📚 Addition Formulae in Trigonometry | 三角函数加法公式
s(A ± B) and tan(A ± B) into products of the sine and cosine of the single angles A and B.
加法公式（又称复合角公式）是把 sin(A ± B)、cos(A ± B) 和 tan(A ± B) 展开为单角 A、B 的正弦与余弦乘积的恒等式。
In Edexcel A-Level Mathematics they appear in proofs, in exact-value calculations done without a calculator, and in the solution of equations that reference angles alone cannot handle, so a fluent command of them is required for high marks.
在 Edexcel A-Level 数学中，它们出现在证明题、不使用计算器的精确值计算，以及仅靠参考角无法求解的方程中，因此熟练运用这些公式是取得高分的关键。
1. What the Addition Formulae Are | 什么是加法公式
An identity is an equation that is true for every value of the variable for which both sides are defined. The addition formulae are identities in the angle variables A and B, not equations to be solved: substitute any angle, in degrees or radians, and the two sides agree exactly.
恒等式是指对变量的一切取值（只要两边都有定义）都成立的等式。加法公式是关于角 A 和 B 的恒等式，而不是需要求解的方程：无论用角度还是弧度代入任意角，两边都严格相等。
They exist for all three trigonometric functions, and each one has a "plus" version and a "minus" version. The relationship between the sign on the left and the sign on the right is the single most heavily examined detail in this topic.
三个三角函数都有对应的加法公式，且每个公式都有"加"版与"减"版。左端符号与右端符号之间的关系，是本考点中被考查得最频繁的细节。
You will meet the same formulae written with different letters: sin(x + y), sin(θ + φ), sin(2x + 3y), or even sin(A + 30°). The names of the angles are irrelevant; only the structure matters.
你会看到同一公式写成不同的字母：sin(x + y)、sin(θ + φ)、sin(2x + 3y)，甚至 sin(A + 30°)。角的名称无关紧要，重要的是结构。
One more point of vocabulary: the expressions sin(A + B) and sin A + sin B are completely different. The addition formula is exactly the tool that tells you how to break the first one open.
还有一点术语上的要点：sin(A + B) 与 sin A + sin B 完全不同。加法公式正是用来拆开前者的工具。
2. The Addition Formulae at a Glance | 加法公式一览
Below are the forms you must have at your fingertips for the Edexcel specification. They should be reproducible from memory in seconds, because they underpin every later result in this topic.
以下是 Edexcel 考纲要求你烂熟于心的形式。它们必须能在几秒内凭记忆写出，因为本主题后面的每一个结论都建立在它们之上。
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)
The symbol ∓ means that when the left-hand side carries a plus sign, the right-hand side carries a minus sign, and vice versa. It is a compact way of writing two identities at once.
符号 ∓ 表示：当左端取加号时，右端对应取减号，反之亦然。它是一行写出两个恒等式的简写方式。
ed. Since tan θ = sin θ / cos θ, the tangent formula fails when cos A = 0, cos B = 0, or 1 ∓ 0, that is when A or B is an odd multiple of 90°, or when A + B is an odd multiple of 90°.
这些恒等式对任意实数角 A 和 B 都成立，但正切表达式必须有定义。由于 tan θ = sin θ / cos θ，当 cos A = 0、cos B = 0 或 1 ∓ tan A tan B = 0 时正切公式失效，即 A 或 B 为 90° 的奇数倍，或 A + B 为 90° 的奇数倍时。
n 40° cos 20° + cos 40° sin 20° = 0.6428 × 0.9397 + 0.7660 × 0.3420 = 0.8660, as expected.
做一个快速数值验证可以增强信心：取 A = 40°、B = 20°，sin 60° = 0.8660，而 sin 40° cos 20° + cos 40° sin 20° = 0.6428 × 0.9397 + 0.7660 × 0.3420 = 0.8660，与预期一致。
3. Reading the ± and ∓ Signs Correctly | 正确理解 ± 与 ∓ 的符号规律
The sign pattern is not random, and it follows three simple rules. For sine, the sign on the right matches the sign on the left. For cosine, the sign on the right is the opposite of the sign on the left. For tangent, the sign in the numerator matches the left, and the sign in the denominator is the opposite.
符号规律并非随意，它遵循三条简单规则。对正弦来说，右端符号与左端相同；对余弦来说，右端符号与左端相反；对正切来说，分子符号与左端相同，分母符号与左端相反。
- sin: sin(A + B) has a plus between the two products; sin(A − B) has a minus. | 正弦：sin(A + B) 两项之间取加号；sin(A − B) 取减号。
- cos: cos(A + B) has a minus; cos(A − B) has a plus. | 余弦：cos(A + B) 取减号；cos(A − B) 取加号。
- tan: ) = (tan A + tan B) / (1 − tan A tan B); tan(A − B) = (tan A − tan B) / (1 + tan A tan B). | 正切：tan(A + B) = (tan A + tan B) / (1 − tan A tan B)；tan(A − B) = (tan A − tan B) / (1 + tan A tan B)。
A reliable way to avoid slips is to think of the cosine formula as the "awkward" heck the middle sign before you move on, because a single flipped sign can turn a correct method into a zero on that part of the question.
避免失误的可靠方法是把余弦公式当作"特殊"的那一个。每次写出 cos(A ± B) 时，先检查中间那个符号再继续，因为一个符号写反就可能让正确的方法得零分。
30° = 0.75 − 0.25 = 0.5. If your version gives 1.0 instead, you have used the wrong sign.
