📚 Trigonometric Identities and Equations for Edexcel A-Level Maths | Edexcel A-Level 数学三角恒等式与方程
Trigonometric identities and equations form one of the most heavily examined topics in Edexcel A-Level Mathematics. A strong command of these identities allows you to simplify expressions, prove relationships, and solve equations that model periodic behaviour. This revision article covers the core identities, standard solution methods, and common exam pitfalls you need to master.
三角恒等式与方程是 Edexcel A-Level 数学中考查最频繁的主题之一。扎实掌握这些恒等式能帮助你化简表达式、证明关系并求解描述周期行为的方程。本文涵盖核心恒等式、标准求解方法以及你需要掌握的常见考试陷阱。
1. The Pythagorean Identities | 勾股恒等式
The Pythagorean identities arise directly from the unit circle. For any angle θ, the coordinates (cos θ, sin θ) lie on a circle of radius 1, so cos²θ + sin²θ = 1. Dividing through by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives cot²θ + 1 = cosec²θ. These three identities are the foundation for simplifying many trigonometric expressions.
勾股恒等式直接来源于单位圆。对于任意角 θ，坐标 (cos θ, sin θ) 位于半径为 1 的圆上，因此 cos²θ + sin²θ = 1。两边同除以 cos²θ 可得 1 + tan²θ = sec²θ，同除以 sin²θ 可得 cot²θ + 1 = cosec²θ。这三个恒等式是化简许多三角表达式的基础。
You should be able to use these identities both forwards and backwards. For example, replacing 1 – cos²θ with sin²θ can immediately simplify an expression, while recognising 1 + tan²θ as sec²θ often helps when dealing with reciprocal functions.
你需要能够正反两个方向使用这些恒等式。例如，将 1 – cos²θ 替换为 sin²θ 可以立即化简表达式，而将 1 + tan²θ 识别为 sec²θ 在处理倒数函数时通常很有帮助。
cos²θ + sin²θ = 1
1 + tan²θ = sec²θ
cot²θ + 1 = cosec²θ
2. Tangent and Reciprocal Identities | 正切与倒数恒等式
The tangent function is defined as tan θ = sin θ / cos θ. Its reciprocal identities are sec θ = 1/cos θ, cosec θ = 1/sin θ, and cot θ = 1/tan θ = cos θ / sin θ. These definitions allow you to rewrite expressions in terms of sin and cos only, which is often the first step in proving an identity.
正切函数定义为 tan θ = sin θ / cos θ。其倒数恒等式为 sec θ = 1/cos θ、cosec θ = 1/sin θ 以及 cot θ = 1/tan θ = cos θ / sin θ。这些定义使你能够将表达式仅用 sin 和 cos 表示，这通常是证明恒等式的第一步。
When an expression contains mixed functions such as tan θ and sec θ, converting everything to sines and cosines can reveal hidden simplifications. This strategy is especially useful in proof questions and when integrating trigonometric functions.
当一个表达式含有混合函数，例如 tan θ 和 sec θ 时，将所有内容转换为正弦和余弦可以揭示隐藏的化简。这一策略在证明题和积分三角函数时尤其有用。
tan θ = sin θ / cos θ
sec θ = 1/cos θ, cosec θ = 1/sin θ, cot θ = cos θ / sin θ
3. Compound Angle Formulae | 复合角公式
The compound angle formulae expand sin(A ± B), cos(A ± B), and tan(A ± B). For sine and cosine, remember the order of signs carefully: sin(A ± B) = sin A cos B ± cos A sin B, while cos(A ± B) = cos A cos B ∓ sin A sin B. These formulae are essential for exact value calculations such as sin 75° = sin(45° + 30°).
复合角公式展开 sin(A ± B)、cos(A ± B) 和 tan(A ± B)。对于正弦和余弦，要特别注意符号顺序：sin(A ± B) = sin A cos B ± cos A sin B，而 cos(A ± B) = cos A cos B ∓ sin A sin B。这些公式对于精确值计算至关重要，例如 sin 75° = sin(45° + 30°)。
You must also be confident applying these formulae in reverse. For example, recognising that sin x cos y + cos x sin y can be condensed to sin(x + y) is a skill that frequently appears in both pure mathematics and mechanics questions.
你还必须能够熟练地反向应用这些公式。例如，识别出 sin x cos y + cos x sin y 可以合并为 sin(x + y) 是一项在纯数和力学题中都经常出现的技能。
sin(A ± B) = sin A cos B ± cos 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)
4. Double Angle Formulae | 二倍角公式
The double angle formulae are special cases of the compound angle formulae with A = B. You must know three forms of cos 2θ: cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ. These rearrangements are particularly useful when integrating sin²θ or cos²θ, or when solving equations that involve both single and double angles.
