📚 Trigonometric Equations and Identities for Edexcel A-Level | Edexcel A-Level 三角方程与恒等式
the Pure Mathematics papers. Questions often require you to solve equations such as sin(2x – ner_(4)-062, test your ability to manipulate angles, apply CAST diagrams, and use key identities with confidence. This article provides a thorough, step-by-step revision guide that covers every essential technique you need.
在 Edexcel A-Level 数学大纲中，三角方程和恒等式是纯数试卷的核心内容。试题常要求你解如 sin(2x – 30°) = 0.4 在 0° ≤ x ≤ 360° 的方程，或者证明恒等式并求出精确值。许多历年真题——包括在 pdfjoiner_(4)-062 这类合集中出现的题目——都在考察你是否能熟练地处理角度、运用 CAST 图以及正确使用关键恒等式。本文提供一个完整、逐步的复习指南，涵盖你所需要的所有关键技巧。
1. Understanding the Trigonometric Functions | 理解三角函数
and tangent on the unit circle. For any angle θ measured anticlockwise from the positive x‑ igin: tan θ = sin θ / cos θ. Knowing the exact values for 0°, 30°, 45°, 60° and 90° is non‑negotiable — these appear in almost every paper.
在解方程之前，你必须对单位圆上正弦、余弦和正切的定义了如指掌。对于从正 x 轴逆时针测量的任意角 θ，圆上点的坐标为 (cos θ, sin θ)。tan θ 则是经过原点的直线的斜率：tan θ = sin θ / cos θ。牢记 0°、30°、45°、60° 和 90° 的精确值是没有商量余地的——它们几乎在每份试卷中都会出现。
For angles greater than 90°, the signs of sin, cos and tan are determined by the quadrant. In the first quadrant (0° to 90°) all are positive; in the second (90° to 180°) only sin is positive; in the third (180° to 270°) only tan is positive; and in the fourth (270° to 360°) only cos is positive — the CAST rule.
对于大于 90° 的角，sin、cos 和 tan 的符号由象限决定。第一象限（0° 到 90°）全为正；第二象限（90° 到 180°）仅 sin 为正；第三象限（180° 到 270°）仅 tan 为正；第四象限（270° 到 360°）仅 cos 为正——这就是 CAST 规则。
2. The Fundamental Trigonometric Identities | 基本三角恒等式
+ cos²θ = 1, and tan θ = sin θ / cos θ. From the first identity, you can derive sin²θ = 1 – cos²θ and cos²θ = 1 – ften used to convert a mixed equation into a quadratic in a single trigonometric function.
两个恒等式是几乎所有方程求解与证明工作的基础：sin²θ + cos²θ = 1，以及 tan θ = sin θ / cos θ。由第一个恒等式可推出 sin²θ = 1 – cos²θ 和 cos²θ = 1 – sin²θ。它们常常用来将混合方程转化为关于单一三角函数的二次方程。
For example, to solve 2 sin²x + 3 cos x = 0, you replace sin²x with 1 – cos²x to obtain 2(1 – cos²x) + 3 cos x = 0, which simplifies to 2 cos²x – 3 cos x – 2 = 0, a quadratic in cos x.
例如，要解 2 sin²x + 3 cos x = 0，用 1 – cos²x 代替 sin²x 得到 2(1 – cos²x) + 3 cos x = 0，化简为 2 cos²x – 3 cos x – 2 = 0，这是一个关于 cos x 的二次方程。
3. Solving Basic Equations: sin x = k, cos x = k, tan x = k | 解基本方程：sin x = k, cos x = k, tan x = k
terval, identify the acute principal value α, then use symmetry properties: sin x = sin(180° – x) and sin x = sin(x + 360°). For cos x = k, the symmetric angle is 360° – α; for tan x = k, the period is 180°.
当你看到 sin x = 0.5 时，计算器给出的 x = 30°。但有无穷多个角具有相同的正弦。要想找到给定区间内的所有解，先找出锐角主值 α，然后利用对称性：sin x = sin(180° – x) 且 sin x = sin(x + 360°)。对于 cos x = k，对称角是 360° – α；对于 tan x = k，周期为 180°。
Example: solve cos x = 1/2 for 0° ≤ x ≤ 360°. α = 60°. The second solution is 360° – 60° = 300°. Thus x = 60°, 300°. Always state answers clearly and in the required interval.
例子：解 cos x = 1/2 ，0° ≤ x ≤ 360°。 α = 60°。第二个解是 360° – 60° = 300°。因此 x = 60°, 300°。务必清楚写出答案并限制在指定区间内。
4. Using the CAST Diagram Effectively | 有效使用 CAST 图
The CAST diagram (or the ‘ stakes. Draw the four quadrants and label the positive functions. When given, say, sin x = – and fourth quadrants, the solutions between 0° and 360° are 180° + 44.4° = 224.4° and 360° – 44.4° = 315.6°.
