Mastering Trigonometric Equations 0°-180° for Edexcel A-Level | 掌握 Edexcel A-Level 三角函数方程：0° 到 180°

📚 Mastering Trigonometric Equations 0°-180° for Edexcel A-Level | 掌握 Edexcel A-Level 三角函数方程：0° 到 180°

Trigonometric equations are a high-frequency topic in Edexcel A-Level Mathematics. Many questions restrict solutions to the interval 0° ≤ θ ≤ 180°, and success depends on understanding exact values, graph symmetry and the link between degrees and radians.

三角方程是 Edexcel A-Level 数学中的高频考点。许多题目将解限制在 0° ≤ θ ≤ 180° 的区间内，解题成功的关键在于掌握精确值、图像对称性以及角度制与弧度制的联系。

1. Why the 0° to 180° Range Matters | 为什么 0° 到 180° 范围很重要

The range 0° to 180° covers the first two quadrants of the unit circle. In this interval, sin θ is non-negative, cos θ changes from 1 to -1, and tan θ changes from 0 through infinity to 0.

0° 到 180° 的范围覆盖单位圆的前两个象限。在这个区间内，sin θ 为非负，cos θ 从 1 变为 -1，而 tan θ 从 0 经过无穷大再回到 0。

Examiners frequently use this restricted interval to test whether candidates can identify both the principal solution and the supplementary solution for equations involving sine. Missing the second solution is one of the most common errors in A-Level trigonometry.

考官经常使用这个受限区间来考查考生能否找出正弦方程的主解和补角解。漏掉第二个解是 A-Level 三角函数中最常见的错误之一。

Understanding why the 0° to 180° interval behaves differently for sine, cosine and tangent is therefore the foundation for accurate equation solving.

因此，理解 0° 到 180° 区间对正弦、余弦和正切的不同表现，是准确解方程的基础。

2. Angles, Radians and the 180° Benchmark | 角度、弧度与 180° 基准

In Edexcel A-Level, angles may be given in degrees or radians. The key conversion is that 180° equals π radians, so 90° equals π/2 rad and 45° equals π/4 rad.

在 Edexcel A-Level 中，角度可以以度或弧度给出。关键换算是 180° 等于 π 弧度，因此 90° 等于 π/2 弧度，45° 等于 π/4 弧度。

180° = π rad, 90° = π/2 rad, 60° = π/3 rad, 45° = π/4 rad, 30° = π/6 rad

Always check whether the question asks for answers in degrees or radians. Writing 30° instead of

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