Trigonometric Identities and Equations | 三角恒等式与方程

📚 Trigonometric Identities and Equations | 三角恒等式与方程

Trigonometric identities and equations are a central part of Edexcel A-Level Pure Mathematics. You need to know the exact trigonometric ratios, the Pythagorean identity, compound angle formulas, and how to solve equations in both degrees and radians. This article revisits the key concepts, worked examples, and common mistakes so you can approach exam questions with confidence.

三角恒等式与方程是 Edexcel A-Level 纯数学的核心内容。你需要掌握特殊角的精确三角比、勾股恒等式、复合角公式，并能在角度制与弧度制下解三角方程。本文系统梳理核心概念、例题讲解与常见错误，帮助你自信应对考试题型。

1. Trigonometric Ratios Refresher | 三角函数比快速回顾

In a right-angled triangle, the three basic trigonometric ratios are defined as follows: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, and tan θ = opposite/adjacent. These definitions work for acute angles, but the unit circle extends them to all real angles.

在直角三角形中，三个基本三角比定义为：sin θ = 对边/斜边，cos θ = 邻边/斜边，tan θ = 对边/邻边。这些定义适用于锐角，而单位圆将其扩展到任意角。

You must memorise the exact values of sin θ, cos θ and tan θ for the angles 0°, 30°, 45°, 60° and 90°. These values appear frequently in non-calculator exam questions.

你必须牢记 0°、30°、45°、60° 和 90° 的 sin θ、cos θ 和 tan θ 精确值。这些数值经常出现在非计算器考试题中。

θ	sin θ	cos θ	tan θ
0°	0	1	0
30°	1/2	√3/2	√3/3
45°	√2/2	√2/2	1
60°	√3/2	1/2	√3
90°	1	0	undefined

Using these exact values, you can quickly simplify expressions such as sin 60° cos 30° + cos 60° sin 30° without a calculator.

利用这些精确值，你可以无需计算器快速化简诸如 sin 60° cos 30° + cos 60° sin 30° 的表达式。

2. The Pythagorean Identity | 勾股恒等式

The most important trigonometric identity in A-Level Mathematics is derived from the unit circle: for any angle θ, the square of the sine plus the square of the cosine equals 1.

A-Level 数学中最重要的三角恒等式来自单位圆：对于任意角 θ，正弦平方与余弦平方之和等于 1。

sin²θ + cos²θ ≡ 1

This identity is true for all values of θ, not just some specific angles, which is why the symbol ≡ is used rather than =. From it, you can rearrange to obtain useful forms: sin²θ = 1 − cos²θ and cos²θ = 1 − sin²θ.

该恒等式对所有 θ 值都成立，而不仅仅对某些特定角度成立，因此使用 ≡ 号而不是 = 号。由此可以移项得到常用形式：sin²θ = 1 − cos²θ 和 cos²θ = 1 − sin²θ。

These rearrangements are especially useful when a quadratic equation contains both sin²θ and cos²θ, because you can replace one of them to leave a single trigonometric function.

这些变形在方程同时含有 sin²θ 和 cos²θ 时特别有用，因为你可以替换其中一个，使方程只含一个三角函数。

3. Tangent and Reciprocal Identities | 正切与倒数恒等式

The tangent function is defined by the identity tan θ ≡ sin θ / cos θ. This relationship is fundamental when proving further identities or simplifying expressions.

正切函数由恒等式 tan θ ≡ sin θ / cos θ 定义。这一关系在证明其他恒等式或化简表达式时非常基本。

tan θ ≡ sin θ / cos θ

In addition, Edexcel A-Level includes the reciprocal trigonometric functions: sec θ = 1/cos θ, cosec θ = 1/sin θ, and cot θ = 1/tan θ. These lead to two further identities that are often tested.

此外，Edexcel A-Level 还包含倒数三角函数：sec θ = 1/cos θ，cosec θ = 1/sin θ，cot θ = 1/tan θ。它们导出两个经常考查的恒等式。

1 + tan²θ ≡ sec²θ

1 + cot²θ ≡ cosec²θ

These can be derived by dividing sin²θ + cos²θ ≡ 1 by cos²θ or sin²θ. They are useful when you need to convert between tangent and secant, or cotangent and cosecant.

它们可以通过用 cos²θ 或 sin²θ 除以 sin²θ + cos²θ ≡ 1 得到。当你需要在正切与正割，或余切与余割之间转换时，这些恒等式非常有用。

4. Compound Angle and Double Angle Formulas | 复合角与倍角公式

Compound angle formulas allow you to expand sin(A ± B), cos(A ± B) and tan(A ± B). They are listed in the Edexcel formula booklet, but you should be able to use them fluently.

复合角公式可用来展开 sin(A ± B)、cos(A ± B) 和 tan(A ± B)。它们列在 Edexcel 公式手册中，但你必须能够熟练运用。

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

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

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

By setting A = B = θ, you obtain the double angle formulas: sin 2θ = 2 sin θ cos θ, cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ, and tan 2θ = 2 tan θ / (1 − tan²θ).

令 A = B = θ，可得到倍角公式：sin 2θ = 2 sin θ cos θ，cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ，以及 tan 2θ = 2 tan θ / (1 − tan²θ)。

The multiple forms of cos 2θ are often used to integrate powers of sine or cosine, and to solve equations involving sin²θ or cos²θ.

cos 2θ 的多种形式常用于积分正弦或余弦的幂，以及解含有 sin²θ 或 cos²θ 的方程。

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

To solve a basic equation such as sin θ = 1/2 for 0° ≤ θ ≤ 360°, first find the principal value using the inverse function: θ = sin⁻¹(1/2) = 30°. Then use the symmetry of the sine graph to find the second solution in the given interval.

