📚 G-3-3: Solving Trigonometric Equations Step by Step | G-3-3：逐步攻克三角方程
Trigonometric equations often appear intimidating, but with a structured approach they become one of the most rewarding topics in A‑Level Mathematics. This guide unpacks the G-3-3 skill set, showing you how to move from basic single‑ratio equations to those involving identities, multiple angles, and interval restrictions.
三角方程常让人望而生畏，但只要方法得当，它就会成为 A‑Level 数学中最有成就感的知识点之一。本文梳理 G-3-3 所要求的技能，带你从单三角函数方程起步，一路攻克恒等式、多倍角与区间限定等各个难关。
1. What Are Trigonometric Equations? | 什么是三角方程？
A trigonometric equation is any equation that contains a trigonometric function of an unknown angle, such as sin x = 0.5 or 2cos²θ − cos θ − 1 = 0. Our goal is to find all angle values that make the equation true, usually within a specified domain like 0° ≤ x ≤ 360° or 0 ≤ θ < 2π.
三角方程是含有未知角三角函数关系的等式，例如 sin x = 0.5 或 2cos²θ − cos θ − 1 = 0。我们的目标是找出使等式成立的所有角度，通常要求在给定范围（如 0° ≤ x ≤ 360° 或 0 ≤ θ < 2π）内求解。
Because trig functions are periodic, a single equation can have infinitely many solutions. In an exam, you will almost always be asked to list all solutions within a given interval.
由于三角函数具有周期性，一个方程通常有无数个解。考试中几乎总会要求你列出某一特定区间内的全部解。
2. The Unit Circle and Key Angles | 单位圆与关键角
Before solving equations, you need instant recall of exact values for 0°, 30°, 45°, 60° and 90° (or 0, π/6, π/4, π/3, π/2 in radians). These values come directly from the unit circle or the two standard right‑angled triangles.
解方程之前，你必须对 0°、30°、45°、60° 和 90°（或弧度制 0、π/6、π/4、π/3、π/2）的精确三角函数值烂熟于心。这些值源于单位圆或两个标准直角三角形。
| Angle θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° (0) | 0 | 1 | 0 |
| 30° (π/6) | 1/2 | √3/2 | 1/√3 |
| 45° (π/4) | 1/√2 | 1/√2 | 1 |
| 60° (π/3) | √3/2 | 1/2 | √3 |
| 90° (π/2) | 1 | 0 | Undefined |
Memorising the signs of each function in the four quadrants is equally important. A popular mnemonic is ‘All Students Take Calculus’ (All, Sine, Tangent, Cosine positive in QI, QII, QIII, QIV respectively).
同样重要的是记住各函数在四个象限的正负号。常见口诀是“All Students Take Calculus”（第一象限全正，第二象限仅正弦正，第三象限仅正切正，第四象限仅余弦正）。
3. General Solutions for Sine and Cosine | 正弦与余弦的通解
For an equation of the form sin θ = k, start by finding the principal value θ₁ = arcsin(k) using your calculator. Then obtain the second solution in the 0°–360° range using the symmetry of the sine curve: θ₂ = 180° − θ₁ (if working in degrees) or θ₂ = π − θ₁ (in radians).
对于 sin θ = k 型方程，先用计算器求主值 θ₁ = arcsin(k)。然后利用正弦曲线的对称性，找到 0°–360° 范围内的第二个解：θ₂ = 180° − θ₁（角度制）或 θ₂ = π − θ₁（弧度制）。
sin θ = k ⇒ θ = θ₁, 180° − θ₁ (plus multiples of 360°)
sin θ = k ⇒ θ = θ₁, 180° − θ₁（加上 360° 的整数倍）
For cos θ = k, the second solution lies symmetrically on the other side of 0° or 360°: θ₂ = 360° − θ₁ (or 2π − θ₁). Remember that cosine is positive in QI and QIV, giving two solutions per cycle.
对于 cos θ = k，第二个解出现在 0° 或 360° 的另一侧对称位置：θ₂ = 360° − θ₁（或 2π − θ₁）。记住余弦在第一、四象限为正，每个周期有两个解。
4. Solving tan Equations | 解正切方程
Tangent equations behave differently because tan has a period of 180° (π rad). If tan θ = k, the calculator gives the principal value θ₁. All other solutions are obtained by adding or subtracting multiples of 180°: θ = θ₁ + 180°·n, n ∈ ℤ.