另一个保险做法是用 A = B = 30° 检验写出的公式。此时 cos 60° = 0.5，而 cos 30° cos 30° − sin 30° sin 30° = 0.75 − 0.25 = 0.5。如果你的版本算出 1.0，说明符号用错了。
4. Geometric Derivation of cos(A − B) | cos(A − B) 的几何推导
To understand why the formulae work, start with cos(A − B) and a unit circle. Take two points P and Q on the circle, where P has angle A and Q has angle B, measured anticlockwise from the positive x-axis.
为了理解公式的来源，先从 cos(A − B) 和单位圆开始。在圆上取两点 P 和 Q，P 的角度为 A，Q 的角度为 B，均从 x 轴正方向逆时针度量。
quare of the distance PQ can be written in two equivalent ways, and setting the two expressions equal will produce the identity.
因此它们的坐标为 P = (cos A, sin A)，Q = (cos B, sin B)。距离 PQ 的平方可以用两种等价方式表示，令两式相等即可得到恒等式。
First, directly from the coordinates using the distance formula:
第一种，直接用距离公式从坐标写出：
PQ² = (cos A − cos B)² + (sin A − sin B)²
nce sin²θ + cos²θ = 1 for each angle, this simplifies to 2 − 2(cos A cos B + sin A sin B).
展开得 cos²A − 2 cos A cos B + cos²B + sin²A − 2 sin A sin B + sin²B。由于每个角都满足 sin²θ + cos²θ = 1，式子化简为 2 − 2(cos A cos B + sin A sin B)。
Second, rotate the whole diagram clockwise by B. Rotation preserves distances, so PQ is unchanged, but now Q has moved to (1, 0) and P has moved to the point at angle A − B. Hence:
第二种，把整个图形顺时针旋转 B。旋转保持距离不变，所以 PQ 不变，但此时 Q 移到 (1, 0)，P 移到角度为 A − B 的点。于是：
PQ² = (cos(A − B) − 1)² + (sin(A − B) − 0)²
Expanding this gives cos²(A − B) − 2 cos(A − B) + 1 + sin²(A − B), which simplifies to 2 − 2 cos(A − B).
展开得 cos²(A − B) − 2 cos(A − B) + 1 + sin²(A − B)，化简为 2 − 2 cos(A − B)。
Equating the two expressions for PQ² and dividing by −2 leaves the key result, valid for all A and B:
令 PQ² 的两个表达式相等并两边除以 −2，即得核心结论，对一切 A、B 成立：
cos(A − B) = cos A cos B + sin A sin B
5. Building the Other Formulae from cos(A − B) | 由 cos(A − B) 推出其余公式
Once cos(A − B) is established, the remaining five results follow by substitution and by using the symmetry of the trigonometric functions, so nothing else needs to be memorised from scratch.
一旦确立了 cos(A − B)，其余五个结论都可以通过代换以及三角函数的对称性推得，无需从零记忆。
e cosine formula, and using cos(−B) = cos B with sin(−B) = −sin B, gives the plus version:
在余弦公式中用 −B 替换 B，并利用 cos(−B) = cos B、sin(−B) = −sin B，即得"加"版：
cos(A + B) = cos A cos B − sin A sin B
a gives sin A cos B + cos A sin B after substituting cos(90° − A) = sin A and sin(90° − A) = cos A.
对于正弦，使用余角恒等式 sin θ = cos(90° − θ)。于是 sin(A + B) = cos(90° − A − B) = cos((90° − A) − B)，应用余弦公式并代入 cos(90° − A) = sin A、sin(90° − A) = cos A，即得 sin A cos B + cos A sin B。
B) / cos(A + B), then divide both numerator and denominator by cos A cos B. The numerator becomes tan A + tan B and the denominator becomes 1 − tan A tan B, provided cos A cos B ≠ 0.
对于正切，写出 tan(A + B) = sin(A + B) / cos(A + B)，再分子分母同除以 cos A cos B。在 cos A cos B ≠ 0 的条件下，分子变成 tan A + tan B，分母变成 1 − tan A tan B。
The chain of reasoning is worth remembering, because a "show that" question on the derivation of the sine formula from the cosine formula is a realistic exam task.
这条推理链值得记住，因为"由余弦公式推导正弦公式"这类证明题是很现实的考试任务。
6. The Double-Angle Formulae as a Special Case | 二倍角公式作为特例
converts it into a double-angle formula, which explains why these results are never listed separately as magic facts to learn.
在每个加法公式中令 B = A，即得到二倍角公式，这也解释了为什么这些结论无需作为孤立的事实单独记忆。
sin 2A = 2 sin A cos A
cos 2A = cos²A − sin²A = 1 − 2 sin²A = 2 cos²A − 1
tan 2A = 2 tan A / (1 − tan²A)
atters: 1 − 2 sin²A is convenient when the rest of the expression is in sines, and 2 cos²A − 1 when it is in cosines.
cos 2A 的三种形式都是用 sin²A + cos²A = 1 改写同一个恒等式。选择合适的形式很重要：当式子其余部分是正弦时用 1 − 2 sin²A 方便，当其余部分是余弦时用 2 cos²A − 1 方便。
ngent version is undefined when tan²A = 1, that is when A = 45° + 90°n. It also requires A
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