📚 Secant, Cosecant and Cotangent | 正割、余割和余切
In A-Level Mathematics, extending the family of trigonometric functions to their reciprocals unlocks a richer set of tools for modelling, calculus, and solving equations. Secant (sec), cosecant (cosec) and cotangent (cot) are defined directly from sine, cosine and tangent, and they appear frequently in identities, differentiation, integration and trigonometric equations. A solid grasp of these three functions is essential for success in the Edexcel Pure Mathematics syllabus.
在 A-Level 数学中，将三角函数族扩展到它们的倒数，为建模、微积分和方程求解提供了更丰富的工具。正割 (sec)、余割 (cosec) 和余切 (cot) 直接由正弦、余弦和正切定义，它们频繁出现在恒等式、微分、积分和三角方程中。扎实掌握这三个函数对于 Edexcel 纯数学课程的成功至关重要。
1. Introduction to Reciprocal Trigonometric Functions | 倒数三角函数简介
The three basic trigonometric ratios sine, cosine and tangent each have a corresponding reciprocal function. Historically, these were used to simplify calculations in navigation and astronomy before the era of electronic calculators. Today they are standard functions in calculus and analysis, helping to handle expressions where division by a trigonometric function is more natural than dealing with the original ratio.
三个基本三角比正弦、余弦和正切各自有一个对应的倒数函数。历史上，这些函数在电子计算器时代之前被用于简化航海和天文学中的计算。如今它们是微积分和分析中的标准函数，有助于处理那些除以一个三角函数比使用原始比值更自然的表达式。
In the Edexcel specification, you are expected to know the definitions, graphs, key identities, derivatives and integrals of sec x, cosec x and cot x. You must also be able to use them to solve equations and simplify expressions.
在 Edexcel 大纲中，要求掌握 sec x、cosec x 和 cot x 的定义、图像、关键恒等式、导数和积分，并能够用它们求解方程和化简表达式。
2. Definitions of Secant, Cosecant and Cotangent | 正割、余割和余切的定义
The reciprocal functions are defined for angles where the original function is not zero. This immediately introduces restrictions on their domains because division by zero is undefined.
倒数函数在原始函数不为零的角度上有定义。这立即引入了对它们定义域的限制，因为除以零是未定义的。
sec θ = 1 / cos θ, defined for cos θ ≠ 0, i.e. θ ≠ π/2 + kπ.
sec θ = 1 / cos θ，定义域为 cos θ ≠ 0，即 θ ≠ π/2 + kπ。
cosec θ = 1 / sin θ (often written csc θ), defined for sin θ ≠ 0, i.e. θ ≠ kπ.
cosec θ = 1 / sin θ（也常写作 csc θ），定义域为 sin θ ≠ 0，即 θ ≠ kπ。
cot θ = 1 / tan θ = cos θ / sin θ, defined for sin θ ≠ 0, i.e. θ ≠ kπ.
cot θ = 1 / tan θ = cos θ / sin θ，定义域为 sin θ ≠ 0，即 θ ≠ kπ。
It is crucial to remember that cot θ can also be expressed as cos θ divided by sin θ, which is often more useful than 1/tan θ when simplifying expressions involving sine and cosine.
记住 cot θ 也可以表示为 cos θ 除以 sin θ，这非常关键，因为相比 1/tan θ，在处理包含正弦和余弦的表达式时，这种形式通常更有用。
3. Graphs of sec x, cosec x and cot x | sec x、cosec x 和 cot x 的图像
Understanding the shapes of these graphs helps with visualising solutions to equations and identifying asymptotes and ranges. All three functions are periodic and have vertical asymptotes where their denominators are zero.
理解这些函数的图像有助于直观地看出方程的解，并识别渐近线和值域。这三个函数都是周期函数，且在其分母为零处有垂直渐近线。
The graph of y = sec x d its range is y ≤ −1 or y ≥ 1. The minimum positive value is 1, achieved where cos x = 1.
y = sec x 的图像周期为 2π。在 x = π/2 + kπ 处有垂直渐近线。曲线由伸向 +∞ 或 −∞ 的分支组成，其值域为 y ≤ −1 或 y ≥ 1。最小正值为 1，在 cos x = 1 处取得。
The graph of y = cosec x t x = kπ. Its range is y ≤ −1 or y ≥ 1. It is essentially the secant graph shifted by π/2.