二倍角公式是复合角公式在 A = B 时的特例。你必须掌握 cos 2θ 的三种形式：cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ。这些变形在积分 sin²θ 或 cos²θ，或求解同时包含单角和二倍角的方程时尤为有用。
The different forms of cos 2θ allow you to choose the version that best matches the rest of the equation. For example, if an equation already contains cos²θ, using cos 2θ = 2cos²θ – 1 may lead to a simpler quadratic. If it contains sin²θ, use cos 2θ = 1 – 2sin²θ instead.
cos 2θ 的不同形式允许你选择与方程其余部分最匹配的版本。例如，如果方程已含有 cos²θ，使用 cos 2θ = 2cos²θ – 1 可能导出更简单的二次方程。如果含有 sin²θ，则使用 cos 2θ = 1 – 2sin²θ。
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ – sin²θ = 2cos²θ – 1 = 1 – 2sin²θ
tan 2θ = 2 tan θ / (1 – tan²θ)
5. Solving Basic Trigonometric Equations | 解基本三角方程
For Edexcel A-Level, you must be able to solve equations like sin x = k, cos x = k, and tan x = k for a given interval, usually 0° ≤ x ≤ 360° or 0 ≤ x ≤ 2π. Start by finding the principal value using your calculator. Then use the symmetry of the unit circle to find all other solutions.
在 Edexcel A-Level 中，你必须能够求解给定区间上的方程，如 sin x = k、cos x = k 和 tan x = k，通常区间为 0° ≤ x ≤ 360° 或 0 ≤ x ≤ 2π。首先用计算器求出主值，然后利用单位圆的对称性找出所有其他解。
For sin x = k, the second solution in the interval 0° to 360° is 180° – θ. For cos x = k, the second solution is 360° – θ. The tangent function, however, repeats every 180°, so if θ is a solution of tan x = k, then θ + 180° is also a solution within the expanded interval.
对于 sin x = k，在 0° 到 360° 区间内的第二个解是 180° – θ。对于 cos x = k，第二个解是 360° – θ。然而，正切函数每 180° 重复一次，因此如果 θ 是 tan x = k 的一个解，那么 θ + 180° 在扩大后的区间内也是一个解。
The table below summarises the exact values you are expected to memorise for common angles.
下表总结了你需要熟记的常见角度的精确值。
| θ | 0° | 30° | 45° | 60° | 90° |
|---|---|---|---|---|---|
| sin θ | 0 | 1/2 | √2/2 | √3/2 | 1 |
| cos θ | 1 | √3/2 | √2/2 | 1/2 | 0 |
| tan θ | 0 | 1/√3 | 1 | √3 | undefined |
6. Solving Equations Using Identities | 利用恒等式解方程
Many exam questions require you to use identities to reduce an equation to a solvable form. For example, an equation with both sin²θ and cos θ can be rewritten using sin²θ = 1 – cos²θ to produce a quadratic in cos θ. Similarly, equations with sin 2θ can be expanded as 2 sin θ cos θ, and then factorised.
许多考试题要求你利用恒等式将方程化为可解形式。例如，同时含有 sin²θ 和 cos θ 的方程可用 sin²θ = 1 – cos²θ 改写，得到关于 cos θ 的二次方程。类似地，含 sin 2θ 的方程可展开为 2 sin θ cos θ，然后进行因式分解。
Always check for potential division by zero if you divide through by a trigonometric function. For instance, dividing both sides of an equation by sin θ will cause you to lose the solutions where sin θ = 0. Instead, bring all terms to one side and factorise.
如果你除以某个三角函数，务必检查可能出现的除零情况。例如，方程两边同除以 sin θ 会导致你丢失 sin θ = 0 的解。正确做法是把所有项移到一边并进行因式分解。
After substituting and factorising, set each factor equal to zero and solve each resulting basic trigonometric equation separately. Then collect all solutions that lie within the required interval.
代换和因式分解后，令每个因式等于零，并分别求解每个得到的基本三角方程。然后收集所有位于要求区间内的解。
7. R cos(θ ± α) Method | R cos(θ ± α) 方法
The R cos(θ ± α) method transforms an expression of the form a sin θ + b cos θ into a single cosine or sine wave. You set R cos(θ – α) = a cos θ + b sin θ, where R = √(a² + b²) and tan α = b/a. This technique is essential for solving equations like 3 sin θ + 4 cos θ = 2 and for finding maximum and minimum values of such expressions.
R cos(θ
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