📚 Solving Trigonometric Equations | 解三角方程
h as mechanics. Mastering the techniques to find all solutions within a specified interval – typically 0° ≤ x ≤ 360° or 0 ≤ x ≤ 2π – ss in the examination. This article brings together key methods, common pitfalls, and step‑by‑step strategies to help you approach any trigonometric equation with confidence.
三角方程贯穿 Edexcel A‑Level 数学的整个教学大纲，既出现在纯数学部分，也出现在力学等应用场景中。掌握在指定区间（通常是 0° ≤ x ≤ 360° 或 0 ≤ x ≤ 2π）内求出所有解的方法，对考试成功至关重要。本文汇集了关键方法、常见错误和循序渐进的策略，帮助你自信地应对任何三角方程。
1. Building from the Unit Circle | 从单位圆出发
ngle α, obtained from a calculator, the other solutions in the range 0° to 360° are found using symmetry properties: sin(180° – θ) = sin θ, cos(360° – the correct quadrants before listing final answers is the single most reliable technique.
所有三角方程都依赖于正弦、余弦和正切的周期性，而单位圆能最直观地展示这种性质。对于由计算器得出的某个锐角 α，可以利用对称性在 0° 到 360° 范围内找出其余的解：sin(180° – θ) = sin θ，cos(360° – θ) = cos θ，tan(180° + θ) = tan θ。在列出最终答案之前，先把角度放到正确的象限中，是最为可靠的方法。
Key quadrant rules can be summarised in a simple table to avoid sign errors:
关键的象限规则可以总结在一个简单的表格中，以避免符号错误：
| Function / 函数 | Quadrant I / 第一象限 (0°‑90°) | Quadrant II / 第二象限 (90°‑180°) | Quadrant III / 第三象限 (180°‑270°) | Quadrant IV / 第四象限 (270°‑360°) |
|---|---|---|---|---|
| sin | + | + | − | − |
| cos | + | − | − | + |
| tan | + | − | + | − |
2. First‑step procedure: isolate the trig function | 第一步操作：将三角比分离
that one side contains only a single trigonometric ratio, such as sin x = 0.5, cos(2x) = −√3/2 or tan(x/3) = 1. Having a ‘ steps far simpler. Do not be tempted to dive into identities until this isolation is done.
在考虑角度之前，先把方程重新整理，使一边只含一个三角比，例如 sin x = 0.5、cos(2x) = −√3/2 或 tan(x/3) = 1。将“纯净”的三角比等于一个数值，能让后续步骤简单得多。在完成这一步隔离开之前，不要急着引入恒等式。
3. Using principal values and the CAST diagram | 利用主值和 CAST 图
sing the inverse trigonometric function on your calculator. For positive k, the calculator gives an acute angle. Next, use the CAST diagram (or quadrant rules) to determine all angles within the given domain that have the same sign. This method works equally well for negative k, where the calculator returns a negative angle – just add 360° or 2π to obtain a positive working value before applying CAST.
一旦方程变为 trig(θ) = k 的形式，首先用计算器上的反三角函数求出主值 θ₁。当 k 为正时，计算器给出的是锐角。接着，利用 CAST 图（或象限规则）确定给定范围内符号相同的所有角度。当 k 为负时，计算器会返回负角 —— 只需加上 360° 或 2π 得到正的工作角度，再应用 CAST 图即可。
A common layout for the CAST diagram is shown below; mark the quadrants where the relevant function is positive.
下面给出了 CAST 图的常见布局；在相关函数为正的象限上做标记。
S (sin +) | A (all +)
---------------|---------------
T (tan +) | C (cos +)
4. Dealing with multiple angles: 2x, 3x, x/2 | 处理复合角：2x、3x、x/2
When the argument is a multiple of the variable – for example sin 2x = 0.7, 0° ≤ x ≤ 360° – n divide by 2 to obtain x. Failing to adjust the interval is one of the most frequent errors in A‑Level scripts.
当变量是复合角时 —— 例如 sin 2x = 0.7，0° ≤ x ≤ 360° —— 必须首先扩大复合角的区间。这里的 2x 范围为 0° 到 720°。在扩大后的区间内找出所有关于 2x 的解，然后除以 2 得到 x。忘记调整区间是 A‑Level 答卷中最常见的错误之一。
A systematic method is to write “let u = 2x, so 0° ≤ u ≤ 720°” and finally convert back to x. This reduces mental load and helps avoid missing solutions.
一个系统的方法是写出“令 u = 2x，则 0° ≤ u ≤ 720°”，求解 u，最后再转换回 x。这样能减轻思维负担，并有助于避免漏解。
5. Solving quadratic trigonometric equations | 解二次型三角方程
Many examination questions present an equation such as 2 sin² x + sin x – 1 = 0 or 3 cos² x – 5 cos x + 2 = 0. Treat the trig function as an algebraic variable – for instance let y = sin x – and solve the quadratic. Once you have values for y, discard any solutions outside [–1, 1] for sine and cosine, then proceed to solve the resulting simple trig equations.
许多考试题目会给出像 2 sin² x + sin x – 1 = 0 或 3 cos² x – 5 cos x + 2 = 0 这样的方程。把三角函数当作代数变量 —— 例如令 y = sin x —— 然后解二次方程。得到 y 的值后，舍弃正弦和余弦中落在 [–1, 1] 之外的解，接着求解剩下的简单三角方程。
For tangent, the range is all real numbers, so no quadratic root needs to be rejected; however, always check the final angles lie within the requested interval.
