A-Level Maths: Applying the Addition Formulas | A-Level数学：和角公式在解题中的应用

📚 A-Level Maths: Applying the Addition Formulas | A-Level数学：和角公式在解题中的应用

The addition formulas (also called compound angle formulas) are among the most useful tools in A-Level trigonometry. They allow us to express the sine, cosine, and tangent of sums or differences of angles in terms of the trigonometric functions of the individual angles. Mastering these formulas is essential for solving a wide range of problems, from evaluating exact values to proving identities and solving equations.

和角公式（又称复合角公式）是A-Level三角学中最实用的工具之一。它们让我们能够将两个角的和或差的正弦、余弦、正切，用这两个角各自的三角函数值来表示。掌握这些公式，对于解决各种问题至关重要，无论是求精确值、证明恒等式，还是解方程。

1. The Addition Formulas | 和角公式回顾

The standard addition formulas are as follows. For any angles A and B:

对于任意角 A 和 B，标准和角公式如下：

sin(A + B) = sin A cos B + cos A sin B

sin(A − B) = sin A cos B − cos A sin B

cos(A + B) = cos A cos B − sin 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)

tan(A − B) = (tan A − tan B) / (1 + tan A tan B)

These formulas are valid for all angles where the expressions are defined (for tangent, the denominators must not be zero).

这些公式在表达式有定义的所有角度范围内均成立（对于正切，分母不能为零）。

2. Derivation and Memory Aids | 公式的推导与记忆技巧

In the Edexcel, AQA, and OCR A-Level specifications, you are expected to know these formulas, but understanding their derivation from the unit circle or geometric diagrams can help you remember them. The cosine difference formula cos(A − B) = cos A cos B + sin A sin B is often proved first, and the others can be derived from it using symmetry and the identities sin θ = cos(90° − θ) and tan θ = sin θ / cos θ.

在Edexcel、AQA和OCR的A-Level考纲中，你需要熟记这些公式，但理解它们从单位圆或几何图形的推导过程有助于记忆。通常会先证明余弦差公式 cos(A − B) = cos A cos B + sin A sin B，然后利用对称性以及 sin θ = cos(90° − θ) 和 tan θ = sin θ / cos θ 等恒等式推导出其他公式。

A common memory aid is the phrase “All Silly Tom Cats” for the signs: for sine, the sign stays the same; for cosine, the sign changes. Another popular method is to write the formulas out repeatedly until they become automatic.

一个常见的记忆口诀是“正弦同号，余弦变号”：即对于正弦，运算符号保持不变；对于余弦，运算符号要改变。另一种常见方法是反复书写这些公式，直到熟练掌握。

3. Application: Evaluating Exact Values | 应用：求精确值

One of the most straightforward applications of the addition formulas is finding the exact value of trigonometric functions of angles that are not standard, such as 15°, 75°, or 105°. These angles can be expressed as sums or differences of special angles like 30°, 45°, and 60°.

和角公式最直接的应用之一是求非标准角的三角函数精确值，例如15°、75°或105°。这些角可以表示为30°、45°、60°等特殊角的和或差。

For example, to find sin 75°, write 75° = 45° + 30°. Then:

例如，求 sin 75°，将 75° 写成 45° + 30°。于是：

sin 75° = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°

Substituting the known exact values:

代入已知的精确值：

= (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4

This technique is particularly useful in non-calculator exam papers and in further calculus problems where exact surd forms are required.

这种技巧在非计算器考试中以及需要精确根式形式的进阶微积分问题中尤其有用。

4. Application: Simplifying Expressions | 应用：化简表达式

The addition formulas can also be used in reverse to simplify trigonometric expressions. For instance, the expression sin x cos 30° + cos x sin 30° can be recognized as sin(x + 30°), immediately reducing its complexity.

和角公式也可以反向使用来化简三角表达式。例如，表达式 sin x cos 30° + cos x sin 30° 可以识别为 sin(x + 30°)，从而迅速降低其复杂度。

Consider the expression cos A cos B − sin A sin B. Using the formula for cos(A + B), this simplifies to cos(A + B). Recognising such patterns allows you to condense a long expression into a single trigonometric term, which is especially helpful when solving equations or integrating.