要解基本方程如 sin θ = 1/2，其中 0° ≤ θ ≤ 360°，先用反函数求主值：θ = sin⁻¹(1/2) = 30°。然后利用正弦图像的对称性求出给定区间内的第二个解。

Because sin θ is positive in the first and second quadrants, the second solution is 180° − 30° = 150°. Therefore the full solution set is θ = 30°, 150°.

由于 sin θ 在第一和第二象限为正，第二个解为 180° − 30° = 150°。因此完整解集为 θ = 30°, 150°。

Always check the interval. If no interval is given, you may need to give the general solution using n ∈ ℤ, such as θ = 30° + 360°n or θ = 150° + 360°n.

始终检查给定区间。如果未给出区间，可能需要用 n ∈ ℤ 表示通解，例如 θ = 30° + 360°n 或 θ = 150° + 360°n。

6. The CAST Diagram | CAST 图

The CAST diagram is a mnemonic that tells you which trigonometric functions are positive in each quadrant. Starting from the fourth quadrant and moving anticlockwise: C stands for cos positive, A for all positive, S for sin positive, and T for tan positive.

CAST 图是一种记忆方法，告诉你各象限中哪些三角函数为正。从第四象限开始逆时针依次为：C 表示 cos 为正，A 表示全部为正，S 表示 sin 为正，T 表示 tan 为正。

C: cos positive, sin and tan negative
A: all positive
S: sin positive, cos and tan negative
T: tan positive, sin and cos negative

对应的中文说明：C 象限中 cos 为正，sin 和 tan 为负；A 象限中全部为正；S 象限中 sin 为正，cos 和 tan 为负；T 象限中 tan 为正，sin 和 cos 为负。

Using CAST, if sin θ = −0.5, you know that θ must lie in the third or fourth quadrant, so you can work out the relevant angles from the principal value 30°.

利用 CAST，如果 sin θ = −0.5，你知道 θ 必在第三或第四象限，因此可以从主值 30° 求出相关角。

7. Solving Quadratic Trigonometric Equations | 解二次三角方程

Quadratic trigonometric equations appear regularly in Edexcel exams. For example, solve 2sin²θ − sinθ − 1 = 0 for 0° ≤ θ ≤ 360°. Start by substituting u = sinθ, giving 2u² − u − 1 = 0.

二次三角方程在 Edexcel 考试中经常出现。例如，在 0° ≤ θ ≤ 360° 内解 2sin²θ − sinθ − 1 = 0。首先令 u = sinθ，得到 2u² − u − 1 = 0。

(2u + 1)(u − 1) = 0

So u = −1/2 or u = 1. Returning to sinθ, we solve sinθ = −1/2 and sinθ = 1. The first gives θ = 210°, 330°; the second gives θ = 90°. Hence the solutions are θ = 90°, 210°, 330°.

因此 u = −1/2 或 u = 1。代回 sinθ，解 sinθ = −1/2 和 sinθ = 1。前者给出 θ = 210°, 330°；后者给出 θ = 90°。因此解为 θ = 90°, 210°, 330°。

If an equation contains both sin²θ and cos²θ, use the Pythagorean identity to rewrite it in terms of one function before factoring.

如果方程同时含有 sin²θ 和 cos²θ，先用勾股恒等式把它改写成只含一个函数的形式，再进行因式分解。

8. Working with Radians | 使用弧度制

In Edexcel A-Level Maths, many questions require answers in radians rather than degrees. The key conversion is π radians = 180°, so 1° = π/180 and 1 rad = 180°/π.

在 Edexcel A-Level 数学中，许多题目要求用弧度制而不是角度制作答。关键换算是 π 弧度 = 180°，因此 1° = π/180，1 弧度 = 180°/π。

π rad = 180°

You should also be able to use the arc length formula s = rθ and the sector area formula A = ½r²θ, where θ is measured in radians.

你还需要会用弧长公式 s = rθ 和扇形面积公式 A = ½r²θ，其中 θ 以弧度为单位。

For example, solve sin θ = √3/2 for 0 ≤ θ ≤ 2π. The principal value is θ = sin⁻¹(√3/2) = π/3. Since sine is positive in the first and second quadrants, the second solution is π − π/3 = 2π/3. Thus θ = π/3, 2π/3.

例如，在 0 ≤ θ ≤ 2π 内解 sin θ = √3/2。主值为 θ = sin⁻¹(√3/2) = π/3。由于正弦在第一和第二象限为正，第二个解为 π − π/3 = 2π/3。因此 θ = π/3, 2π/3。

9. Proving Trigonometric Identities | 证明三角恒等式

Proof questions ask you to show that one trigonometric expression is identical to another. A reliable strategy is to start with the more complicated side, express everything in terms of sin θ and cos θ, and apply the Pythagorean identity.

证明题要求你说明一个三角表达式与另一个表达式恒等。可靠策略是从较复杂的一边入手，将所有函数用 sin θ 和 cos θ 表示，并应用勾股恒等式。

For example, prove that (1 − cos²θ)/sin θ = sin θ. Starting with the left-hand side

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Trigonometric Identities and Equations | 三角恒等式与方程