📚 Using sec x, cosec x and cot x: Definitions, Graphs, Identities and Calculus | 使用 sec x、cosec x 与 cot x：定义、图像、恒等式与微积分
In A-Level Mathematics, the reciprocal trigonometric functions sec x, cosec x and cot x extend the familiar sin, cos and tan toolkit. They appear in identities, differentiation, integration, equation solving and graph transformations, so a strong command of their definitions and behaviour is essential for Edexcel Pure Mathematics.
在 A-Level 数学中，倒数三角函数 sec x、cosec x 和 cot x 扩展了我们熟悉的正弦、余弦和正切工具。它们出现在恒等式、微分、积分、解方程和图像变换中，因此牢固掌握它们的定义和性质对 Edexcel 纯数学至关重要。
1. Why These Three Functions Matter | 这三个函数为什么重要
These three functions are not a separate branch of trigonometry. Each one is simply 1 divided by a function you already know, so every fact about them can be derived from sin x, cos x and tan x. Examiners therefore expect you to move fluently between the two notations instead of memorising the reciprocals in isolation.
这三个函数并不是三角学的一个独立分支。每一个都只是"1 除以你已经熟悉的函数"，因此关于它们的所有结论都可以从 sin x、cos x 和 tan x 推导出来。正因如此，考官期望你能在两种记号之间自由切换，而不是孤立地死记硬背。
²x = cosec²x are used to prove identities and solve equations. They reappear later in differentiation, and again in Further Mathematics when you integrate them, so the same knowledge earns marks across several chapters.
在 Edexcel A-Level 数学大纲中，sec x、cosec x 和 cot x 首次出现在 Year 2 纯数学，用恒等式 1 + tan²x = sec²x 与 1 + cot²x = cosec²x 来证明恒等式、解方程。它们随后在微分中再次出现，并在进阶数学（Further Mathematics）的积分中第三次出现，因此同一份知识能在多个章节中反复得分。
A useful mental model is to ask a question about size: how many times does cos x fit into 1? When cos x is very small, sec x is very large; when cos x is close to 1, sec x is close to 1. Exactly the same reasoning applies to cosec x and cot x.
一个有用的思维模型是问"大小关系"：1 里面能装下多少个 cos x？当 cos x 很小时，sec x 会很大；当 cos x 接近 1 时，sec x 接近 1。完全相同的推理也适用于 cosec x 和 cot x。
2. Definitions and Exact Values | 定义与特殊角的精确值
The three reciprocal trigonometric functions are defined as follows.
三个倒数三角函数的定义如下。
sec x = 1 / cos x, cosec x = 1 / sin x, cot x = cos x / sin x = 1 / tan x
Read the notation carefully. cosec x means 1 / sin x, while sin⁻¹x means arcsin x, the inverse function. Confusing the reciprocal with the inverse is one of the most expensive slips in the exam, because the two answers have nothing in common.
请仔细看记号。cosec x 表示 1 / sin x，而 sin⁻¹x 表示 arcsin x，即反函数。把"倒数"与"反函数"混为一谈是考试中最昂贵的失误之一，因为两者的答案毫无关系。
Exact values at the standard angles are worth memorising, since they let you finish a question without a calculator. Reciprocate the familiar sin, cos and tan values, and remember that a reciprocal of zero does not exist.
标准角的精确值值得背下来，因为它们让你不必依赖计算器就能完成题目。把熟悉的 sin、cos、tan 值取倒数即可，同时记住"零的倒数不存在"。
| x | 0 | π/6 | π/4 | π/3 | π/2 |
| sec x | 1 | 2/√3 | √2 | 2 | undefined |
| cosec x | undefined | 2 | √2 | 2/√3 | 1 |
| cot x | undefined | √3 | 1 | 1/√3 | 0 |
y valid at points where both sides exist, which is a subtlety that examiners like to test.
特别注意 cot(π/2) = 0，而 tan(π/2) 无定义。因此恒等式 cot x = 1 / tan x 只在两边都有定义的点成立，这是考官喜欢考查的一个细节。
3. Domains: Where the Functions Break Down | 定义域：函数在哪里失效
Because these definitions involve division by sin x or cos x, the functions are undefined whenever the corresponding denominator is zero. There is no value to plot at those points, only an asymptote.
由于这些定义涉及除以 sin x 或 cos x，因此只要相应分母为零，函数就没有定义。这些点上没有可描绘的函数值，只有一条渐近线。
x are undefined where sin x = 0, that is at x = nπ, which is 180°n. Here n is any integer.
sec x 在 cos x = 0 处无定义，即 x = π/2 + nπ（也就是 90° + 180°n）。cosec x 和 cot x 在 sin x = 0 处无定义，即 x = nπ（也就是 180°n）。其中 n 为任意整数。
- Domain of sec x: x ∈ ℝ, x ≠ π/2 + nπ
- Domain of cosec x: x ∈ ℝ, x ≠ nπ
- Domain of cot x: x ∈ ℝ, x ≠ nπ
- sec x 的定义域：x ∈ ℝ，x ≠ π/2 + nπ
- cosec x 的定义域：x ∈ ℝ，x ≠ nπ
- cot x 的定义域：x ∈ ℝ，x ≠ nπ
When a question asks for solutions in an interval, always compare your answers with the domain. A solution that makes sin x or cos x zero must be discarded, because the original equation was never defined there.
当题目要求在某个区间内求解时，务必把答案与定义域比对。任何使 sin x 或 cos x 为零的解都必须舍去，因为原方程在那一点根本没有定义。
4. Ranges: |sec x| ≥ 1 and |cosec x| ≥ 1 | 值域：|sec x| ≥ 1 与 |cosec x| ≥ 1
ends and produces values of magnitude at least 1. The range of sec x is therefore (-∞, -1] ∪ [1, ∞), and exactly the same argument gives |cosec x| ≥ 1.
由于 -1 ≤ cos x ≤ 1，取倒数后端点不等式方向反转，得到的值绝对值至少为 1。因此 sec x 的值域是 (-∞, -1] ∪ [1, ∞)，完全相同的论证给出 |cosec x| ≥ 1。
|sec x| ≥ 1 |cosec x| ≥ 1 cot x ∈ ℝ
cot x behaves differently. As sin x approaches zero the ratio cos x / sin x grows without bound, and as x sweeps through one period cot x takes every real value, so its range is all real numbers.
cot x 的行为不同。当 sin x 趋于零时，比值 cos x / sin x 无限增大；而当 x 扫过一个周期时，cot x 取遍所有实数值，所以它的值域是全体实数。
his is why the graphs of sec x and cosec x never enter the horizontal strip between y = -1
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