📚 Solving Trigonometric Equations | 解三角方程
Trigonometric equations are a fundamental component of the Edexcel A-Level Pure Mathematics syllabus. They require you to find all angles that satisfy a given trigonometric relationship within a specified domain or as a general solution. This article systematically covers the key methods and strategies needed to solve these equations confidently.
三角方程是 Edexcel A-Level 纯数学大纲的基础组成部分。它要求你找出在指定定义域内或作为通解的所有角度，满足给定的三角关系。本文系统地涵盖了解决这类方程所需的关键方法和策略。
1. Basic Trigonometric Equations and General Solutions | 基本三角方程与通解
he inverse function sin⁻¹ gives the principal value, which lies in the range −90° ≤ sin⁻¹ ≤ 90°.
首先回忆正弦、余弦和正切的图像及主值。例如 sin x = 1/2 的主解为 x = 30°（或 π/6 弧度）。反函数 sin⁻¹ 给出的主值落在 −90° 到 90° 之间。
The general solution for sin x = k (|k| ≤ 1) is
x = nπ + (−1)ⁿα, n ∈ ℤ
where α is the principal value in radians. For cos x = k,
x = 2nπ ± α
and for tan x = k,
x = nπ + α
sin x = k (|k| ≤ 1) 的通解为 x = nπ + (−1)ⁿα，其中 α 是以弧度表示的主值，n ∈ ℤ。cos x = k 的通解为 x = 2nπ ± α；tan x = k 的通解为 x = nπ + α。
uired domain, typically 0 ≤ x < 2π or 0° ≤ x < 360°, unless the general solution is asked.
除非题目要求通解，否则依据要求的定义域给出答案，通常为 0 ≤ x < 2π 或 0° ≤ x < 360°。
2. The Unit Circle and CAST Diagram | 单位圆与 CAST 图
The CAST diagram is a powerful tool for identifying the signs of trigonometric functions in each quadrant: Cosine positive in the 4th quadrant, All positive in the 1st, Sine positive in the 2nd, Tangent positive in the 3rd.
CAST 图是判断各象限内三角函数符号的强大工具：第四象限余弦为正，第一象限全正，第二象限正弦为正，第三象限正切为正。
s also positive in Quadrant II, so the second solution in 0° to 360° is 180° − 30° = 150°.
对于 sin x = 0.5，主角为 30°（第一象限）。利用 CAST，正弦在第二象限也为正，因此在 0° 到 360° 内的第二个解是 180° − 30° = 150°。
rence angle is 60° (cos⁻¹(1/2)), so solutions are 180° − 60° = 120° and 180° + 60° = 240°.
对于 cos x = −1/2，余弦在第二和第三象限为负。参考角为 60°（cos⁻¹(1/2)），因此解为 180° − 60° = 120° 和 180° + 60° = 240°。
3. Equations with Multiple Angles | 含多重角的方程
a suitable range (e.g., 0° ≤ 2x < 720°). Then divide by 2 to get x = 15°, 75°, 195°, 255°.
要解 sin(2x) = 0.5 在 0° ≤ x < 360° 内的解，首先求解 2x：在合适的范围内（如 0° ≤ 2x < 720°），2x = 30°, 150°, 390°, 510°, ... 然后除以 2，得到 x = 15°, 75°, 195°, 255°。
Always extend the range for the multiple angle first, then find all solutions within that extended range before dividing.
务必先将多重角的范围进行扩展，找出该扩展范围内的所有解，再除以系数得到最终 x 的值。
ve coefficients, e.g., sin(−x) = 0.5 could be rewritten using odd symmetry as −sin x = 0.5 ⇒ sin x = −0.5.
当系数为负时要小心，例如 sin(−x) = 0.5 可利用奇对称性改写为 −sin x = 0.5 ⇒ sin x = −0.5。
4. Using Trigonometric Identities | 使用三角恒等式
Essential identities include the Pythagorean identity sin²x + cos²x ≡ 1 and the tangent identity tan x ≡ sin x / cos x. Reciprocal identities: sec x ≡ 1/cos x, cosec x ≡ 1/sin x, cot x ≡
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图