📚 Trigonometric Equations & Identities: Edexcel A-Level Maths (pdfjoiner_(4)-144) | 三角函数方程与恒等式：Edexcel A-Level 数学（pdfjoiner_(4)-144）
This revision guide focuses on the trigonometric equations and identities that appear frequently in Edexcel A-Level Pure Mathematics, especially in the type of practice set referenced by pdfjoiner_(4)-144. Mastering these techniques is essential for both Paper 1 and Paper 2, as trig questions often combine algebraic manipulation, exact values, and interval solutions.
本复习指南聚焦于 Edexcel A-Level 纯数学中频繁出现的三角函数方程与恒等式，特别是练习文件 pdfjoiner_(4)-144 所对应的题型。掌握这些技巧对 Paper 1 和 Paper 2 都至关重要，因为三角题通常综合了代数变形、精确值以及在给定区间内求解。
1. Radian Measure, Arc Length and Sector Area | 弧度制、弧长与扇形面积
In Edexcel A-Level Pure Mathematics, angles are normally measured in radians unless stated otherwise. The radian is defined so that an arc of length r on a circle of radius r subtends an angle of 1 radian at the centre. Conversions are given by π rad = 180°.
在 Edexcel A-Level 纯数学中，除特别说明外，角度通常以弧度为单位。弧度的定义是：在半径为 r 的圆上，长度为 r 的弧所对的圆心角为 1 弧度。换算关系为 π 弧度 = 180°。
The arc length is l = rθ and the sector area is A = ½ r²θ, where θ must be in radians. Many exam questions require converting between degrees and radians before substituting into these formulae.
弧长公式为 l = rθ，扇形面积公式为 A = ½ r²θ，其中 θ 必须用弧度。很多考题要求先完成角度与弧度的换算，再代入公式。
l = rθ | A = ½ r²θ
2. The Pythagorean and Reciprocal Identities | 平方恒等式与倒数恒等式
In Edexcel A-Level Pure Mathematics, the trigonometric identities are not just isolated results; they form a toolkit for simplifying expressions, proving statements, and solving equations in a given interval. The most common starting point is the identity sin² θ + cos² θ = 1, which can be rearranged to give sin² θ = 1 − cos² θ or cos² θ = 1 − sin² θ.
在 Edexcel A-Level 纯数学中，三角恒等式不是孤立的结论，而是一套用于化简表达式、证明命题以及在给定区间内解方程的工具。最常见的出发点是恒等式 sin² θ + cos² θ = 1，它可以变形为 sin² θ = 1 − cos² θ 或 cos² θ = 1 − sin² θ。
Two further identities are derived by dividing the Pythagorean identity by cos² θ or sin² θ. These are 1 + tan² θ = sec² θ and 1 + cot² θ = cosec² θ. They are especially useful when an equation contains mixed functions such as tan θ and sec θ.
将平方恒等式分别除以 cos² θ 或 sin² θ，可以得到另外两个恒等式：1 + tan² θ = sec² θ 以及 1 + cot² θ = cosec² θ。当方程中同时出现 tan θ 和 sec θ 等混合函数时，这两个恒等式尤为有用。
sin² θ + cos² θ = 1
1 + tan² θ = sec² θ | 1 + cot² θ = cosec² θ
3. Double Angle Formulae | 二倍角公式
The double angle formulae are essential for changing the argument of a trig function. The three forms of cos 2θ are particularly important because exam questions often require you to choose the form that matches the rest of the equation. For example, if the equation contains sin² θ, use cos 2θ = 1 − 2 sin² θ.
二倍角公式对于改变三角函数的角频率至关重要。cos 2θ 的三种形式尤其重要，因为考题往往要求你选择与方程其余部分相匹配的形式。例如，如果方程中含有 sin² θ，就应使用 cos 2θ = 1 − 2 sin² θ。
| Formula | Equivalent form |
|---|---|
| sin 2θ | 2 sin θ cos θ |
| cos 2θ | cos² θ − sin² θ = 2 cos² θ − 1 = 1 − 2 sin² θ |
| tan 2θ | 2 tan θ / (1 − tan² θ) |
When solving equations such as sin 2θ = sin θ, replacing sin 2θ with 2 sin θ cos θ can reduce the equation to a factorisable form. This often leads to sin θ = 0 or cos θ = ½, which can then be solved over the required interval.