考虑表达式 cos A cos B − sin A sin B。利用 cos(A + B) 的公式，可以化简为 cos(A + B)。识别出这种模式，可以将冗长的表达式压缩成一个单一的三角项，这在解方程或积分时特别有帮助。

In A-Level problems, you may be asked to express sin x + cos x in the form R sin(x + α). This is a direct application of the addition formula and will be discussed in more detail later.

在A-Level题目中，你可能会被要求将 sin x + cos x 表达为 R sin(x + α) 的形式。这是和角公式的直接应用，稍后将详细讨论。

5. Application: Proving Identities | 应用：证明恒等式

Proving trigonometric identities is a classic exam question type. The addition formulas are often the key step in establishing more complex identities. For example, to prove that sin(90° − θ) = cos θ, we can use the sine subtraction formula:

证明三角恒等式是经典的考试题型。和角公式常常是建立更复杂恒等式的关键步骤。例如，要证明 sin(90° − θ) = cos θ，我们可以使用正弦差公式：

sin(90° − θ) = sin 90° cos θ − cos 90° sin θ = 1 · cos θ − 0 · sin θ = cos θ

Another example: prove that tan(A + B) = (tan A + tan B) / (1 − tan A tan B). Start from the definitions:

另一个例子：证明 tan(A + B) = (tan A + tan B) / (1 − tan A tan B)。从定义出发：

tan(A + B) = sin(A + B) / cos(A + B)

Then expand both the numerator and denominator using the addition formulas, and divide every term by cos A cos B to obtain the result. When proving identities, always show every step clearly and state which formula you are using.

然后用和角公式展开分子和分母，并将每一项除以 cos A cos B 即可得到结果。证明恒等式时，务必清晰地写出每一步，并说明使用了哪个公式。

6. Application: Solving Trigonometric Equations | 应用：求解三角方程

The addition formulas are frequently needed to solve equations that involve sums or differences of angles. For instance, the equation sin(x + 30°) = cos x can be solved by expanding the left-hand side using the addition formula.

在求解涉及角之和或差的方程时，经常需要用到和角公式。例如，方程 sin(x + 30°) = cos x 可以通过使用和角公式展开左边来求解。

Expanding gives:

展开得到：

sin x cos 30° + cos x sin 30° = cos x

(√3/2) sin x + (1/2) cos x = cos x

Then rearrange and solve for tan x:

然后整理并求解 tan x：

(√3/2) sin x = (1/2) cos x ⇒ tan x = 1/√3

x = 30° + 180°n or x = 210° + 180°n, n ∈ ℤ

Remember to use the general solution or restrict the domain as required by the question.

记住要根据题目要求使用通解或限定定义域。

7. The Double-Angle Formulas as Special Cases | 倍角公式作为特例

Setting B = A in the addition formulas yields the double-angle formulas:

在和角公式中令 B = A，就得到倍角公式：

sin 2A = 2 sin A cos A

cos 2A = cos² A − sin² A = 2 cos² A − 1 = 1 − 2 sin² A

tan 2A = 2 tan A / (1 − tan² A)

These are essential for integration, solving equations, and proving further identities. In many A-Level papers, solving an equation such as sin 2x = cos x requires you to replace sin 2x with 2 sin x cos x, a direct consequence of the addition formula.

这些公式在积分、解方程和证明进一步恒等式中至关重要。在许多A-Level试卷中，解 sin 2x = cos x 这类方程需要将 sin 2x 替换为 2 sin x cos x，这正是和角公式的直接推论。

Furthermore, the double-angle formulas are the foundation for the half-angle formulas, which are sometimes covered in further mathematics modules.

此外，倍角公式是半角公式的基础，而半角公式有时在进阶数学模块中会涉及。

8. The Harmonic Form (R sin and R cos) | 辅助角公式

An important extension of the addition formulas is expressing a sin x + b cos x in the form R sin(x + α) or R cos(x + α). This is known as the harmonic form or the auxiliary angle method. Expanding R sin(x + α) gives:

和角公式的一个重要推广是将 a sin x + b cos x 表达为 R sin(x + α) 或 R cos(x + α) 的形式。这称为辅助角公式或辅助角法。展开 R sin(x + α) 得到：

R sin(x + α) = R sin x cos α + R cos x sin α

Comparing coefficients with a sin x + b cos x, we get:

与 a sin x + b cos x 比较系数，得到：

R cos α = a, R sin α = b

Hence R = √(a² + b²) and tan α = b/a. The angle α is chosen in the correct quadrant based on the signs of a and b.

