📚 Secant, Cosecant and Cotangent | 正割、余割与余切
In A-Level Edexcel Mathematics, secant (sec), cosecant (cosec) and cotangent (cot) are the three reciprocal trigonometric functions. They are built directly from cosine, sine and tangent, and they appear throughout identities, equations, graph sketching and further calculus work. This guide covers their definitions, domains, exact values, graphs, the Pythagorean and quotient identities, worked proof and equation techniques, and the exam habits that protect your marks.
在 A-Level Edexcel 数学中，正割（sec）、余割（cosec）和余切（cot）是三个倒数三角函数。它们由余弦、正弦和正切直接构造而来，广泛出现在恒等式、方程、图像以及后续的微积分内容中。本文将系统梳理它们的定义、定义域与值域、精确值、图像、平方恒等式与商数关系、证明与解方程的实战技巧，以及能帮你守住分数的考试习惯。
1. Definitions of the Reciprocal Ratios | 三个倒数比的定义
For any angle θ where the relevant denominator is non-zero, the three functions are defined as follows:
对于任意使得分母不为零的角 θ，三个函数的定义如下：
| Function | Definition | Equivalent form | Undefined when |
|---|---|---|---|
| sec θ | 1 / cos θ | 1 / cos θ | cos θ = 0, i.e. θ = 90° + 180°n |
| cosec θ | 1 / sin θ | 1 / sin θ | sin θ = 0, i.e. θ = 180°n |
| cot θ | 1 / tan θ | cos θ / sin θ | sin θ = 0, i.e. θ = 180°n |
Note that sec θ is paired with cosine, cosec θ with sine, and cot θ with tangent or the quotient of cosine and sine. This pairing is often the first point to check in algebraic manipulation, and it explains why the word "co" appears in every one of the three names.
请注意，sec θ 与余弦配对，cosec θ 与正弦配对，cot θ 与正切配对，也可写成余弦除以正弦。这种配对关系通常是代数变形时第一个要检查的地方，也解释了为什么这三个名称中都带有 "co" 这个前缀。
nd, sec²θ always means (sec θ)², never sec(θ²). The same convention applies to cosec²θ and cot²θ, and examiners rely on it constantly.
有两个记号陷阱必须一开始就讲清楚。第一，sec θ 是一个整体的函数值，不是 s × e × c 的乘积。第二，sec²θ 始终表示 (sec θ)²，绝不表示 sec(θ²)。cosec²θ 与 cot²θ 遵循同样的约定，而考官正是反复利用这一点命题。
ec θ fails wherever sin θ = 0, and cot θ fails wherever sin θ = 0 as well. In radians the same restrictions read θ = π/2 + πn for sec, and θ = πn for cosec and cot.
由于不允许除以零，每个函数的"禁区"都直接来自它的分母。只要 cos θ = 0，sec θ 就无定义；只要 sin θ = 0，cosec θ 无定义；cot θ 同样在 sin θ = 0 处无定义。用弧度表示，即为：sec 在 θ = π/2 + πn 处无定义，cosec 与 cot 在 θ = πn 处无定义。
2. Where These Functions Come From | 这三个函数从哪里来
djacent. These are exactly the ratios that appear in the formulae for similar triangles, which is why the reciprocal names were invented before calculators existed.
在直角三角形中，sin θ = 对边 / 斜边，cos θ = 邻边 / 斜边，因此 cosec θ = 斜边 / 对边，sec θ = 斜边 / 邻边。这些恰好是相似三角形公式中自然出现的比，这也是为什么在计算器被发明之前，人们就给这三个倒数起了专名。
The names carry their history. "Secant" comes from the Latin for "cutting", because in a unit circle diagram the line from the origin to the tangent at P cuts the circle. "Tangent" means "touching" and "cosecant" is the secant of the complementary angle. You do not need the etymology in the exam, but it makes the reciprocal pairing far easier to remember.
这些名称自带历史痕迹。"正割"（secant）源自拉丁语"切割"，因为在单位圆图中，从原点出发到 P 点切线的线段会切割圆；"正切"（tangent）意为"相切"；"余割"则是余角的正割。考试中并不需要词源知识，但它能让你更轻松地记住这组倒数配对关系。
θ is the y-coordinate, so sec θ = 1 / x and cosec θ = 1 / y. When P sits exactly on the y-axis, x = 0 and sec θ blows up, which is precisely the vertical asymptote you later see on the graph.
