A-Level Maths: Trigonometric Addition Formulas Explained | A-Level数学：三角函数和角公式详解

📚 A-Level Maths: Trigonometric Addition Formulas Explained | A-Level数学：三角函数和角公式详解

The addition formulas for sine, cosine and tangent are among the most important tools in A-Level trigonometry. They allow us to expand expressions like sin(A + B) or cos(A − B), and they form the foundation for double-angle formulas, R-formulas and solving trigonometric equations.

正弦、余弦和正切的和角公式是A-Level三角函数中最重要的工具之一。它们帮助我们展开如 sin(A + B) 或 cos(A − B) 这样的表达式，也是二倍角公式、R公式和三角方程求解的基础。

1. Why Do We Need Addition Formulas? | 为什么需要和角公式？

Many trigonometric expressions involve the sum or difference of two angles. Without addition formulas, we could only evaluate such expressions numerically, not algebraically. These formulas let us rewrite sin(A + B) in terms of sin A, cos A, sin B and cos B.

许多三角表达式涉及两个角的和或差。如果没有和角公式，我们只能数值计算，而无法进行代数变形。和角公式让我们能用 sin A、cos A、sin B 和 cos B 来重写 sin(A + B)。

They are essential for simplifying identities, solving equations, and deriving other results such as double-angle and compound-angle formulas.

它们在化简恒等式、解方程以及推导二倍角和复合角公式时必不可少。

2. The Three Core Addition Formulas | 三大核心和角公式

The three most important identities are listed below.

最重要的三个恒等式如下。

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)

These are valid for all angles A and B where the tangent expression is defined.

这些公式对所有角 A 和 B 都成立（正切公式分母不为零时）。

3. Subtraction Formulas | 差角公式

Replacing B with −B in the addition formulas gives the subtraction formulas.

在和角公式中用 −B 代替 B，即可得到差角公式。

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)

Note the sign changes: sine changes sign, cosine keeps plus, tangent changes the sign of the denominator.

注意符号变化：正弦中间变号，余弦中间变加号，正切分母变号。

4. Proof of sin(A + B) Using the Unit Circle | 用单位圆证明 sin(A + B)

We can prove the formula geometrically using the unit circle. Place points P and Q on the unit circle at angles A and −B respectively, with coordinates (cos A, sin A) and (cos B, −sin B). Use the distance formula to compare PQ² in two ways.

我们可以用单位圆从几何上证明该公式。在单位圆上取点 P 和 Q，分别对应角 A 和 −B，坐标为 (cos A, sin A) 和 (cos B, −sin B)。用距离公式以两种方式比较 PQ²。

One way gives: PQ² = 2 − 2[cos A cos B − sin A sin B] = 2 − 2 cos(A + B).

一种方式得到：PQ² = 2 − 2[cos A cos B − sin A sin B] = 2 − 2 cos(A + B)。

The other way uses the coordinates directly, and equating them leads to the cosine addition formula. Then using sin θ = cos(90° − θ) gives the sine formula.

另一种方式直接使用坐标，联立即可得到余弦和角公式。再利用 sin θ = cos(90° − θ) 可推出正弦公式。

5. Proof of cos(A + B) from Euler’s Formula | 从欧拉公式证明 cos(A + B)

For A-Level Further Maths, Euler’s formula provides a very elegant proof. We know that e^(i(A+B)) = e^(iA) · e^(iB).

对于进阶数学，欧拉公式提供了非常优雅的证明。我们知道 e^(i(A+B)) = e^(iA) · e^(iB)。

Expanding both sides using e^(iθ) = cos θ + i sin θ gives:

利用 e^(iθ) = cos θ + i sin θ 展开两边：

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

Equating real and imaginary parts immediately yields both addition formulas.

比较实部和虚部即可立即得到两个和角公式。

6. Deriving the Tangent Addition Formula | 推导正切和角公式

Since tan θ = sin θ / cos θ, we write tan(A + B) = sin(A + B) / cos(A + B).

因为 tan θ = sin θ / cos θ，所以 tan(A + B) = sin(A + B) / cos(A + B)。

Substitute the sine and cosine addition formulas, then divide numerator and denominator by cos A cos B.