正切方程规律不同，因为 tan 的周期为 180°（π 弧度）。若 tan θ = k，计算器给出主值 θ₁，其余解只需加或减 180° 的整数倍：θ = θ₁ + 180°·n，n ∈ ℤ。
tan θ = k ⇒ θ = θ₁ + 180°n (n ∈ ℤ)
tan θ = k ⇒ θ = θ₁ + 180°n (n 为整数)
Never extend the sine/cosine symmetry logic to tan; always use the 180° periodicity. Students who confuse this often lose marks by introducing spurious solutions.
切勿将正余弦对称规则套用到正切上，务必使用 180° 周期性。混淆这一点的同学常会引入增根，白白丢分。
5. Quadratic Form Trigonometric Equations | 二次型三角方程
Equations like 2sin²x − sin x − 1 = 0 can be solved by substituting u = sin x, giving 2u² − u − 1 = 0. Factorise to (2u + 1)(u − 1) = 0, yielding u = 1 and u = −1/2. Then revert to sin x = 1 and sin x = −1/2, solving each simple trig equation separately.
像 2sin²x − sin x − 1 = 0 这样的方程，可用代换 u = sin x 得到 2u² − u − 1 = 0。分解因式得 (2u + 1)(u − 1) = 0，推出 u = 1 和 u = −1/2。然后依次解 sin x = 1 和 sin x = −1/2 这两个简单三角方程。
Always check that the substituted value is in the valid range for the trig function (e.g., −1 ≤ sin x ≤ 1). Discard any impossible values to avoid wasted work.
务必检查代换值是否在三角函数的值域内（例如 −1 ≤ sin x ≤ 1）。舍弃不可能的值，避免做无用功。
6. Using Trigonometric Identities | 使用三角恒等式
Many equations mix different functions. If you see both sin and cos, the identity sin²θ + cos²θ = 1 is your first tool. For example, 2sin²θ + 3cos θ = 0 can be rewritten entirely in terms of cos θ by substituting sin²θ = 1 − cos²θ.
许多方程混合了不同三角函数。同时出现 sin 和 cos 时，第一个工具是恒等式 sin²θ + cos²θ = 1。例如 2sin²θ + 3cos θ = 0，可用 sin²θ = 1 − cos²θ 把方程全部转化为关于 cos θ 的式子。
sin²θ = 1 − cos²θ or cos²θ = 1 − sin²θ
sin²θ = 1 − cos²θ 或 cos²θ = 1 − sin²θ
Double‑angle identities such as sin 2A = 2 sin A cos A and cos 2A = cos²A − sin²A are also frequently tested. Spotting when to apply them can turn a messy equation into a quadratic you already know how to solve.
倍角公式如 sin 2A = 2 sin A cos A 和 cos 2A = cos²A − sin²A 也经常考查。一旦识别出应用时机，原本复杂的方程就能化为你已掌握的二次型。
7. Equations with Multiple Angles | 多角方程
When the argument is 2x, 3θ or (x + 30°), first solve for the compound angle as if it were a single variable, then adjust the interval. For instance, to solve sin 2θ = 0.5 for 0° ≤ θ ≤ 360°, let X = 2θ. The condition becomes 0° ≤ X ≤ 720°. Solve sin X = 0.5 over this doubled interval, giving X = 30°, 150°, 390°, 510°. Finally divide each by 2 to recover θ.
当角度以 2x、3θ 或 (x + 30°) 等形式出现时，应先将复合角视作一个整体求解，再调整区间。例如，解 sin 2θ = 0.5（0° ≤ θ ≤ 360°），设 X = 2θ，则 X 的范围变为 0° ≤ X ≤ 720°。在此扩大区间内解 sin X = 0.5，得 X = 30°、150°、390°、510°，最后每个值除以 2 得到 θ。
Students frequently forget to expand the interval, which leads to missing solutions. Write the new interval explicitly in your working to avoid this common mistake.
同学们常忘记将区间同步扩大，导致漏解。在草稿中明确写出新区间，可有效避免这一常见错误。
8. Finding Solutions Within a Given Interval | 在给定区间内求解
Once you have the general solution, list values systematically by adding or subtracting the period until you leave the required interval. Always check the endpoints carefully: a solution equal to an endpoint is included if the interval uses ≤, but excluded with <.
得到通解后，通过逐步加或减周期，依次列举符合条件的值，直至超出给定区间。需仔细核对端点：若区间用 ≤，等于端点的解要保留；若用 <，则应剔除。
Organise your final answers in ascending order and, unless instructed otherwise, provide them to the required degree of accuracy (often 1 decimal place or exact values in terms of π).
