Trigonometric Identities and Equations for Edexcel A-Level Pure Maths | Edexcel A-Level 纯数学：三角恒等式与方程

📚 Trigonometric Identities and Equations for Edexcel A-Level Pure Maths | Edexcel A-Level 纯数学：三角恒等式与方程

This revision guide covers the core trigonometric identities and equation-solving techniques required for Edexcel A-Level Pure Mathematics. Mastery of these skills is essential for Paper 1 and Paper 2 questions involving proof, algebraic manipulation, exact values and periodic modelling.

本复习指南涵盖 Edexcel A-Level 纯数学中核心的三角恒等式与方程求解技巧。掌握这些技能对于 Paper 1 和 Paper 2 中涉及证明、代数变形、精确值以及周期建模的题目至关重要。

1. Radian Measure and Arc Length | 弧度制与弧长

Radians are the standard angle unit used throughout A-Level calculus. One radian is the angle subtended at the centre of a circle by an arc whose length is equal to the radius of the circle.

弧度是 A-Level 微积分中使用的标准角度单位。1 弧度是指圆心角所对的弧长恰好等于圆的半径时的角度。

For a circle of radius r, the arc length s and sector area A are given by s = rθ and A = ½r²θ, where θ must be expressed in radians. This is why angle conversion is a frequent first step in exam questions.

对于半径为 r 的圆，弧长 s 和扇形面积 A 分别由 s = rθ 和 A = ½r²θ 给出，其中 θ 必须以弧度为单位。这就是为什么角度换算经常是考试题的第一步。

s = rθ, A = ½r²θ, 180° = π rad

Convert 30° to radians: 30 × π/180 = π/6 | 30° 转换为弧度：30 × π/180 = π/6
Arc length for r = 5 and θ = π/3: s = 5 × π/3 = 5π/3 | 半径 5、弧度 π/3 时的弧长：s = 5 × π/3 = 5π/3
Sector area for r = 4 and θ = π/2: A = ½ × 4² × π/2 = 4π | 半径 4、弧度 π/2 时的扇形面积：A = ½ × 4² × π/2 = 4π

2. Sine, Cosine and Tangent Identities | 正弦、余弦与正切恒等式

The tangent identity links the three primary trigonometric ratios: tanθ = sinθ / cosθ. This is used constantly to simplify expressions and to solve equations involving tan.

正切恒等式将三个基本三角比联系起来：tanθ = sinθ / cosθ。它常用于化简表达式以及解含 tan 的方程。

The reciprocal ratios are secθ = 1/cosθ, cosecθ = 1/sinθ and cotθ = 1/tanθ. These appear frequently in Edexcel trigonometry proofs and in simplifying rational trigonometric expressions.

倒数比包括 secθ = 1/cosθ、cosecθ = 1/sinθ 和 cotθ = 1/tanθ。它们在 Edexcel 三角证明题和化简有理三角函数式中经常出现。

tanθ = sinθ / cosθ, cotθ = cosθ / sinθ

sin(−θ) = −sinθ | 正弦是奇函数
cos(−θ) = cosθ | 余弦是偶函数
tan(−θ) = −tanθ | 正切是奇函数

3. Pythagorean Identities | 勾股恒等式

The core Pythagorean identity is sin²θ + cos²θ = 1. It follows directly from the unit circle definition and is the key tool in many proof and simplification questions.

核心勾股恒等式为 sin²θ + cos²θ = 1。它直接由单位圆定义得出，是许多证明题和化简题的关键工具。

Dividing this identity by cos²θ gives 1 + tan²θ = sec²θ, while dividing by sin²θ gives 1 + cot²θ = cosec²θ. These two versions are especially useful when the expression involves tanθ or cotθ.

将该恒等式除以 cos²θ 可得 1 + tan²θ = sec²θ；除以 sin²θ 可得 1 + cot²θ = cosec²θ。这两种形式在涉及 tanθ 或 cotθ 时尤其有用。

sin²θ + cos²θ = 1

For example, 2sin²θ − cos²θ can be rewritten as 3sin²θ − 1 by replacing cos²θ with 1 − sin²θ. This kind of manipulation is common when proving identities.

例如，通过将 cos²θ 替换为 1 − sin²θ，2sin²θ − cos²θ 可以改写为 3sin²θ − 1。这种变形在证明恒等式时很常见。

4. Compound Angle Formulae | 复合角公式

Compound angle formulae expand expressions such as sin(A ± B) and cos(A ± B). They are provided in the Edexcel formula booklet, but you must apply the signs correctly, especially for cos(A ± B).

复合角公式可展开 sin(A ± B) 和 cos(A ± B) 等表达式。Edexcel 公式手册会提供这些公式，但你必须正确使用符号，尤其是 cos(A ± B)。

sin(A ± B) = sinA cosB ± cosA sinB

cos(A ± B) = cosA cosB ∓ sinA sinB

For tan(A ± B), the formula is tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB). The sign in the denominator is always opposite to the sign in the numerator.

对于 tan(A ± B)，公式为 tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)。分母中的符号总是与分子中的符号相反。

For example, sin75° can be found exactly: sin75° = sin(45° + 30°) = sin45° cos30° + cos45° sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4.

例如，sin75° 可以通过公式求精确值：sin75° = sin(45° + 30°) = sin45° cos30° + cos45° sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4。

5. Double Angle Formulae | 二倍角公式

The double angle formulae follow directly by setting A = B in the compound angle formulae. They are particularly important later in the course when integrating squared trigonometric functions.

二倍角公式可通过在复合角公式中令 A = B 直接得到。它们在课程后续对三角函数平方进行积分时尤为重要。

sin2θ = 2sinθ cosθ

cos2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ

tan2θ = 2tanθ / (1 − tan²θ)

Choose the version of cos2θ that best matches the given expression. For example, 1 − cos2θ = 2sin²θ is useful when integrating sin²θ, while cos2θ = 2cos²θ − 1 is useful when integrating cos²θ.

选择与给定表达式最匹配的 cos2θ 形式。例如，1 − cos2θ = 2sin²θ 在积分 sin²θ 时很有用，而 cos2θ = 2cos²θ − 1 在积分 cos²θ 时很有用。

6. Solving Basic Trigonometric Equations | 解基本三角方程

To solve an equation such as sinθ = k, first find the principal value from a calculator, then use the CAST diagram or the symmetry of the sine curve to find all solutions in the required interval.

解 sinθ = k 这类方程时，先通过计算器求出主值，然后利用 CAST 图或正弦曲线的对称性，在指定

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Trigonometric Identities and Equations for Edexcel A-Level Pure Maths | Edexcel A-Level 纯数学：三角恒等式与方程