代入正弦和余弦和角公式，然后分子分母同时除以 cos A cos B。

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

= (tan A + tan B) / (1 − tan A tan B)

This derivation also shows why the formula is undefined when cos A cos B = 0 or when 1 − tan A tan B = 0.

这个推导也解释了为何在 cos A cos B = 0 或 1 − tan A tan B = 0 时公式无定义。

7. Double-Angle Formulas as Special Cases | 二倍角公式作为特例

Setting B = A in the addition formulas gives the double-angle identities.

在和角公式中令 B = A，即可得到二倍角恒等式。

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)

The three forms of cos 2A are especially useful for integration and for converting between powers and double angles.

cos 2A 的三种形式在积分和幂与倍角之间的转换中尤其有用。

8. Triple-Angle Formulas | 三倍角公式

Using the addition formula with A + B + C or by combining double and single angles, we obtain:

利用和角公式处理 A + B + C，或结合二倍角与单角，可得：

sin 3A = 3 sin A − 4 sin³ A

cos 3A = 4 cos³ A − 3 cos A

These are rarely needed for pure A-Level, but they appear in Further Maths and Olympiad-style problems.

这些在纯A-Level中很少需要，但在进阶数学和竞赛风格题目中会出现。

9. Strategy: Simplifying Expressions | 策略：化简表达式

When simplifying expressions such as sin(θ + 30°) = 2 cos θ, first expand the left side using the addition formula.

化简如 sin(θ + 30°) = 2 cos θ 的表达式时，先用和角公式展开左边。

For example, sin(θ + 30°) = sin θ cos 30° + cos θ sin 30° = (√3/2) sin θ + (1/2) cos θ.

例如，sin(θ + 30°) = sin θ cos 30° + cos θ sin 30° = (√3/2) sin θ + (1/2) cos θ。

Then collect coefficients of sin θ and cos θ to rewrite the expression as R sin(θ + α).

然后合并 sin θ 和 cos θ 的系数，将表达式改写成 R sin(θ + α)。

10. Application: Finding Exact Values | 应用：求精确值

Addition formulas let us find exact values for angles that are not standard, such as 15° or 75°.

和角公式能帮助我们求出非标准角（如 15° 或 75°）的精确值。

For example, 15° = 45° − 30°, so:

例如，15° = 45° − 30°，所以：

sin 15° = sin(45° − 30°) = sin 45° cos 30° − cos 45° sin 30°

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

Similarly, cos 75° = cos(45° + 30°) = (√6 − √2)/4 after simplification.

类似地，cos 75° = cos(45° + 30°)，化简后也等于 (√6 − √2)/4。

11. Common Mistakes | 常见错误

Writing sin(A + B) = sin A + sin B. This is false; you must use the full formula.

把 sin(A + B) 写成 sin A + sin B。这是错误的；必须使用完整公式。
Incorrectly applying the sign in cos(A − B): a common error is writing cos A cos B − sin A sin B, but the correct sign is plus.

在 cos(A − B) 中弄错符号：常见错误是写成 cos A cos B − sin A sin B，但正确的符号是加号。
Forgetting to divide by cos A cos B when deriving tan(A + B).

推导 tan(A + B) 时忘记除以 cos A cos B。

Always check your formula sheet and practice with small angles to spot these errors.

时刻对照公式表并用小角度检验，可以避免这些错误。

12. Practice and Summary | 练习与总结

Try these quick checks:

试试以下快速练习：

Expand sin(x + π/3).

展开 sin(x + π/3)。
Simplify cos(θ − 45°) − cos(θ + 45°).

化简 cos(θ − 45°) − cos(θ + 45°)。
Solve sin 2x = cos x for 0 ≤ x ≤ 360°.

解方程 sin 2x = cos x，其中 0 ≤ x ≤ 360°。

The key to mastering addition formulas is repeated practice. Memorise the structure, understand the proof once, and then apply them in countless identities and equations.

掌握和角公式的关键是反复练习。记住结构，理解一次证明，然后在无数恒等式和方程中应用它们。

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A-Level Maths: Trigonometric Addition Formulas Explained | A-Level数学：三角函数和角公式详解