最终答案要按升序排列，并按题目要求保留精度（通常是一位小数，或以 π 表示的精确值）。
9. Common Pitfalls and How to Avoid Them | 常见陷阱与避免方法
- Pitfall: Applying sine symmetry to tangent equations.
Fix: Always use the 180° period for tan; sketch the tan graph if unsure. - 陷阱：将正弦对称规则套用到正切方程。
对策：正切一律使用 180° 周期；不确定时勾画正切图像。 - Pitfall: Forgetting negative angles when using inverse trig functions.
Fix: Consider the quadrant signs and generate all solutions, even if the calculator gives a negative principal value. - 陷阱：使用反三角函数时忽略负角。
对策：关注象限正负号，即使计算器给出的主值为负，也要生成全部解。 - Pitfall: Dividing both sides of an equation by a trig term (e.g., dividing by cos θ) and losing a factor.
Fix: Bring all terms to one side and factorise; never casually cancel a trig expression. - 陷阱：方程两边同除以一个三角函数项（如除以 cos θ），导致因子丢失。
对策：将所有项移至同侧后分解因式，绝不随意约去三角函数表达式。
10. Worked Examples from Past Papers | 真题示例详解
Example 1: Solve 2cos²x − 1 = 0 for 0° ≤ x < 360°.
Rewrite as cos²x = 1/2, so cos x = ±1/√2. For cos x = 1/√2, x = 45°, 315°. For cos x = −1/√2, x = 135°, 225°. The full solution set is {45°, 135°, 225°, 315°}.
例 1：解 2cos²x − 1 = 0，0° ≤ x < 360°。
改写为 cos²x = 1/2，∴ cos x = ±1/√2。cos x = 1/√2 时 x = 45°, 315°；cos x = −1/√2 时 x = 135°, 225°。解集为 {45°, 135°, 225°, 315°}。
Example 2: Solve sin 3θ = −0.5 for 0 ≤ θ < 2π.
Let X = 3θ, so 0 ≤ X < 6π. sin X = −0.5 ⇒ X = 7π/6, 11π/6, 19π/6, 23π/6, 31π/6, 35π/6. Dividing by 3: θ = 7π/18, 11π/18, 19π/18, 23π/18, 31π/18, 35π/18.
例 2：解 sin 3θ = −0.5，0 ≤ θ < 2π。
设 X = 3θ，则 0 ≤ X < 6π。sin X = −0.5 ⇒ X = 7π/6, 11π/6, 19π/6, 23π/6, 31π/6, 35π/6。各项除以 3 得 θ = 7π/18, 11π/18, 19π/18, 23π/18, 31π/18, 35π/18。
11. Visualising the Solutions Graphically | 图解辅助理解
Drawing a quick sketch of the relevant trig function and a horizontal line y = k can instantly confirm the number of solutions expected. For sin or cos, the intersections in one full period immediately show the pattern. This visual check is invaluable under exam pressure.
快速画出相关三角函数的草图，再画一条水平线 y = k，能立即确认应当有几个解。对 sin 或 cos 而言，一个完整周期内的交点布局一目了然。考试紧张时，这种图示检查尤为珍贵。
If you are solving on a graphics calculator, use the ‘analyse graph’ feature to find intersections, but always show your analytical method to gain full method marks.
若使用图形计算器，可利用“图像分析”功能求交点，但务必展示代数推导过程，才能获得完整的方法分。
12. Practising for Mastery – The G-3-3 Routine | 精熟练习——G-3-3 训练流程
Mastery of trigonometric equations comes through deliberate practice. Start with single‑ratio equations, progress to quadratic forms, then tackle those requiring identities or multiple angles. After each problem, check your solutions by substitution and by sketching.
三角方程的掌握离不开刻意练习。从单一函数方程起步，进阶到二次型，再攻克需要恒等式和多倍角的题目。每做完一题，通过代回检验和草图核实答案。
The G-3-3 animated exercises accompanying this guide are designed to build exactly this fluency. Revisit the animations whenever you feel stuck; seeing the angles move around the unit circle makes the algebra feel natural.
配套本文的 G-3-3 动画练习正是为此流畅度而设计。但凡遇到卡顿，不妨回看动画——亲眼观察角度在单位圆上移动，代数运算便会豁然开朗。
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