y = cosec x 的图像周期也是 2π，渐近线在 x = kπ 处。值域为 y ≤ −1 或 y ≥ 1。它本质上是正割图像平移 π/2 的结果。
The graph of y = cot x otes at x = kπ. It decreases from +∞ to −∞ between each pair of asymptotes, crossing the x‑axis at x = π/2 + kπ. Its range is all real numbers.
y = cot x 的图像周期为 π，渐近线在 x = kπ 处。它在每对渐近线之间从 +∞ 下降到 −∞，并在 x = π/2 + kπ 处穿过 x 轴。其值域为全体实数。
When sketching, always label asymptotes and indicate the behaviour as x approaches them. These sketches are frequently tested on Edexcel papers.
作图时，务必标注渐近线并标明 x 趋近渐近线时的函数行为。Edexcel 试卷中常会考察这些草图。
4. Key Properties and Asymptotes | 关键性质和渐近线
The reciprocal trigonometric functions inherit symmetries from sine and cosine. Knowing whether a function is even, odd or neither can simplify many problems.
倒数三角函数继承了正弦和余弦的对称性。了解一个函数是偶函数、奇函数还是非奇非偶，可以简化许多问题。
osec (−x) = −cosec x, making cosecant odd. cot (−x) = −cot x, so cotangent is odd as well.
sec (−x) = sec x，因此正割是偶函数。cosec (−x) = −cosec x，余割是奇函数。cot (−x) = −cot x，余切也是奇函数。
ing on the sign of cos x. Near an asymptote, the behaviour can be determined by considering the sign of the reciprocal of a very small number.
渐近线对应于分母的零点。对于 sec x，垂直渐近线为 x = π/2 + nπ，函数趋近 +∞ 还是 −∞ 取决于 cos x 的符号。在渐近线附近，可通过考虑一个非常小的数的倒数的符号来判断行为。
The range of both sec x and cosec x is (−∞, −1] ∪ ake values strictly between −1 and 1. This property is often used in solving inequalities.
sec x 和 cosec x 的值域均为 (−∞, −1] ∪ [1, ∞)。这意味着它们永远不取严格介于 −1 和 1 之间的值。这一性质常用于求解不等式。
range is all real numbers. Because cot x = cos x / sin x, it is undefined when sin x = 0.
对于 cot x，定义域排除了 π 的整数倍，值域为全体实数。由于 cot x = cos x / sin x，当 sin x = 0 时无定义。
5. Fundamental Identities | 基本恒等式
The reciprocal functions lead to new versions of the Pythagorean identities, which are essential for simplifying trigonometric expressions and proving other results.
倒数函数导出了毕达哥拉斯恒等式的新形式，这些形式对于化简三角表达式和证明其他结论至关重要。
Starting from sin²θ + cos²θ = 1, we can divide through by cos²θ to obtain: 1 + tan²θ = sec²θ. This is valid for cos θ ≠ 0.
从 sin²θ + cos²θ = 1 出发，两边同除以 cos²θ 可得：1 + tan²θ = sec²θ。该式在 cos θ ≠ 0 时成立。
Dividing sin²θ + cos²θ = 1 by sin²θ gives: 1 + cot²θ = cosec²θ. This holds for sin θ ≠ 0.
将 sin²θ + cos²θ = 1 除以 sin²θ 可得：1 + cot²θ = cosec²θ。该式在 sin θ ≠ 0 时成立。
or instance, an equation like sec²x − 3 tan x = 1 can be transformed into a quadratic in tan x using 1+tan²x = sec²x.
这两个恒等式在求解方程时极有价值，当需要将包含 sec² 或 cosec² 的项替换为 tan 或 cot 的表达式时就派上用场。例如，方程 sec²x − 3 tan x = 1 可利用 1+tan²x = sec²x 转化为关于 tan x 的二次方程。
Additionally, from the definitions we have useful reciprocal relationships: sin θ cosec θ = 1, cos θ sec θ = 1 and tan θ cot θ = 1.
此外，由定义可得到一些有用的倒数关系：sin θ cosec θ = 1、cos θ sec θ = 1 和 tan θ cot θ = 1。
6. Simplifying Expressions | 化简表达式
When an expression contains a mixture of trigonometric functions, rewriting everything in terms of sine and cosine is often the safest strategy. This is especially true for rational expressions involving sec, cosec or cot.
