Graphs of sec x, cosec x and cot x | sec x、cosec x 与 cot x 的图像

📚 Graphs of sec x, cosec x and cot x | sec x、cosec x 与 cot x 的图像

The reciprocal trigonometric functions — secant (sec), cosecant (cosec) and cotangent (cot) — are essential tools in A-Level Mathematics. Although each is simply the reciprocal of a basic trigonometric function, their graphs display vertical asymptotes, U-shaped branches and periodic patterns that deserve careful attention. A clear visual understanding of these curves helps you solve equations, verify identities and work confidently with calculus.

倒数三角函数——正割（sec）、余割（cosec）与余切（cot）——是 A-Level 数学中的重要工具。尽管它们分别是基本三角函数的倒数，但它们的图像却展现出垂直渐近线、U 形分支和周期性模式，需要仔细理解。清晰掌握这些曲线的图像形态，有助于你解方程、验证恒等式，并自信地处理微积分问题。

1. Definitions and Fundamental Identities | 定义与基本恒等式

For any angle x measured in radians, the three reciprocal functions are defined in terms of sin x, cos x and tan x. The domain of each function must exclude any angle where the denominator becomes zero.

对于以弧度为单位的角度 x，三个倒数函数分别由 sin x、cos x 和 tan x 定义。每个函数的定义域都必须排除使分母为零的角度。

sec x = 1 ÷ cos x (cos x ≠ 0)

cosec x = 1 ÷ sin x (sin x ≠ 0)

cot x = 1 ÷ tan x = cos x ÷ sin x (sin x ≠ 0)

Notice that cot x is not best thought of as the reciprocal of tan x alone; the equivalent form cos x ÷ sin x is far more useful for sketching and solving equations.

注意，cot x 最好不要仅仅看成 tan x 的倒数；等价形式 cos x ÷ sin x 在作图和解方程时更为实用。

Two Pythagorean identities arise directly from these definitions and appear constantly in exam questions:

以下两个毕达哥拉斯恒等式直接来自这些定义，在考试题目中频繁出现：

sec² x = 1 + tan² x

cosec² x = 1 + cot² x

These identities are indispensable for simplifying expressions and are frequently

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