📚 Edexcel A Level Maths: Trigonometric Equations and Identities | 爱德思 A Level 数学：三角方程与恒等式
Trigonometric equations and identities appear in both AS and A Level Edexcel Pure Mathematics. Questions often combine exact values, interval solving, and proof-style manipulation, so a clear method is essential.
三角方程与恒等式在爱德思 AS 和 A Level 纯数学中都会出现。题目经常综合考查精确值、区间求解和证明式变形，因此清晰的方法至关重要。
1. The Edexcel Trigonometry Blueprint | 爱德思三角考点蓝图
Edexcel Pure Mathematics tests trigonometry through three stages: exact value recall, equation solving on a restricted interval, and proof or modelling using identities.
爱德思纯数学通过三个阶段考查三角学：精确值记忆、在规定区间上解方程，以及使用恒等式进行证明或建模。
Most questions are worth 4 to 8 marks. The mark scheme rewards the method, not just the final answer, so always show the principal value, the relevant symmetry lines or CAST diagram, and every solution inside the given interval.
大多数题目为 4 到 8 分。评分标准奖励方法而不仅是最终答案，因此始终展示主值、相关对称线或 CAST 图，以及区间内的所有解。
2. Core Identities You Must Know | 核心恒等式清单
These identities are printed in the Edexcel formula booklet, but exam success depends on quick recognition and accurate rearrangement.
这些恒等式印在爱德思公式表中，但考试成功取决于快速识别和准确变形。
- sin² θ + cos² θ = 1
- 1 + tan² θ = sec² θ
- 1 + cot² θ = cosec² θ
- sin(A ± B) = sin A cos B ± cos A sin B
- cos(A ± B) = cos A cos B ∓ sin A sin B
- sin 2θ = 2 sin θ cos θ
- cos 2θ = cos² θ − sin² θ = 2 cos² θ − 1 = 1 − 2 sin² θ
For sec, cosec and cot identities, Edexcel often asks you to prove or simplify expressions such as tan θ + cot θ = sec θ cosec θ.
对于 sec、cosec 和 cot 恒等式，爱德思经常要求你证明或化简诸如 tan θ + cot θ = sec θ cosec θ 的表达式。
3. Solving Basic Trigonometric Equations | 解基本三角方程
Always begin by isolating sin θ, cos θ or tan θ. For example, 4 sin θ = 1 becomes sin θ = 1/4.
始终先分离出 sin θ、cos θ 或 tan θ。例如，4 sin θ = 1 变为 sin θ = 1/4。
Use the inverse function for the principal value, then use the graph or CAST diagram to find all solutions in the required interval.
使用反函数求出主值，然后利用图像或 CAST 图找到要求区间内的所有解。
sin θ = 1/2 ➔ θ = 30° or 150° for 0° ≤ θ ≤ 360°
For cosine, the second solution is 360° minus the principal value; for tangent, add 180° to the principal value.
对于余弦，第二个解是 360° 减去主值；对于正切，在主值基础上加 180°。
4. Quadratic Trigonometric Equations | 二次型三角方程
Replace sin θ or cos θ with a single variable such as x to reveal a quadratic. Always check whether the roots lie between -1 and 1.
将 sin θ 或 cos θ 替换为单个变量如 x，以转化出二次方程。始终检查根是否在 -1 到 1 之间。
Example: 2 sin² θ − 3 sin θ + 1 = 0. Let x = sin θ, then 2x² − 3x + 1 = 0, so x = 1 or x = 1/2.
示例：2 sin² θ − 3 sin θ + 1 = 0。令 x = sin θ，则 2x² − 3x + 1 = 0，所以 x = 1 或 x = 1/2。
Then solve sin θ = 1 and sin θ = 1/2 separately within the given range.
然后分别在给定范围内解 sin θ = 1 和 sin θ = 1/2。
5. Using Angle Transformations | 角度变换问题
When the angle is not simply θ, rewrite it as u = ax + b or u = aθ + b and adjust the interval before solving.
当角度不是简单的 θ 时，将其改写为 u = ax + b 或 u = aθ + b，并在求解前调整区间。
For sin(2θ − 30°) = 0.5 with 0° ≤ θ ≤ 360°, the interval for u is −30° ≤ u ≤ 690°. Solve for u, then divide and add to recover θ.
对于 sin(2θ − 30°) = 0.5，且 0° ≤ θ ≤ 360°，u 的区间为 −30° ≤ u ≤ 690°。解出 u 后再除以系数并回代得到 θ。
This prevents missing solutions that arise from the stretched angle range.
这可以防止因角度范围被拉伸而遗漏解。
6. R sin(θ ± α) and Harmonic Form | R sin(θ ± α) 与辅助角形式
Expressions like a sin θ + b cos θ can be written as R sin(θ ± α) or R cos(θ ± α). Edexcel expects you to find R and α exactly or to one decimal place.
形如 a sin θ + b cos θ 的表达式可以写成 R sin(θ ± α) 或 R cos(θ ± α)。爱德思要求你求出精确或保留一位小数的 R 和 α。
R = √(a² + b²), tan α = b/a
Choose the correct quadrant for α by considering the signs of a and b. Then solve R sin(θ + α) = c as a single sine equation.
根据 a 和 b 的符号选择 α 的正确象限。然后将 R sin(θ + α) = c 当作单个正弦方程求解。
7. Harder Equations with Double Angles | 二倍角相关难题
When cos 2θ or sin 2θ appears with sin θ or cos θ, use the double-angle identities to reduce the equation to one trigonometric function.
当 cos 2θ 或 sin 2θ 与 sin θ 或 cos θ 同时出现时，利用二倍角恒等式将方程化为只含一个三角函数的形式。
Example: cos 2θ + sin θ = 0 becomes 1 − 2 sin² θ + sin θ = 0, then solve as a quadratic in sin θ.
示例：cos 2θ + sin θ = 0 变为 1 − 2 sin² θ + sin θ = 0，然后按 sin θ 的二次方程求解。
8. General Solutions and Interval Restrictions | 通解与区间限制
If no interval is given, you may need the general solution. For sine and cosine, add 360°n or 2πn; for tangent, add 180°n or πn, where n is an integer.
如果没有给定区间，你可能需要写出通解。对于正弦和余弦，加上 360°n 或 2πn；对于正切，加上 180°n 或 πn，其中 n 为整数。
In Edexcel papers the interval is usually explicit, such as 0° ≤ θ < 360° or −π ≤ θ ≤ π, so only list solutions inside that range.
在爱德思试卷中，区间通常是明确的，例如 0° ≤ θ < 360° 或 −π ≤ θ ≤ π，因此只列出该范围内的解。
9. Exam-Style Worked Example | 考试级例题演示
Solve 3 cos 2θ + 2 = 0 for 0° ≤ θ < 360°, giving answers to one decimal place.
解方程 3 cos 2θ + 2 = 0，其中 0° ≤ θ < 360°，答案保留一位小数。
Rearrange to cos 2θ = −2/3. Let u = 2θ, so 0° ≤ u < 720°. The principal value of cos⁻¹(−2/3) is about 131.8°.
变形得到 cos 2θ = −2/3。令 u = 2θ，则 0° ≤ u < 720°。cos⁻¹(−2/3) 的主值约为 131.8°。
Cosine is negative in the second and third quadrants, so u = 131.8°, 228.2°, 491.8°, 588.2° within 0° to 720°.
余弦在第二和第三象限为负，因此在 0° 到 720°
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