当一个表达式中混有各种三角函数时，将所有函数都写成正弦和余弦的形式通常是稳妥的策略。对于包含 sec、cosec 或 cot 的有理表达式来说尤其如此。
cos x. Dividing by cot x (i.e. multiplying by sin x / cos x) yields sin²x / cos x × sin x / cos x = sin³x / cos²x, which can be left as sin³x sec²x or further simplified.
例如，化简 (sec x − cos x) / cot x。将 sec x 写成 1/cos x，cot x 写成 cos x / sin x。分子变为 (1/cos x) − cos x = (1 − cos²x) / cos x = sin²x / cos x。再除以 cot x（即乘以 sin x / cos x），得到 sin²x / cos x × sin x / cos x = sin³x / cos²x，该结果可保留为 sin³x sec²x 或进一步化简。
Another common trick is to multiply numerator and denominator by a conjugate or by a factor that creates a known identity. For instance, to simplify 1/(1+sec x), multiply top and bottom by cos x to get cos x/(cos x + 1). This often leads to cancellations.
另一个常用技巧是将分子和分母同时乘以共轭因式或能产生已知恒等式的因子。例如，化简 1/(1+sec x)，将上下同乘 cos x 得到 cos x/(cos x + 1)。这常常能导致约分。
Always check that your simplified expression is valid for the same set of x‑values as the original. Domain restrictions introduced by denominators must remain.
务必检查化简后的表达式与原表达式对于同一组 x 值有效。分母引入的定义域限制必须保留。
7. Solving Equations with Reciprocal Functions | 用倒数函数解方程
Equations involving sec, cosec and cot appear frequently in A‑Level exams. The method usually involves converting to sine, cosine and tangent, or using the squared identities to form quadratic equations.
包含 sec、cosec 和 cot 的方程经常出现在 A-Level 考试中。解法通常涉及转化为正弦、余弦和正切，或利用平方恒等式构造二次方程。
= 0. Solve this quadratic in tan θ, then find all solutions within the required interval.
考虑方程 2 sec²θ = 5 tan θ + 3。将 sec²θ 替换为 1 + tan²θ，得到 2(1 + tan²θ) = 5 tan θ + 3，化简为 2 tan²θ − 5 tan θ − 1 = 0。求解这个关于 tan θ 的二次方程，然后在指定区间内找出所有解。
sin x. Bring all terms to one side and factorise: cos x (1 − 2 sin x) = 0. This gives cos x = 0 or sin x = 1/2. Solve each case and reject any solutions where sin x = 0, as that would make the original equation undefined.
对于像 cot x = 2 cos x 这样的方程，将 cot x 改写为 cos x / sin x，得到 cos x / sin x = 2 cos x。两边乘以 sin x（注意 sin x ≠ 0），得到 cos x = 2 cos x sin x。将所有项移到一边并因式分解：cos x (1 − 2 sin x) = 0。得到 cos x = 0 或 sin x = 1/2。分别求解，并舍去任何使 sin x = 0 的解，因为那会使原方程无定义。
Always check that your solutions do not fall on the asymptotes of the reciprocal functions present in the original equation. Examiners expect this final check.
务必检查所求得的解不能落在原方程中倒数函数的渐近线上。阅卷人期望看到这最后一步检验。
8. Differentiation of sec x, cosec x and cot x | sec x、cosec x 和 cot x 的微分
The derivatives of these three functions are standard results that must be memorised. They can be derived using the quotient rule or chain rule from the definitions, but for the exam it is far more efficient to know them by heart.
这三个函数的导数是必须记住的标准结果。它们可以从定义出发利用商法则或链式法则推导出来，但在考试中直接记住它们要高效得多。
The key derivatives are:
d/dx (sec x) = sec x tan x
关键导数公式为：
d/dx (sec x) = sec x tan x
d/dx (cosec x) = −cosec x cot x
d/dx (cosec x) = −cosec x cot x
d/dx (cot x) = −cosec² x
d/dx (cot x) = −cosec² x
It is worth noting the sign patterns: secant's derivative is positive and contains tan, while cosecant and cotangent have negative derivatives involving their complementary functions. The derivative of cot x is similar to that of tan x (which is sec² x) but with a minus sign and cosec² instead of sec².
