Using sec x, cosec x and cot x | 使用 sec x、cosec x 与 cot x

📚 Using sec x, cosec x and cot x | 使用 sec x、cosec x 与 cot x

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. Definitions, Domains and Ranges | 定义、定义域与值域

The three reciprocal functions are defined by sec x = 1/cos x, cosec x = 1/sin x, and cot x = cos x / sin x = 1/tan x. Because these definitions involve division by sin x or cos x, the functions are undefined whenever the corresponding denominator is zero.

三个倒数三角函数定义为 sec x = 1/cos x，cosec x = 1/sin x，cot x = cos x / sin x = 1/tan x。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。

For sec x, the excluded values are x = π/2 + nπ, or 90° + 180°n, where cos x = 0. For cosec x and cot x, the excluded values are x = nπ, or 180°n, where sin x = 0. In set notation, sec x is undefined at x = π/2 + nπ and cosec x, cot x are undefined at x = nπ.

sec x 的定义域排除 x = π/2 + nπ，即 90° + 180°n，此时 cos x = 0；cosec x 和 cot x 的定义域排除 x = nπ，即 180°n，此时 sin x = 0。用集合符号表示，sec x 在 x = π/2 + nπ 处无定义，cosec x 和 cot x 在 x = nπ 处无定义。

The ranges are also worth noting: |sec x| ≥ 1 and |cosec x| ≥ 1, while cot x can take all real values because it is the ratio cos x / sin x. This explains why the graphs of sec x and cosec x never enter

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