines of A and B. They are essential for deriving double-angle results and for handling equations where angles are combined.
和差公式可将 sin(A ± B)、cos(A ± B) 和 tan(A ± B) 展开为 A 和 B 的正弦、余弦形式。它们是推导倍角公式和处理组合角方程的基础。
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)
Notice the sign changes: for cosine, the sign in the formula is opposite to the sign of the compound angle (cos(A+B) has a minus). For tangent, the numerator follows the compound angle sign, while the denominator flips it.
注意符号变化：对于余弦，公式中的符号与复合角的符号相反（cos(A+B) 是减号）。对于正切，分子遵循复合角符号，分母则相反。
4. Double-Angle Formulae | 倍角公式
ae gives the double-angle identities. The cosine double-angle formula has three useful variations, all of which appear regularly in Edexcel exam questions.
在和角公式中令 A = B = θ，即得到倍角恒等式。余弦倍角公式有三种有用的变形，在 Edexcel 考题中频繁出现。
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ - sin²θ
cos 2θ = 2 cos²θ - 1
cos 2θ = 1 - 2 sin²θ
tan 2θ = 2 tan θ / (1 - tan²θ)
Choosing the right form of cos 2θ can quickly reduce an expression to a single trig function. For instance, 1 - 2 sin²θ is ideal when you want to express everything in terms of sin θ, while 2 cos²θ - 1 works for cos θ.
选择合适的 cos 2θ 形式可以迅速将表达式化为单个三角函数。例如，1 - 2 sin²θ 适合将所有项化为 sin θ 的形式，而 2 cos²θ - 1 则适合化为 cos θ 的形式。
5. The R-Formula (Harmonic Form) | R 公式（辅助角形式）
gle sine or cosine function with a phase shift. This technique, often called the R-formula, is extremely useful for solving equations and finding maximum and minimum values.
形如 a sin θ ± b cos θ 或 a cos θ ± b sin θ 的表达式可以写为一个带相移的正弦或余弦函数。这种方法通常称为 R 公式，在解方程以及求最大值和最小值时极为有用。
a sin θ + b cos θ = R sin(θ + α)
a sin θ - b cos θ = R sin(θ - α)
a cos θ + b sin θ = R cos(θ - α)
a cos θ - b sin θ = R cos(θ + α)
Here R = √a² + b² b/a (with the sides adjusted depending on the chosen form). Always check which quadrant α lies in based on the signs of a and b when using a specific form.
其中 R = √a² + b²，α 是满足 tan α = b/a 的锐角（具体关系依所选形式而定）。在使用特定形式时，务必根据 a 和 b 的符号确定 α 所在的象限。
6. Simplifying Trigonometric Expressions | 化简三角表达式
A typical exam question asks you to simplify an expression like (sin θ + cos θ)² - 1. Using the identities systematically reduces it to a simpler form.
典型的考题要求你化简类似 (sin θ + cos θ)² - 1 的表达式。系统地运用恒等式可将其化为更简单的形式。
Example: simplify (sin θ + cos θ)² - 1.
示例：化简 (sin θ + cos θ)² - 1。
Expand: sin²θ + 2 sin θ cos θ + cos²θ - 1.
展开：sin²θ + 2 sin θ cos θ + cos²θ - 1。
n²θ + cos²θ = 1, so the expression becomes 1 + 2 sin θ cos θ - 1 = 2 sin θ cos θ = sin 2θ.
将 sin²θ + cos²θ = 1 合并，表达式变为 1 + 2 sin θ cos θ - 1 = 2 sin θ cos θ = sin 2θ。
This process highlights a common strategy: replace mixed terms with a single function using Pythagorean and double-angle identities.
这个过程突出了一个常用策略：利用毕达哥拉斯恒等式和倍角恒等式将混合项替换为单一函数。
7. Proving Trigonometric Identities | 证明三角恒等式
When proving an identity, start from one side (usually the more complicated one) and manipulate it step by step until it matches the other side. Writing everything in terms of sine and cosine is often a reliable first move.
证明恒等式时，从其中一边（通常是较复杂的一边）入手，逐步变形直到与另一边相同。将所有项改写为正弦和余弦的形式通常是可靠的第一步。
Example: prove that (1 + tan²θ) / sec θ = sec θ.
示例：证明 (1 + tan²θ) / sec θ = sec θ。
de: use 1 + tan²θ = sec²θ, giving sec²θ / sec θ = sec θ, which equals the right-hand side.