值得注意符号规律：正割的导数为正且包含 tan，而余割和余切的导数为负且涉及它们的互补函数。cot x 的导数类似于 tan x 的导数（sec² x），但带有一个负号且用 cosec² 替换 sec²。
These derivatives can be combined with the chain, product and quotient rules to differentiate more complicated expressions like eˣ sec 2x or ln(cot x).
这些导数可以与链式法则、乘积法则和商法则结合，用于对更复杂的表达式求导，如 eˣ sec 2x 或 ln(cot x)。
9. Integration of Reciprocal Trigonometric Functions | 倒数三角函数的积分
Integration of secant, cosecant and cotangent involves some standard results and occasionally clever manipulation. The two integrals you must know are:
正割、余割和余切的积分涉及一些标准结果，有时还需要巧妙的变形。必须掌握的两个积分是：
∫ sec x dx = ln |sec x + tan x| + C
∫ sec x dx = ln |sec x + tan x| + C
∫ cosec x dx = −ln |cosec x + cot x| + C = ln |cosec x − cot x| + C
∫ cosec x dx = −ln |cosec x + cot x| + C = ln |cosec x − cot x| + C
can simply write it as cos x / sin x. The integral ∫ cot x dx = ∫ (cos x / sin x) dx = ln |sin x| + C.
对于 cot x，只需将其写为 cos x / sin x。积分 ∫ cot x dx = ∫ (cos x / sin x) dx = ln |sin x| + C。
x sec²x dx and then replace one sec²x with 1 + tan²x, leading to a substitution u = tan x.
要积分幂次或乘积，通常需要先利用恒等式。例如，∫ sec⁴x dx 可写为 ∫ sec²x sec²x dx，然后将其中一个 sec²x 替换为 1 + tan²x，从而使用代换 u = tan x。
tan x, so the integral is simply sec x + C. Similarly, ∫ cosec x cot x dx = −cosec x + C.
遇到 ∫ sec x tan x dx 时，要能认出 sec x 的导数正是 sec x tan x，因此积分就是 sec x + C。类似地，∫ cosec x cot x dx = −cosec x + C。
Edexcel often tests these integrals in the context of finding areas, volumes of revolution, or solving first‑order differential equations.
Edexcel 常会在求面积、旋转体体积或求解一阶微分方程的题目中考查这些积分。
10. Applying to Real Problems and Exam Tips | 实际应用与考试技巧
Beyond pure manipulation, reciprocal trigonometric functions appear in parametric equations, polar coordinates, and in modelling oscillatory behaviour where amplitude or phase leads to natural use of sec or cosec. Being fluent with these functions will give you extra speed and confidence in a wide range of problems.
除了纯粹的代数操作，倒数三角函数还出现在参数方程、极坐标以及建模振荡行为中，那里振幅或相位的变化会自然地用到 sec 或 cosec。熟练运用这些函数能让你在面对各类问题时速度更快、信心更足。
Here are some exam-focused tips:
Memorise the definitions, graphs, derivative and integral results precisely. Mixing up the sign in the derivative of cot x is a common mistake.
精确记忆定义、图像、导数和积分结果。混淆 cot x 导数中的符号是常见错误。
When solving equations, always state the domain restrictions explicitly. Marks are often allocated for discarding invalid solutions.
求解方程时，务必明确写出定义域限制。舍去无效解通常会有评分点。
Convert to sine and cosine as a default first step if you are stuck. This often reveals hidden simplifications.
如果卡住了，默认第一步就转化为正弦和余弦。这往往能揭示隐藏的化简途径。
+ tan²x = sec²x and 1 + cot²x = cosec²x as soon as you see sec² or cosec² in an equation.
一旦在方程中看到 sec² 或 cosec²，立即使用平方恒等式 1 + tan²x = sec²x 和 1 + cot²x = cosec²x。
For integration, practice spotting the derivatives of the reciprocal functions. This will allow you to reverse chain rule quickly.
对于积分，要练习识别倒数函数的导数。这样你就能快速运用逆向链式法则。
By internalising these functions as naturally as you know sine and cosine, you will handle the most challenging Edexcel questions with clarity and precision.
当你像熟知正弦和余弦一样将这些函数内化于心时，你就能清晰而准确地应对最具挑战性的 Edexcel 题目。
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