从三角形过渡到单位圆，一切都被推广了。若 P 是单位圆上对应角 θ 的点，则 cos θ 是它的横坐标，sin θ 是纵坐标，于是 sec θ = 1 / x，cosec θ = 1 / y。当 P 恰好落在 y 轴上时 x = 0，sec θ 趋于无穷，这正是你之后在图像上看到的竖直渐近线。
3. Domain and Range | 定义域与值域
Because each reciprocal function is one divided by something, its range can never contain a value between −1 and 1. A reciprocal of a number whose modulus is at most 1 must have modulus at least 1. This single observation answers a large number of multiple-choice and "explain why" questions.
由于每个函数都是"1 除以某个量"，其值域中绝不可能出现 −1 与 1 之间的数。一个模长不超过 1 的数的倒数，其模长必然不小于 1。仅凭这一条观察，就能答对大量选择题和"请说明理由"类问题。
sec θ ∈ (−∞, −1] ∪ [1, ∞) cosec θ ∈ (−∞, −1] ∪ [1, ∞) cot θ ∈ ℝ
ry real value, because cos θ / sin θ can be arbitrarily large, arbitrarily small, positive or negative, and it passes through zero at θ = 90° + 180°n.
sec θ 的定义域排除 θ = 90° + 180°n；cosec θ 与 cot θ 的定义域都排除 θ = 180°n。在值域方面 cot 是特例：它可以取到所有实数值，因为 cos θ / sin θ 可以任意大、任意小、取正或取负，并且在 θ = 90° + 180°n 处取零。
nterlace rather than coincide. Similarly, sec θ = 0 and cosec θ = 0 have no solutions at all, so an equation such as sec θ = 0.5 is immediately impossible and should be rejected before any working.
注意一个常让学生出错的"不对称性"：cot θ 无定义之处，正是 tan θ = 0 之处；而 tan θ 无定义之处，正是 cot θ = 0 之处。两条图像是交错而非重合的。同理，sec θ = 0 与 cosec θ = 0 完全没有解，因此像 sec θ = 0.5 这样的方程一开始就不可能成立，应当在动笔前就判定无解。
4. Exact Values You Must Know | 必须掌握的精确值
You should be able to quote the reciprocal values at 0°, 30°, 45°, 60° and 90° instantly, without a calculator. Build them by taking the standard sine and cosine values and flipping them.
你应当能够不借助计算器，立即说出 0°、30°、45°、60°、90° 处的倒数值。方法是先写出标准的正弦、余弦值，然后取倒数即可。
| θ | sec θ | cosec θ | cot θ |
|---|---|---|---|
| 0° | 1 | undefined | undefined |
| 30° | 2 / √3 = 2√3 / 3 | 2 | √3 |
| 45° | √2 | √2 | 1 |
| 60° | 2 | 2 / √3 = 2√3 / 3 | 1 / √3 = √3 / 3 |
| 90° | undefined | 1 | 0 |
are complementary. That is the complementary-angle relationship at work, and it is a fast self-check on any value you produce under pressure.
沿对角线读这张表，你会看到对称性：sec 30° = cosec 60°，sec 60° = cosec 30°，因为 30° 与 60° 互余。这正是余角关系的体现，也是你在考场上快速自查的好办法。
20° = 1 / cos 120° = 1 / (−½) = −2, and cosec 210° = 1 / sin 210° = 1 / (−½) = −2. The sign of sec always matches the sign of cos, and the sign of cosec always matches the sign of sin.
对于其他象限的角，先对底层的正弦或余弦施用 CAST 符号规则，再取倒数即可。由于"负数的倒数仍为负"，sec 120° = 1 / cos 120° = 1 / (−½) = −2，cosec 210° = 1 / sin 210° = 1 / (−½) = −2。sec 的符号始终与 cos 一致，cosec 的符号始终与 sin 一致。
5. Graphs and Asymptotes | 图像与渐近线
ere sin x = 0, the cosec graph has a vertical asymptote; where sin x = 1 or −1, the cosec graph touches y = 1 or y = −1 at a vertex.