对于正切，其值域是全体实数，因此无需舍弃任何二次根；但要始终检查最终角度是否位于题目要求的区间内。
6. Applying fundamental identities to simplify | 运用基本恒等式进行化简
to express everything in terms of a single function. In this case, replace sin² x with (1 – . Similarly, equations involving both tan and sec can be tackled with 1 + tan² x = sec² x.
当方程混合了不同的三角函数时，例如 2 sin² x + 3 cos x = 0，可以利用勾股恒等式 sin² x + cos² x = 1，将所有项用同一个函数表示。在这个例子中，将 sin² x 替换为 (1 – cos² x)，得到关于 cos x 的二次方程。同样，涉及 tan 和 sec 的方程可以用 1 + tan² x = sec² x 来处理。
Key identities to memorise:
需要牢记的关键恒等式：
- sin² θ + cos² θ = 1 → sin² θ = 1 – cos² θ, cos² θ = 1 – sin² θ
- tan θ = sin θ / cos θ
- 1 + tan² θ = sec² θ
- 1 + cot² θ = cosec² θ
7. Equations involving sec, cosec and cot | 涉及 sec、cosec 和 cot 的方程
ritten. For example, sec x = 2 becomes 1/cos x = 2, i.e. cos x = 1/2. Similarly, cosec x = –√2 gives sin x = –1/√2. Always convert to sine, cosine or tangent before solving.
含有倒数三角函数的方程常常让学生感到恐惧，但一旦重写就能化为熟悉的形式。例如，sec x = 2 变成 1/cos x = 2，即 cos x = 1/2。同理，cosec x = –√2 给出 sin x = –1/√2。在求解之前，总要先把方程化为正弦、余弦或正切的形式。
8. Using the sine and cosine wave (graphical approach) | 利用正弦波和余弦波的图像法
intersections. This method is particularly useful when checking that no solution is missed and when dealing with transformations such as y = 3 sin(x + 30°).
在所需区间上画出相关三角函数的图像，可以提供强大的视觉检查。标出直线 y = k，找出交点。在检查是否有遗漏的解以及处理 y = 3 sin(x + 30°) 这类变换时，图像法特别有用。
9. Radian measure and exact values | 弧度制和精确值
Edexcel A-Level requires fluency in both degrees and radians. Many equations set in radians demand exact solutions involving π/6, π/4, π/3, etc. Ensure you know the exact values of sine, cosine and tangent for these standard angles without relying on a calculator. A quick reference table is invaluable for revision.
Edexcel A‑Level 要求熟练掌握角度制和弧度制。许多以弧度设置的方程需要给出包含 π/6、π/4、π/3 等的精确解。务必在不依赖计算器的情况下，记住这些标准角的正弦、余弦和正切精确值。快速查阅表格对复习极有帮助。
| Angle (rad) / 角度（弧度） | sin | cos | tan |
|---|---|---|---|
| 0 | 0 | 1 | 0 |
| π/6 | ½ | √3/2 | 1/√3 |
| π/4 | √2/2 | √2/2 | 1 |
| π/3 | √3/2 | ½ | √3 |
| π/2 | 1 | 0 | undefined / 不存在 |
10. Factoring out trig functions (avoid dividing by zero) | 提取三角函数的公因式（避免除以零）
s x = sin x, never cancel sin x immediately. Instead, bring all terms to one side: 2 sin x cos x – sin x = 0, then factor: sin x (2 cos x – 1) = 0. This yields all possibilities: sin x = 0 or cos x = ½.
一个危险的错误是两边同除以可能为零的三角表达式，从而丢失解。例如，对于 2 sin x cos x = sin x，绝不能立即约去 sin x。而应将所有项移到一边：2 sin x cos x – sin x = 0，然后分解因式：sin x (2 cos x – 1) = 0。这样就能得到所有可能情况：sin x = 0 或 cos x = ½。
11. Handling double‑angle and compound‑angle identities | 处理倍角与和角恒等式
Equations such as cos 2x + sin x = 0 or sin(x + 60°) = 2 cos x require the use of double‑angle or compound‑angle formulas. Choose the version of cos 2x (e.g. 1 – 2 sin² x or 2 cos² x – angle sum appears, expand using appropriate addition formulas and then combine like terms.
像 cos 2x + sin x = 0 或 sin(x + 60°) = 2 cos x 这类方程，需要使用倍角或和角公式。选择 cos 2x 的某种表达形式（例如 1 – 2 sin² x 或 2 cos² x – 1），以最高效地化简方程。当出现角度和时，用相应的加法公式展开，然后合并同类项。
Standard expansions to recall:
需要记住的标准展开式：
- sin(A ± B) = sin A cos B ± cos A sin B
- cos(A ± B) = cos A cos B ∓ sin A sin B
- tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)
12. Writing the final answer set | 写出最终解集
After solving, present your solutions as a list of values, clearly indicating the unit (degrees or radians). Check that every solution falls strictly inside the specified interval – endpoints may be included or excluded depending on the inequality sign. A final scan of the CAST diagram or a graphical sketch can catch any overlooked angles.
求解后，将答案以数值列表的形式呈现，清楚地标明单位（度或弧度）。检查每个解是否严格落在指定的区间内 —— 端点可能根据不等号而被包含或排除。最后再扫一眼 CAST 图或草图，可以揪出任何漏掉的角度。
Published by TutorHao | Mathematics Revision Series | aleveler.com
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