左边：利用 1 + tan²θ = sec²θ，得到 sec²θ / sec θ = sec θ，与右边相等。
When the numerator and denominator are both sums or differences, it can be helpful to factorise or rationalise, always aiming to reveal a known identity.
当分子和分母均为和差形式时，可尝试因式分解或有理化，始终以揭示已知恒等式为目标。
8. Solving Trigonometric Equations | 解三角方程
sin 2θ = cos θ can be tackled by applying the double-angle formula sin 2θ = 2 sin θ cos θ.
三角恒等式的运用常可将方程转化为基本正弦、余弦或正切方程。例如，方程 sin 2θ = cos θ 可通过倍角公式 sin 2θ = 2 sin θ cos θ 来处理。
→ bring terms to one side: 2 sin θ cos θ - cos θ = 0 → factorise: cos θ (2 sin θ - 1) = 0.
于是：2 sin θ cos θ = cos θ → 移项：2 sin θ cos θ - cos θ = 0 → 因式分解：cos θ (2 sin θ - 1) = 0。
each case within the given interval, remembering to add multiples of the period (360° or 2π) and to check for extraneous roots if squaring was involved earlier.
由此得到 cos θ = 0 或 sin θ = 1/2。在给定区间内分别求解，记住要加上周期倍数（360° 或 2π），并且若之前有平方操作则需验根。
θ = c, use the R-formula to rewrite as R sin(θ ± α) = c, then solve for θ ± α and hence θ.
对于形如 a sin θ + b cos θ = c 的方程，使用 R 公式改写为 R sin(θ ± α) = c，然后求解 θ ± α 进而得到 θ。
9. Common Pitfalls and Exam Tips | 常见陷阱与考试技巧
cos θ is undefined where cos θ = 0. Always state restrictions if required by the question.
错误：忘记恒等式仅在两边有定义时成立。例如，tan θ = sin θ / cos θ 在 cos θ = 0 处无定义。若题目要求，务必说明限制条件。
s(2θ) = cos²θ - sin²θ, not cos²θ + sin²θ. It is easy to confuse because sin²θ + cos²θ = 1.
错误：余弦倍角符号记错。记住 cos(2θ) = cos²θ - sin²θ，而不是 cos²θ + sin²θ。由于 sin²θ + cos²θ = 1，极易混淆。
Mistake: dividing both sides of an equation by a trigonometric function without considering the case where that function equals zero. This can cause you to lose solutions.
错误：将方程两边除以一个三角函数而没有考虑该函数等于零的情形，这可能导致丢解。
Tip: when proving identities, avoid moving terms from one side to the other unless you are working on both sides separately. It is safer to manipulate one side only.
技巧：证明恒等式时，除非两边分别处理，否则不要将项从一边移到另一边。仅从一边入手变形更为稳妥。
Tip: use a table of key identities as a quick reference, but your real security comes from practising their applications in varied contexts.
技巧：可用关键恒等式表格作为速查，但真正的把握来自于在多种情境下的应用练习。
10. Summary Table of Essential Identities | 核心恒等式汇总表
| sin²θ + cos²θ = 1 | 毕达哥拉斯基本恒等式 |
| 1 + tan²θ = sec²θ | 由除以 cos²θ 推出 |
| 1 + cot²θ = cosec²θ | 由除以 sin²θ 推出 |
| sin(A ± B) = sinA cosB ± cosA sinB | 正弦和差公式 |
| cos(A ± B) = cosA cosB ∓ sinA sinB | 余弦和差公式（注意符号相反） |
| tan(A ± B) = (tanA ± tanB)/(1 ∓ tanA tanB) | 正切和差公式 |
| sin 2θ = 2 sinθ cosθ | 正弦倍角 |
| cos 2θ = cos²θ - sin²θ = 2cos²θ - 1 = 1 - 2sin²θ | 余弦倍角（三种形式） |
| tan 2θ = 2tanθ/(1 - tan²θ) | 正切倍角 |
| a sinθ ± b cosθ = R sin(θ ± α) | R 公式（正弦形式） |
| a cosθ ± b sinθ = R cos(θ ∓ α) | R 公式（余弦形式） |
Print this table and keep it in your revision folder, but more importantly, practise using each identity in context until they become second nature.
将这张表格打印出来放在复习文件夹里，但更重要的是在具体情境下反复练习运用每一个恒等式，直到它们变得像本能一样自然。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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