CAST 图（或“All Students Take Calculus”助记法）可以帮助你避免符号错误。画出四个象限并标出正函数。当给出如 sin x = –0.7 时，锐角参考角为 arcsin(0.7) ≈ 44.4°。因为正弦在第三和第四象限为负，在 0° 到 360° 之间的解是 180° + 44.4° = 224.4° 和 360° – 44.4° = 315.6°。
Always sketch a quick CAST diagram in your exam — it takes seconds and saves marks. Mark the specified interval clearly and adjust by adding or subtracting 360° if solutions fall outside.
考试中一定要快速画出 CAST 图——只需几秒却能挽回分数。清楚标出指定区间，若解落在区间外则通过加减 360° 进行调整。
5. Equations Involving Multiple Angles | 含倍角的方程
Equations like sin 2x = √3/2 or cos(3x – el papers. The key is to treat the entire angle argument as a new variable. Solve for the inner expression first, then rearrange for x at the end.
像 sin 2x = √3/2 或 cos(3x – 45°) = 0.2 这样的方程经常出现在 Edexcel 试卷中。关键是把整个角参数当作一个新变量。先解出内部的表达式，最后再整理求出 x。
sin 2x = 1/2 for 0° ≤ x ≤ 360°. Let θ = 2x. Then sin θ = 1/2 → θ = 30°, 150°, 390°, 510°, … θ/2 = 15°, 75°, 195°, 255°. Always check that your final x values lie in the given range.
例子：解 sin 2x = 1/2 ，0° ≤ x ≤ 360°。设 θ = 2x。则 sin θ = 1/2 → θ = 30°, 150°, 390°, 510°, …（因为 θ 的周期要算到 720°，因为 x 到达 360°）。然后 x = θ/2 = 15°, 75°, 195°, 255°。务必检查最终的 x 值是否落在给定范围内。
6. Quadratic Trigonometric Equations | 二次三角方程
der problems reduce to a quadratic in sin x, cos x or tan x. Typical forms include 3 sin²x – sin x – ing, factorise or use the quadratic formula, but you must reject any solutions that fall outside the range [–1, 1] for sin and cos.
许多较难的问题都归结为关于 sin x、cos x 或 tan x 的二次方程。典型形式包括 3 sin²x – sin x – 1 = 0 或 2 cos²x + 5 sin x = 4。经过替换后，分解因式或使用求根公式，但一定要舍去任何落在 sin 和 cos 的 [–1, 1] 范围之外的解。
Example: solve 2 sin²x – sin x – 1 = 0 for 0° ≤ x ≤ 360°. Factorising gives (2 sin x + 1)(sin x – 1) = 0. Hence sin x = –1/2 or sin x = 1. For sin x = 1, x = 90°. For sin x = –1/2, reference angle 30°, x = 210°, 330°. Final solutions: x = 90°, 210°, 330°.
例子：解 2 sin²x – sin x – 1 = 0，0° ≤ x ≤ 360°。因式分解得 (2 sin x + 1)(sin x – 1) = 0。因此 sin x = –1/2 或 sin x = 1。对于 sin x = 1，x = 90°。对于 sin x = –1/2，参考角 30°，x = 210°, 330°。最终解：x = 90°, 210°, 330°。
7. Using Identities to Simplify Equations | 利用恒等式简化方程
os x can be divided by cos x (provided cos x ≠ 0) to give tan x = 2/3. More complex problems involve expressions like 2 sin x cos x, which is sin 2x.
当一个方程包含不止一种三角函数时，恒等式就是你将其化为单一函数的工具。比如 3 sin x = 2 cos x 可以除以 cos x（前提是 cos x ≠ 0）得到 tan x = 2/3。更复杂的问题会涉及如 2 sin x cos x 这样的式子，它等于 sin 2x。
Similarly, cos²x – sin²x can be rewritten as cos 2x. Recognising these forms often turns a seemingly difficult equation into a standard type.
类似地，cos²x – sin²x 可改写成 cos 2x。识别出这些形式常常能将看似困难的方程转化为标准类型。
8. Reciprocal Functions: sec, cosec, cot | 倒数函数：sec, cosec, cot
ar in many proofs and equations. Use these to convert expressions, just as you would with sin²x + cos²x = 1.
Edexcel 要求你处理 sec x = 1/cos x，cosec x = 1/sin x 和 cot x = 1/tan x = cos x / sin x。恒等式 1 + tan²x = sec²x 和 1 + cot²x = cosec²x 出现在许多证明和方程中。就像使用 sin²x + cos²x = 1 一样，你要用它们来转换表达式。
c²x = 4 tan x for 0° ≤ x ≤ 360°. Replace sec²x with 1 + tan²x: 1 + tan²x = 4 tan x → tan²x – 4 tan x + 1 = 0. Solve the quadratic to find tan x, then find x.
