Edexcel A Level Maths: Trigonometric Identities and Equations | Edexcel A Level 数学：三角函数恒等式与方程

📚 Edexcel A Level Maths: Trigonometric Identities and Equations | Edexcel A Level 数学：三角函数恒等式与方程

Trigonometric identities and equations appear across the Edexcel A Level Mathematics specification, from solving simple equations in radians to using double-angle identities in integration. This guide brings together the essential formulas, worked techniques and common exam pitfalls.

三角函数恒等式与方程贯穿 Edexcel A Level 数学考试大纲，从弧度制下解简单方程到在积分中运用倍角恒等式。本指南汇总核心公式、解题技巧与常见考试失分点。

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

Radian measure is central to Edexcel trigonometry. One radian is the angle subtended at the centre of a circle by an arc equal in length to the radius.

弧度制是 Edexcel 三角学的核心。一弧度是指圆心角所对的弧长等于半径时的角。

180° = π rad

Conversion follows directly from the fact that a full turn is 2π radians. To convert degrees to radians multiply by π/180; to convert radians to degrees multiply by 180/π.

换算直接来自于圆周角为 2π 弧度。将角度制转换为弧度制时乘以 π/180；将弧度制转换为角度制时乘以 180/π。

s = rθ and A = ½ r²θ

In the formulas for arc length s = rθ and sector area A = ½ r²θ, the angle θ must always be in radians.

在弧长公式 s = rθ 与扇形面积公式 A = ½ r²θ 中，角度 θ 必须始终使用弧度制。

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2. Reciprocal and Pythagorean Identities | 倒数恒等式与毕达哥拉斯恒等式

The reciprocal functions secant, cosecant and cotangent are defined as sec θ = 1/cos θ, cosec θ = 1/sin θ and cot θ = 1/tan θ = cos θ / sin θ.

倒数函数 sec、cosec 与 cot 定义为 sec θ = 1/cos θ、cosec θ = 1/sin θ、cot θ = 1/tan θ = cos θ / sin θ。

sin²θ + cos²θ = 1

Dividing the basic identity by cos²θ gives 1 + tan²θ = sec²θ; dividing by sin²θ gives 1 + cot²θ = cosec²θ.

将基本恒等式除以 cos²θ 得到 1 + tan²θ = sec²θ；除以 sin²θ 得到 1 + cot²θ = cosec²θ。

1 + tan²θ = sec²θ and 1 + cot²θ = cosec²θ

These identities are often used to prove more complex expressions or to rewrite equations in a single trigonometric ratio.

这些恒等式常用于证明更复杂的表达式，或将方程改写为只含一个三角比的形式。

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3. Compound-Angle Identities | 复合角恒等式

sin(A ± B) = sin A cos B ± cos A sin B

cos(A ± B) = cos A cos B ∓ sin A sin B

Note that the sign in the cosine formula is reversed: cos(A + B) has a minus, and cos(A – B) has a plus.

注意余弦公式中的符号相反：cos(A + B) 带负号，cos(A – B) 带正号。

tan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)

These formulas are useful for expanding expressions such as sin(x + π/3) or simplifying exact-value calculations such as cos 75° = cos(45° + 30°).

这些公式可用于展开如 sin(x + π/3) 的表达式，或化简精确值计算，例如 cos 75° = cos(45° + 30°)。

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4. Double-Angle Identities | 倍角恒等式

sin 2A = 2 sin A cos A

cos 2A = cos²A – sin²A = 2 cos²A – 1 = 1 – 2 sin²A

tan 2A = 2 tan A / (1 – tan²A)

The three versions of cos 2A are important for integration: sin²A = ½(1 – cos 2A) and cos²A = ½(1 + cos 2A) convert squared terms into single cosine terms.

cos 2A 的三种形式对积分很重要：sin²A = ½(1 – cos 2A) 与 cos²A = ½(1 + cos 2A) 可将平方项转化为单个余弦项。

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5. Solving Trigonometric Equations | 三角方程求解

A reliable method is to reduce the equation to one trigonometric function, factorise, then use the CAST diagram or graph to find all solutions in the required interval.

一种可靠的方法是先将方程化为只含一个三角函数，进行因式分解，然后利用 CAST

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