因此 R = √(a² + b²)，tan α = b/a。根据 a 和 b 的符号选择 α 所在的正确象限。

For example, sin x + cos x = √2 sin(x + 45°). This form is invaluable for finding maximum and minimum values, sketching graphs, and solving equations like sin x + cos x = 1.

例如，sin x + cos x = √2 sin(x + 45°)。这种形式在求最大值和最小值、绘制图像以及解 sin x + cos x = 1 这类方程时非常有价值。

9. Common Errors and Pitfalls | 常见错误与陷阱

Students often make the following mistakes when applying the addition formulas:

学生在应用和角公式时常犯以下错误：

Assuming sin(A + B) = sin A + sin B, which is incorrect. The sine of a sum is not the sum of the sines.
Assuming sin(A + B) = sin A + sin B，这是错误的。两角之和的正弦不等于两个正弦之和。
Forgetting the sign change in cosine: cos(A + B) = cos A cos B − sin A sin B, not plus.
忘记余弦公式中的符号变化：cos(A + B) = cos A cos B − sin A sin B，而不是加号。
Using the wrong formula for tan(A + B). The denominator is 1 − tan A tan B, not 1 + tan A tan B.
对 tan(A + B) 使用错误的公式。分母是 1 − tan A tan B，而不是 1 + tan A tan B。
Mixing up radians and degrees when applying the formulas. Always work in the unit specified by the question.
在应用公式时混淆弧度和角度。始终使用题目指定的单位。
When using the harmonic form, choosing the wrong quadrant for α can lead to incorrect results.
使用辅助角公式时，α 象限选择错误会导致错误结果。

To avoid these errors, always write down the formula before substituting values, and check the sign carefully.

为避免这些错误，在代入数值前先写出公式，并仔细检查符号。

10. Practice and Exam Tips | 练习与考试技巧

To master the addition formulas, regular practice is essential. In exam questions, look for clues that the formulas are needed: angles expressed as sums or differences, expressions with products of sines and cosines, or questions asking for the value of sin(A + B) given the values of sin A and sin B.

要掌握和角公式，定期练习至关重要。在考试题目中，注意哪些线索暗示需要用到这些公式：角表示为和或差的形式、包含正弦余弦乘积的表达式，或者给定 sin A 和 sin B 的值求 sin(A + B) 的问题。

Here are a few suggested practice problems:

以下是几个建议的练习题：

Find the exact value of cos 15°.
求 cos 15° 的精确值。
Simplify sin(x + 60°) + sin(x − 60°).
化简 sin(x + 60°) + sin(x − 60°)。
Solve 2 sin(x + 30°) = cos x for 0° ≤ x ≤ 360°.
解方程 2 sin(x + 30°) = cos x，其中 0° ≤ x ≤ 360°。
Express 3 sin x − 4 cos x in the form R sin(x − α).
将 3 sin x − 4 cos x 表达为 R sin(x − α) 的形式。

In the exam, always show your method clearly, as method marks are awarded even if the final answer is incorrect. Write formulas in the margin or at the start of the solution to ensure you do not misremember them.

在考试中，务必清晰展示你的解题过程，因为即使最终答案错误，也会给步骤分。在页边或解题开始时写下公式，确保自己不会记错。

11. Summary | 总结

The addition formulas are a cornerstone of A-Level trigonometry. They are used to evaluate exact values, simplify expressions, prove identities, solve equations, and derive the double-angle and harmonic forms. Understanding their derivation, recognising their patterns, and practicing regularly will help you apply them confidently in exams.

和角公式是A-Level三角学的基石。它们用于求精确值、化简表达式、证明恒等式、解方程，以及推导倍角公式和辅助角公式。理解它们的推导过程、识别其模式并定期练习，将帮助你在考试中自信地应用它们。

Remember the key formulas, watch out for sign errors, and always verify your results by testing with a known angle such as 0°, 30°, or 90°.

记住关键公式，注意符号错误，并通过测试已知角（如0°、30°或90°）来验证结果。

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A-Level Maths: Applying the Addition Formulas | A-Level数学：和角公式在解题中的应用