只要想象对应的正弦或余弦曲线，判断它在哪里取大值、小值或零，就能在几秒内画出每个倒数函数的图像。sin x = 0 处，cosec 图像有竖直渐近线；sin x = 1 或 −1 处，cosec 图像在 y = 1 或 y = −1 处形成顶点并相切。
- 0° (2π), range y ≤ −1 or y ≥ 1, vertical asymptotes at x = 90° + 180°n, vertex at (0°, 1).
- od 360° (2π), range y ≤ −1 or y ≥ 1, vertical asymptotes at x = 180°n, vertex at (90°, 1).
- values, vertical asymptotes at x = 180°n, decreasing on every interval between asymptotes.
- y = sec x：周期 360°（2π），值域 y ≤ −1 或 y ≥ 1，竖直渐近线在 x = 90° + 180°n，顶点在 (0°, 1)。
- y = cosec x：周期 360°（2π），值域 y ≤ −1 或 y ≥ 1，竖直渐近线在 x = 180°n，顶点在 (90°, 1)。
- y = cot x：周期 180°（π），值域为全体实数，竖直渐近线在 x = 180°n，在每个渐近线之间的区间上单调递减。
sses the period from 360° to 120°, so its asymptotes become x = 60°n. When a question says "sketch", mark the asymptotes as dashed lines and label the vertex coordinates.
图像变换与基本函数的变换规律完全一致。y = sec(x − 30°) 是将 y = sec x 向右平移 30°；y = cosec 3x 把周期从 360° 压缩到 120°，因此渐近线变为 x = 60°n。当题目要求"画图"时，请用虚线标出渐近线，并标出顶点的坐标。
, so y ≥ 4 or y ≤ 2. If it asks for the range of y = 1 / (2 + sin x), first find 2 + sin x ∈ [1, 3], then take reciprocals: y ∈ [1/3, 1].
随之而来的是值域问题。若题目问 y = 3 + sec 2x 的值域，注意 sec 2x ≥ 1 或 ≤ −1，因此 y ≥ 4 或 y ≤ 2。若问 y = 1 / (2 + sin x) 的值域，先求 2 + sin x ∈ [1, 3]，再取倒数得 y ∈ [1/3, 1]。
6. The Three Pythagorean Identities | 三个平方恒等式
through by cos²θ. The result is the identity you will use more than any other in this topic:
从 sin²θ + cos²θ = 1 出发，两边同除以 cos²θ，就得到本专题中使用频率最高的恒等式：
1 + tan²θ = sec²θ
Now divide the same starting identity by sin²θ instead:
若改为两边同除以 sin²θ，则得到：
1 + cot²θ = cosec²θ
s, and both factorise, which is the key step in many proofs. The factorised versions are worth writing down before you attack any proof question.
这两个恒等式经常需要变形。sec²θ − tan²θ = 1 与 cosec²θ − cot²θ = 1 是它们的差形式，而两者都可以因式分解，这正是许多证明题的关键一步。在做任何证明题之前，先把因式分解形式写下来是值得的。
sec²θ − tan²θ = (sec θ − tan θ)(sec θ + tan θ) = 1
t is safe, and you obtain sec θ − tan θ = 1/3. Adding and subtracting the two equations then gives sec θ = 5/3 and tan θ = 4/3.
这个因式分解可以把 sec θ + tan θ = 3 这类看似棘手的方程变成一行求解。左端永远不为零，因此可以放心地用它作除数，得到 sec θ − tan θ = 1/3。把两个方程相加、相减，即得 sec θ = 5/3、tan θ = 4/3。
7. Quotient, Reciprocal and Symmetry Identities | 商数、倒数与对称恒等式
Beyond the Pythagoreans, three families of identities do most of the remaining work. Learn them as relationships between functions rather than as isolated facts.
除平方恒等式之外，还有三组恒等式承担了其余大部分工作。请把它们当作函数之间的关系来记忆，而不是孤立的事实。
- Reciprocal: sec θ = 1 / cos θ, cosec θ = 1 / sin θ, cot θ = 1 / tan θ.
- Quotient: tan θ = sin θ / cos θ, cot θ = cos θ / sin θ.
- Complementary: sec(90° − θ) = cosec θ, cosec(90° − θ) = sec θ, cot(90° − θ) = tan θ.
