Using the angle addition formulae | 使用角度加法公式

📚 Using the angle addition formulae | 使用角度加法公式

Angle addition formulae let you expand sin(A ± B), cos(A ± B), and tan(A ± B) into expressions involving sin, cos, and tan of single angles. They are essential for exact values, proving identities, and solving equations in Edexcel A-Level Mathematics.

角度加法公式可以把 sin(A ± B)、cos(A ± B) 和 tan(A ± B) 展开为只含单个角三角函数的表达式。在 Edexcel A-Level 数学中，它们是求精确值、证明恒等式和解方程的核心工具。

1. The Core Formulae You Need to Know | 你需要掌握的核心公式

For Edexcel A-Level, the three addition formulae are given in the formula booklet, but you still need to know how to choose signs and apply them quickly.

在 Edexcel A-Level 考试中，三个加法公式会出现在公式表中，但你仍需要快速判断正负号并熟练应用。

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)

Notice that for sin(A ± B) the sign in the expansion matches the sign inside the bracket; for cos(A ± B) the sign is reversed; for tan(A ± B) the numerator sign matches but the denominator sign is reversed.

注意 sin(A ± B) 展开式中的符号与括号内一致；cos(A ± B) 展开式的符号相反；tan(A ± B) 的分子符号一致，但分母符号相反。

2. Why the Formulae Work: A Geometric Idea | 公式的几何直觉

One classic proof uses the unit circle and the distance formula. If P is at angle A and Q at angle B, the distance PQ can be written in two ways: one from coordinates, one from the chord angle A − B.

一个经典证明使用单位圆和距离公式。若点 P 对应角 A，点 Q 对应角 B，则距离 PQ 可以从坐标与弦角 A − B 两种方式表示。

Equating these two forms gives cos(A − B) = cos A cos B + sin A sin B. Replacing B by −B then gives cos(A + B), and using complementary-angle identities gives the sine formulae.

令两种表示相等就得到 cos(A − B) = cos A cos B + sin A sin B。将 B 换成 −B 可得到 cos(A + B)，再利用余角恒等式可得到正弦公式。

You do not need to reproduce the proof in most exam questions, but understanding why the signs appear helps avoid sign errors.

大多数考题不要求复现证明，但理解符号来源有助于避免符号错误。

3. Finding Exact Values of sin75° and cos15° | 求 sin75° 与 cos15° 的精确值

Write 75° as 45° + 30°. Then apply sin(A + B):

把 75° 写成 45° + 30°，然后应用 sin(A + B)：

sin75° = sin(45° + 30°) = sin45°cos30° + cos45°sin30°

sin75° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4

Similarly, cos15° can be written as cos(45° − 30°):

类似地，cos15° 可以写成 cos(45° − 30°)：

cos15° = cos(45° − 30°) = cos45°cos30° + sin45°sin30°

cos15° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4

Because cos(45° − 30°) uses the plus sign in the cosine formula, cos15° and sin75° have the same exact value; this matches the identity cos θ = sin(90° − θ).

由于 cos(45° − 30°) 使用余弦公式中的加号，cos15° 与 sin75° 的精确值相同；这与恒等式 cos θ = sin(90° − θ) 一致。

For angles like 105°, 165°, or 12° from 45° and 30°, you can build exact values even if the angles are not in the standard list.

对于由 45° 与 30° 组合出的 105°、165° 等角，即使不在标准角列表中，也可以构造精确值。

4. Expanding and Simplifying Expressions | 展开并化简表达式

Many exam questions ask you to expand an expression such as sin(θ + 60°) and then simplify it in terms of sin θ and cos θ.

很多考题要求展开 sin(θ + 60°) 等表达式，并用 sin θ 和 cos θ 化简。

sin(θ + 60°) = sin θ cos60° + cos θ sin60°

sin(θ + 60°) = (1/2)sin θ + (√3/2)cos θ

If the expression is combined with another term, for example sin(θ + 60°) + sin(θ − 60°), expand both and collect like terms.

如果表达式中还有另一项，例如 sin(θ + 60°) + sin(θ − 60°)，就展开两项并合并同类项。

sin(θ + 60°) + sin(θ − 60°) = 2sin θ cos60° = sin θ

This type of simplification is useful because it can remove one trigonometric function and reveal a hidden relationship.

这类化简很有用，因为它可以消去某个三角函数并揭示隐藏关系。

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

Angle addition formulae are often the quickest way to prove identities such as sin(A + B)cos B − cos(A + B)sin B = sin A.

角度加法公式通常是证明恒等式的最快方法，例如 sin(A + B)cos B − cos(A + B)sin B = sin A。

Expand sin(A + B) and cos(A + B), multiply by cos B and sin B, then simplify:

展开 sin(A + B) 和 cos(A + B)，分别乘上 cos B 和 sin B，然后化简：

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

sin(A + B)cos B − cos(A + B)sin B = sin A cos²B + cos A sin B cos B − cos A cos B sin B + sin A sin²B

sin(A + B)cos B − cos(A + B)sin B = sin A (cos²B + sin²B) = sin A

When proving identities, always start with the more complicated side and use sin²θ + cos²θ = 1 to finish the simplification.

证明恒等式时，通常从较复杂的一边开始，并利用 sin²θ + cos²θ = 1 完成化简。

6. Solving Equations with Angle Addition | 用角度加法公式解方程

Some equations contain a product like sin θ cos θ or sin θ cos α + cos θ sin α. Angle addition can convert them into a single sine or cosine term.

有些方程含有 sin θ cos θ 或 sin θ cos α + cos θ sin α 这样的乘积，角度加法公式可以把它们化为单个正弦或余弦项。

For example, solve sin θ cos30° + cos θ sin30° = 1/√2 for 0° ≤ θ < 360°.

例如，解 sin θ cos30° + cos θ sin30° = 1/

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Using the angle addition formulae | 使用角度加法公式