解方程如 sin 2θ = sin θ 时，把 sin 2θ 替换为 2 sin θ cos θ，可以将方程化为可因式分解的形式。这样通常得到 sin θ = 0 或 cos θ = ½，再在指定区间内求解即可。
4. The R-Formula and Harmonic Form | R 公式与简谐形式
The R-formula is used to rewrite expressions of the form a sin θ ± b cos θ or a cos θ ± b sin θ as a single sine or cosine. This technique is crucial for finding the maximum and minimum values of such expressions and for solving equations where the sine and cosine terms cannot be separated easily.
R 公式用于将 a sin θ ± b cos θ 或 a cos θ ± b sin θ 形式的表达式改写为单个正弦或余弦。该技巧对于求这类表达式的最大值和最小值，以及求解难以直接分离正弦项和余弦项的方程至关重要。
a sin θ + b cos θ = R sin(θ + α)
a cos θ + b sin θ = R cos(θ − α)
Here R = √(a² + b²) and α is chosen so that tan α = b/a for the sine form, with α in the correct quadrant. Always check the signs of a and b when determining α.
其中 R = √(a² + b²)，对于正弦形式，选择 α 使得 tan α = b/a，并且 α 必须位于正确的象限。确定 α 时务必检查 a 和 b 的符号。
5. Solving Linear Trigonometric Equations | 解线性三角方程
A linear trigonometric equation contains only one trig function, such as 2 sin θ = 1 or tan θ = −1. The first step is to isolate the trig function, then find the principal value using inverse functions or exact values from the unit circle.
线性三角方程只含一个三角函数，例如 2 sin θ = 1 或 tan θ = −1。第一步是分离三角函数，然后利用反函数或单位圆上的精确值求出主值。
For 2 sin θ = 1 in the interval 0 ≤ θ < 2π, we get sin θ = ½. The principal value is θ = π/6. Since sine is positive in the first and second quadrants, the second solution is θ = π − π/6 = 5π/6. Always draw a CAST diagram or graph to avoid missing solutions.
对于区间 0 ≤ θ < 2π 内的方程 2 sin θ = 1，我们得到 sin θ = ½。主值为 θ = π/6。由于正弦在第一和第二象限为正，第二个解为 θ = π − π/6 = 5π/6。务必画出 CAST 图或函数图像，以免漏解。
sin θ = ½ ⇒ θ = π/6, 5π/6
6. Solving Quadratic Trigonometric Equations | 解二次型三角方程
Quadratic trigonometric equations such as 2 cos² θ − cos θ − 1 = 0 should be treated like ordinary quadratics. Let x = cos θ, solve 2x² − x − 1 = 0, then substitute back and solve each resulting linear equation over the given interval.
如 2 cos² θ − cos θ − 1 = 0 这样的二次型三角方程应当像普通二次方程一样处理。令 x = cos θ，解 2x² − x − 1 = 0，然后代回并在给定区间内求解每个得到的线性方程。
Often the quadratic arises after using an identity. For example, replacing sin² θ by 1 − cos² θ can turn an equation with mixed powers into a quadratic in cos θ. Factorising gives exact values such as cos θ = 1 or cos θ = −½.
这类二次方程通常在使用恒等式后出现。例如，用 1 − cos² θ 替换 sin² θ，可以把混合次数的方程化为关于 cos θ 的二次方程。因式分解后得到精确值，如 cos θ = 1 或 cos θ = −½。
2 cos² θ − cos θ − 1 = 0 ⇒ (2 cos θ + 1)(cos θ − 1) = 0
7. Proving Trigonometric Identities | 证明三角恒等式
Proof questions require a clear and logical sequence of steps. Start with one side of the identity, usually the more complicated side, and rewrite it using known identities until it matches the other side. Avoid moving terms from one side to the other.
证明题要求清晰且有逻辑的步骤。通常从恒等式的一侧入手，一般是较复杂的一侧，利用已知恒等式逐步改写，直到与另一侧相同。避免把项从一侧移到另一侧。
The most common strategy is to express everything in terms of sin θ and cos θ. Then simplify fractions, use the Pythagorean identity sin² θ + cos² θ = 1, and factorise where possible. For example, proving (sin θ + cos θ)² = 1 + sin 2θ uses expansion and the double angle formula.
最常用的策略是把所有函数都表示为 sin θ 和 cos θ。然后化简分式，使用平方恒等式 sin² θ + cos² θ = 1，并在可能的情况下进行因式分解。例如，证明 (sin θ + cos θ)² = 1 + sin 2θ 就用到了
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