📚 Edexcel A-Level Maths: Trigonometric Identities & Equations | Edexcel A-Level 数学：三角恒等式与方程
Trigonometric identities and equations are central to Edexcel A-Level Mathematics, appearing in Pure Paper 1 and Paper 2. This guide covers the key identities, solution methods and common exam pitfalls.
三角恒等式与方程是 Edexcel A-Level 数学的核心内容，在 Pure Paper 1 和 Paper 2 中频繁出现。本指南涵盖关键恒等式、求解方法及常见考试陷阱。
1. Identity vs Equation | 恒等式与方程的区别
An identity is a statement that is true for all values of the variable for which both sides are defined. For example, sin²θ + cos²θ = 1 is an identity because it holds for every θ.
恒等式对变量所有允许值都成立。例如 sin²θ + cos²θ = 1 是恒等式，因为它对任意 θ 都成立。
An equation, by contrast, is only true for particular values, such as sin θ = 0.5. In the exam you must decide whether you are proving an identity or solving an equation.
方程则只在特定取值下成立，例如 sin θ = 0.5。考试中你必须区分是证明恒等式还是求解方程。
2. Reciprocal and Quotient Identities | 倒数与商数恒等式
The reciprocal trigonometric functions are secant, cosecant and cotangent. They are defined by sec θ = 1/cos θ, cosec θ = 1/sin θ and cot θ = 1/tan θ = cos θ/sin θ.
倒数三角函数包括正割、余割和余切。它们定义为 sec θ = 1/cos θ、cosec θ = 1/sin θ、cot θ = 1/tan θ = cos θ/sin θ。
These definitions are especially useful when simplifying rational expressions or changing a problem into sine and cosine.
这些定义在化简分式或将问题化为正弦和余弦时特别有用。
| English | 中文 | Formula |
|---|---|---|
| Secant | 正割 | sec θ = 1/cos θ |
| Cosecant | 余割 | cosec θ = 1/sin θ |
| Cotangent | 余切 | cot θ = cos θ/sin θ |
3. Pythagorean Identities | 毕达哥拉斯恒等式
The main Pythagorean identity is sin²θ + cos²θ = 1. Dividing this identity by cos²θ gives 1 + tan²θ = sec²θ, and dividing by sin²θ gives 1 + cot²θ = cosec²θ.
主要毕达哥拉斯恒等式是 sin²θ + cos²θ = 1。将其除以 cos²θ 得 1 + tan²θ = sec²θ，除以 sin²θ 得 1 + cot²θ = cosec²θ。
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = cosec²θ
These identities let you rewrite expressions such as 2cos²θ − 1 in terms of sin θ or cos θ, which is essential for integration and solving equations.
这些恒等式允许你将 2cos²θ − 1 等表达式用 sin θ 或 cos θ 表示，这对积分和方程求解至关重要。
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