📚 Trigonometric Equations and Identities | 三角方程与恒等式
Trigonometric equations and identities form a core part of the Edexcel A-Level Pure Mathematics syllabus. Mastering techniques to solve equations involving sine, cosine, and tangent is essential for success in Paper 1 and Paper 2. This article will systematically explore the key identities, strategies for solving various types of trig equations, and common exam pitfalls.
三角方程与恒等式是 Edexcel A-Level 纯数课程的核心内容。掌握求解涉及正弦、余弦和正切的方程技巧，对于 Paper 1 和 Paper 2 的成功至关重要。本文将系统地探讨关键恒等式、各类三角方程的解法策略以及考试中的常见陷阱。
1. Understanding Trigonometric Equations | 理解三角方程
ations often have infinitely many solutions due to periodicity. In A-Level exams, you typically solve for θ within a given interval, such as 0° ≤ θ ≤ 360° or 0 ≤ θ ≤ 2π radians.
三角方程是涉及未知角 θ 的三角函数方程。与基本代数方程不同，由于周期性，三角方程通常有无穷多解。在 A-Level 考试中，通常要求在给定区间内求解 θ，例如 0° ≤ θ ≤ 360° 或 0 ≤ θ ≤ 2π 弧度。
lutions. Understanding the symmetry and periods (360° or 2π for sin/cos, 180° or π for tan) helps you find all roots within the required domain.
要画出 y = sin θ、y = cos θ 和 y = tan θ 的图像来直观地找解至关重要。理解对称性和周期（sin/cos 为 360° 或 2π，tan 为 180° 或 π）有助于在指定范围内找出所有根。
2. Key Pythagorean Identities | 关键的毕达哥拉斯恒等式
The fundamental Pythagorean identities link sin²θ, cos²θ and tan²θ. The most important one is:
基本的毕达哥拉斯恒等式将 sin²θ、cos²θ 和 tan²θ 联系起来。最重要的一条是：
sin² θ + cos² θ = 1
From this, we derive two closely related forms:
由此可推导出两个密切相关形式：
1 + tan² θ = sec² θ
1 + cot² θ = cosec² θ
2 sin²θ + cos θ = 1, you can replace sin²θ with 1 - cos²θ to obtain a quadratic in cos θ.
这些恒等式在化简表达式或证明其他结论时不可或缺。例如，解方程 2 sin²θ + cos θ = 1 时，可用 1 - cos²θ 替换 sin²θ，得到关于 cos θ 的二次方程。
Be careful with the domain when taking square roots; always consider both positive and negative solutions unless the quadrant restricts the sign.
开方时注意定义域；除非象限限制了符号，否则务必同时考虑正、负解。
3. Solving Basic Trig Equations | 解基本三角方程
Equations such as sin θ = 0.5, cos θ = -√2 general approach is to find the principal value using an inverse function, then use the graph or CAST diagram to locate all solutions in the given interval.
诸如 sin θ = 0.5、cos θ = -√2/2 或
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