例子：解 sec²x = 4 tan x，0° ≤ x ≤ 360°。用 1 + tan²x 替换 sec²x：1 + tan²x = 4 tan x → tan²x – 4 tan x + 1 = 0。解出二次方程求得 tan x，再求出 x。
9. The R cos(x ± α) and R sin(x ± α) Form | R cos(x ± α) 和 R sin(x ± α) 形式
d into a single sine or cosine function. Write a sin x + b cos x = R sin(x + α), where R = √a² + b² and α = arctan(b/a). The choice between sin(x + α), sin(x – α), cos(x + α) or cos(x – s 3 sin x + 4 cos x = 2, and for finding the maximum and minimum values of the expression.
像 a sin x + b cos x 这样的表达式可以合并成单一的正弦或余弦函数。写作 a sin x + b cos x = R sin(x + α)，其中 R = √a² + b²，α = arctan(b/a)。在 sin(x + α)、sin(x – α)、cos(x + α) 或 cos(x – α) 之间的选择取决于 a 和 b 的符号。这一技巧对于解如 3 sin x + 4 cos x = 2 的方程，以及求该表达式的最大值和最小值来说至关重要。
Example: express 3 sin x + 4 cos x in the form R sin(x + α). R = √3² + 4² = 5; tan α = 4/3 ⇒ α ≈ 53.1°. So 3 sin x + 4 cos x = 5 sin(x + 53.1°). The maximum is 5 and the minimum is –5.
例子：将 3 sin x + 4 cos x 表示成 R sin(x + α) 形式。R = √3² + 4² = 5；tan α = 4/3 ⇒ α ≈ 53.1°。所以 3 sin x + 4 cos x = 5 sin(x + 53.1°)。最大值为 5，最小值为 –5。
10. Proving Trigonometric Identities | 证明三角恒等式
Proof questions require a logical sequence of steps, starting from one side and using known identities to reach the other side. Common strategies include rewriting everything in terms of sin and cos, combining fractions over a common denominator, and using Pythagorean identities.
证明题要求逻辑清晰的步骤，从等式的一边出发，利用已知恒等式推导到另一边。常用的策略包括将各项都写成 sin 和 cos 的形式、通分合并分式，以及使用勾股恒等式。
Example: prove (sin x / (1 + cos x)) + ((1 + cos x) / sin x) ≡ x + (1 + cos x)², simplify using sin²x + cos²x = 1, and finally reach 2/sin x = 2 cosec x.
例子：证明 (sin x / (1 + cos x)) + ((1 + cos x) / sin x) ≡ 2 cosec x。从左式开始，取得公分母 sin x(1 + cos x)，将分子展开为 sin²x + (1 + cos x)²，利用 sin²x + cos²x = 1 化简，最终得到 2/sin x = 2 cosec x。
Always show every algebraic step and state which identity you are using. Marks are awarded for clear reasoning.
务必展示每一步代数推导并说明你使用了哪条恒等式。清晰的推理过程才能得分。
11. Solving Equations in Radians | 用弧度制解方程
ls are in terms of π. For instance, sin x = 1/2 for 0 ≤ x ≤ 2π gives x = π/6 and 5π/6. Always know your exact radian values: π/6, π/4, π/3, π/2 and their multiples. Set your calculator to radian mode to avoid mistakes.
Edexcel 试卷经常在角度制与弧度制之间切换。用弧度解方程时，原理完全相同，但参考角和区间要用 π 来表示。例如，sin x = 1/2 在 0 ≤ x ≤ 2π 时给出 x = π/6 和 5π/6。务必牢记精确的弧度值：π/6, π/4, π/3, π/2 及其倍数。将计算器设为弧度模式可避免出错。
When given an interval such as – efully count the negative rotations. A common mistake is to miss solutions in the negative range.
当给定如 –π ≤ x ≤ π 这样的区间时，画出单位圆并仔细计算顺时针旋转得到的角。常见错误是遗漏负范围内的解。
12. Exam Technique and Common Pitfalls | 考试技巧与常见误区
Many students lose marks by forgetting to check that all final answers are within the specified domain. After solving, circle back to the question and verify each value. Another pitfall is dividing by a trigonometric expression that could be zero — always state the condition or factorise instead. When using the quadratic formula, remember that the variable is the trigonometric function, not x itself.
许多学生因忘记检查最终答案是否全在指定定义域内而失分。解完方程后，要回看题目并验证每个值。另一个陷阱是除以可能为零的三角函数表达式——始终声明条件或改用因式分解。使用求根公式时，记住变量是三角函数，而非 x 本身。
Finally, manage your time: if a question asks for exact values, do not waste time on decimal approximations. Write √2 t questions like those in pdfjoiner_(4)-062 will sharpen your skills and build confidence.
最后，管理好时间：如果题目要求精确值，就不要浪费时间写小数近似。写 √2、π/3 等，并在需要时分母有理化。反复练习像 pdfjoiner_(4)-062 中的历年真题，将提升你的解题技巧并建立信心。
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