- 倒数关系：sec θ = 1 / cos θ，cosec θ = 1 / sin θ，cot θ = 1 / tan θ。
- 商数关系：tan θ = sin θ / cos θ，cot θ = cos θ / sin θ。
- 余角关系：sec(90° − θ) = cosec θ，cosec(90° − θ) = sec θ，cot(90° − θ) = tan θ。
c θ. Tangent is odd, so cot is odd: cot(−θ) = −cot θ. In degrees-to-radians terms nothing changes, since these are properties of the functions, not of the unit of angle.
对称性来自底层函数的奇偶性。余弦是偶函数，所以 sec 也是偶函数：sec(−θ) = sec θ。正弦是奇函数，所以 cosec 是奇函数：cosec(−θ) = −cosec θ。正切是奇函数，所以 cot 是奇函数：cot(−θ) = −cot θ。无论用角度还是弧度，这些结论都不变，因为它们是函数本身的性质，与角的单位无关。
)(cos θ/sin θ) ÷ (1/sin θ) = (1/sin θ)(sin θ) = 1. The whole expression collapses to the constant 1.
一个简单的例子即可展示"全部化成正弦余弦"的威力。若要化简 (sec θ · cot θ) / cosec θ，逐个替换：(1/cos θ)(cos θ/sin θ) ÷ (1/sin θ) = (1/sin θ)(sin θ) = 1。整个式子化为常数 1。
8. Proving Identities: Strategy and Examples | 证明恒等式的策略与例题
ross the equals sign. Second, replace every sec, cosec and cot with sines and cosines or with tan, whichever produces fractions you can combine. Third, use sin²θ + cos²θ = 1 as the final step to collapse the expression.
证明恒等式有一套可靠的三步法。第一，只处理一侧，通常从较复杂的一侧入手，绝不把项移到等号另一边。第二，把所有的 sec、cosec、cot 换成正弦余弦，或换成 tan，选择能让你合并分式的那一种。第三，用 sin²θ + cos²θ = 1 作为最后一步来收束整个表达式。
sec θ − cos θ − 1 = sec θ − cos θ = 1/cos θ − cos θ = (1 − cos²θ)/cos θ = sin²θ / cos θ, which is the right-hand side.
下面是典型的 Edexcel 风格证明。证明 (1 − cos θ)(1 + sec θ) = sin²θ / cos θ。展开左端得 + sec θ − cos θ − 1 = sec θ − cos θ = 1/cos θ − cos θ = (1 − cos²θ)/cos θ = sin²θ / cos θ，即右端。
θ + tan²θ = 1 − 2 sin θ / cos²θ + tan²θ... Instead, a cleaner route is to multiply numerator and denominator of the right-hand side by (1 + sin θ): (1 − sin θ)(1 + sin θ) / (1 + sin θ)² = cos²θ / (1 + sin θ)². Taking square roots on both sides and using sec θ − tan θ = cos θ /(1 + sin θ) completes the argument.
第二个例子直接使用平方恒等式。证明 (sec θ − tan θ)² = (1 − sin θ)/(1 + sin θ)。更简洁的路径是把右端分子分母同乘 (1 + sin θ)：得 (1 − sin θ)(1 + sin θ) / (1 + sin θ)² = cos²θ / (1 + sin θ)²。两边开平方，并利用 sec θ − tan θ = cos θ / (1 + sin θ)，即可完成论证。
If a proof stalls, ask three diagnostic questions. Are there fractions that should be combined over a common denominator? Is there a difference of squares hiding in sec²θ − tan²θ or 1 − cos²θ? Have I accidentally used a value of θ for which a function is undefined? Answering these resolves the overwhelming majority of stuck proofs.
如果证明卡住了，问自己三个诊断性问题：是否存在应当通分合并的分式？在 sec²θ − tan²θ 或 1 − cos²θ 中是否藏着平方差？我是否无意中使用了让某个函数无定义的 θ 值？回答这三个问题，绝大多数"卡壳"的证明都能迎刃而解。
9. Solving Equations with sec, cosec and cot | 用 sec、cosec、cot 解方程
write as cos θ = 1/3, take the principal value θ = 70.5° (3 s.f.), and use the symmetry of cosine to find the second solution θ = 360° − 70.5° = 289.5°.
通用的第一步是把方程化成正弦或余弦，再用熟悉的方法求解。若给定 sec θ = 3，0° ≤ θ < 360°，改写为 cos θ = 1/3，取主值 θ = 70.5°（保留三位有效数字），再利用余弦的对称性求第二个解 θ = 360° − 70.5° = 289.5°。
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