A-Level Maths: Secant, Cosecant & Cotangent Functions | A-Level 数学：正割、余割与余切

📚 A-Level Maths: Secant, Cosecant & Cotangent Functions | A-Level 数学：正割、余割与余切

The secant, cosecant and cotangent functions, often abbreviated as sec, cosec and cot, are the reciprocal trigonometric functions. They play a vital role in A-Level Mathematics, appearing in calculus, identities and equation solving. Mastering these functions is essential for exam success.

正割、余割与余切函数，通常简写为 sec、cosec 和 cot，是三角函数的倒数形式。它们在 A-Level 数学中扮演着重要角色，出现在微积分、恒等式和方程求解中。熟练掌握这些函数是考试成功的关键。

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1. Definitions | 定义

For any angle x, the three reciprocal functions are defined as follows, provided the denominators are not zero:

对于任意角 x，三个倒数函数定义如下（前提是分母不为零）：

• sec x = 1 / cos x, defined when cos x ≠ 0

  sec x = 1 / cos x，在 cos x ≠ 0 时有定义

• cosec x = 1 / sin x, defined when sin x ≠ 0

  cosec x = 1 / sin x，在 sin x ≠ 0 时有定义

• cot x = 1 / tan x = cos x / sin x, defined when sin x ≠ 0

  cot x = 1 / tan x = cos x / sin x，在 sin x ≠ 0 时有定义

Note that cot x is also the reciprocal of tan x, but it is safer to remember it as cos x / sin x.

注意 cot x 也是 tan x 的倒数，但将其记作 cos x / sin x 更为稳妥。

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2. Key Identities | 基本恒等式

Three Pythagorean identities form the foundation for many exam questions involving these functions:

三个毕达哥拉斯恒等式构成了涉及这些函数的许多考题的基础：

sec²x = 1 + tan²x

cosec²x = 1 + cot²x

These follow directly from sin²x + cos²x = 1 by dividing by cos²x or sin²x respectively.

它们分别通过在 sin²x + cos²x = 1 两边除以 cos²x 或 sin²x 直接得到。

A common exam trap is confusing these identities. Remember that sec goes with tan, and cosec goes with cot.

一个常见的考试陷阱是混淆这些恒等式。记住：sec 与 tan 搭配，cosec 与 cot 搭配。

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3. Graphs and Properties | 图像与性质

The graphs of sec x and cosec x have a distinctive U-shaped and inverted-U-shaped pattern, while cot x resembles a decreasing tan x curve.

sec x 和 cosec x 的图像具有独特的 U 形和倒 U 形交替模式，而 cot x 则类似于递减的 tan x 曲线。

Key properties:

关键性质：

• sec x has vertical asymptotes at x = π/2 + nπ, where cos x = 0.

  sec x 在 x = π/2 + nπ（即 cos x = 0）处有垂直渐近线。

• cosec x and cot x have vertical asymptotes at x = nπ, where sin x = 0.

  cosec x 和 cot x 在 x = nπ（即 sin x = 0）处有垂直渐近线。

• The range of sec x and cosec x is (-∞, -1] ∪ [1, ∞).

  sec x 和 cosec x 的值域为 (-∞, -1] ∪ [1, ∞)。

• The range of cot x is (-∞, ∞).

  cot x 的值域为 (-∞, ∞)。

Understanding these graphs helps you avoid sign errors in calculus.

理解这些图像有助于避免在微积分中出现符号错误。

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4. Periodicity and Symmetry | 周期与对称性

Like sin x and cos x, the reciprocal functions have well-defined periodic behaviour:

与 sin x 和 cos x 一样，倒数函数具有明确的周期行为：

• sec x and cosec x have period 2π.

  sec x 和 cosec x 的周期为 2π。

• cot x has period π.

  cot x 的周期为 π。

In addition, the functions have specific parity properties:

此外，这些函数具有特定的奇偶性：

• sec x is even: sec(-x) = sec x.

  sec x 是偶函数：sec(-x) = sec x。

• cosec x and cot x are odd: cosec(-x) = -cosec x, cot(-x) = -cot x.

  cosec x 和 cot x 是奇函数：cosec(-x) = -cosec x，cot(-x) = -cot x。

These properties are useful for simplifying expressions and checking answers.

这些性质在化简表达式和检查答案时非常有用。

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5. Derivatives | 导数

The derivatives of the reciprocal functions appear frequently in calculus questions:

倒数函数的导数经常出现在微积分题目中：

d/dx (sec x) = sec x tan x

d/dx (cosec x) = -cosec x cot x

d/dx (cot x) = -cosec²x

Notice that both cosec x and cot x have a negative sign in their derivatives. A useful memory aid is that only sec x has a positive derivative sign.

注意 cosec x 和 cot x 的导数都带有负号。一个有用的记忆技巧是：只有 sec x 的导数符号为正。

When differentiating composite functions, apply the chain rule. For example, d/dx (sec 3x) = 3 sec 3x tan 3x.

对复合函数求导时，需要应用链式法则。例如，d/dx (sec 3x) = 3 sec 3x tan 3x。

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6. Integrals | 积分

The corresponding integral results are directly related to the derivatives above:

相应的积分结果与上述导数直接相关：

∫ sec²x dx = tan x + C

∫ cosec²x dx = -cot x + C

∫ sec x tan x dx = sec x + C

∫ cosec x cot x dx = -cosec x + C

These four standard results are listed in the A-Level formula booklet, but you must know when to apply them. For example, ∫ sec²(2x) dx = ½ tan(2x) + C.

这四个标准结果列在 A-Level 公式手册中，但你必须知道何时应用它们。例如，∫ sec²(2x) dx = ½ tan(2x)

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