📚 Graphs of sec x, cosec x and cot x | sec x, cosec x 和 cot x 的图像
A-Level Mathematics, a solid understanding of the three reciprocal trigonometric functions—secant (sec), cosecant (cosec), and cotangent (cot)— iliar, their reciprocal counterparts introduce vertical asymptotes, undefined points, and distinct periodicity that must be mastered. This article explores the graphs of y = sec x, y = cosec x, and y = cot x, highlighting their domains, ranges, key features, and how to sketch and transform them for the Edexcel specification.
在A-Level数学中，深入理解三个倒数三角函数——正割(sec)、余割(cosec)和余切(cot)——对于处理三角方程、恒等式和图像变换至关重要。正弦、余弦和正切的图像我们已经熟悉，但它们的倒数函数却引入了垂直渐近线、无定义点以及独特的周期性，这些都必须掌握。本文探讨 y = sec x、y = cosec x 和 y = cot x 的图像，重点介绍它们的定义域、值域、关键特征，以及如何根据Edexcel考试要求进行绘制和变换。
1. Defining the Reciprocal Functions | 定义倒数函数
x = cos x / sin x. These definitions immediately indicate where the functions are undefined— x = sin x / cos x, and its reciprocal cot x = cos x / sin x, making sin x = 0 a problem).
倒数三角函数的定义如下：sec x = 1 / cos x，cosec x = 1 / sin x，cot x = 1 / tan x，也可以表示为 cot x = cos x / sin x。这些定义直接显示了函数在分母为零时没有定义。对于 sec x，当 cos x = 0 时无定义；对于 cosec x，当 sin x = 0 时无定义；对于 cot x，当 sin x = 0 时无定义（因为 tan x = sin x / cos x，其倒数 cot x = cos x / sin x，所以 sin x = 0 使分母为零）。
Understanding where these functions are undefined is the first step in sketching their graphs. The points of undefinedness create vertical asymptotes that divide the graph into distinct branches, and the behaviour close to these asymptotes defines the overall shape of each reciprocal function.
理解这些函数在何处无定义是绘制图像的第一步。无定义点会产生垂直渐近线，将图像分割成不同的分支，而在这些渐近线附近的行为决定了每个倒数函数的整体形状。
2. Domain and Range | 定义域和值域
A precise description of domain and range is needed for sketching and transformation questions. The table below summarises these for all three functions.
精确描述定义域和值域对于绘图和变换题是必需的。下表总结了这三个函数的相关信息。
| Function | Domain | Range |
|---|---|---|
| y = sec x | x ∈ ℝ, x ≠ π/2 + nπ (or x ≠ 90° + 180°n) | y ≤ -1 or y ≥ 1 |
| y = cosec x | x ∈ ℝ, x ≠ nπ (or x ≠ 0°, 180°, 360°, …) | y ≤ -1 or y ≥ 1 |
| y = cot x | x ∈ ℝ, x ≠ nπ | y ∈ ℝ (all real numbers) |
g reciprocals of numbers with absolute value less than 1 yields values with absolute value greater than 1. At the endpoints, where cos x or sin x equals ±1, the reciprocal is also ±1. For cot x, since tan x can take any real value (except where undefined), its reciprocal also sweeps through all non-zero real numbers, and cot x also equals zero when cos x = 0, hence the range is all real numbers.
sec x 和 cosec x 的值域永远不包含 -1 到 1 之间的数。这是因为 cos x 和 sin x 的值在 [-1, 1] 区间内，绝对值小于1的数的倒数绝对值会大于1。在端点处，cos x 或 sin x 等于 ±1，倒数也是 ±1。对于 cot x，由于 tan x 可以取任意实数（无定义点除外），其倒数可以取所有非零实数，而 cot x 在 cos x = 0 时也等于零，因此值域为所有实数。
3. Periodicity of sec, cosec, and cot | 周期性质
larly, y = sin x has period 2π, so y = cosec x has period 2π. However, y = tan x has period π, therefore y = cot x also has period π. This difference is important when solving equations or applying transformations.
倒数函数的周期继承自其母函数。y = cos x 的周期为 2π，所以 y = sec x 也以 2π 为周期重复。类似地，y = sin x 的周期为 2π，故 y = cosec x 的周期也是 2π。然而，y = tan x 的周期为 π，因此 y = cot x 的周期也是 π。在解方程或进行图像变换时，这种周期性差异是非常重要的。
Period of sec x: 2π Period of cosec x: 2π Period of cot x: π
4. Asymptotes and Discontinuities | 渐近线与不连续点
90° + 180°n). For y = cosec x and y = cot x, the asymptotes are at x = nπ (0°, 180°, 360°, … rstanding the behaviour on either side of an asymptote is critical for accurate sketching.
垂直渐近线正好出现在分母 cos x 或 sin x 为零的位置。对于 y = sec x，渐近线在 x = π/2 + nπ（90° + 180°n）。对于 y = cosec x 和 y = cot x，渐近线在 x = nπ（0°, 180°, 360°, …）。这些是不可去不连续点，函数值在此趋向 +∞ 或 -∞。了解渐近线两侧的函数行为对于准确绘制图像至关重要。
ine or sine function. For cot x, as x approaches nπ, the function drops from +∞ (approaching from the right) to -∞ (approaching from the left) in each period interval.
当 x 从左或右趋近渐近线时，sec x 和 cosec x 会冲向 +∞ 或 -∞，取决于相应的余弦或正弦函数的符号。对于 cot x，当 x 趋近 nπ 时，在每个周期区间内，函数从 +∞（从右侧趋近）下降到 -∞（从左侧趋近）。
5. Graph of y = sec x | y = sec x 的图像
The graph of y = sec x consists of a series of U‑ from 1 to 0 then back to 1, so sec x has a minimum point at (0, 1), then rises to +∞ at the asymptotes x = -π/2 and x = π/2. In the interval (π/2, 3π/2), cos x is negative, ranging from just below 0 to -1 and back, so sec x starts from -∞, reaches a maximum point at (π, -1), and returns to -∞ at the asymptotes. This pattern repeats every 2π. The function is even, so its graph is symmetric about the y‑axis.
y = sec x 的图像由一系列朝向 y = 1 上方和 y = -1 下方的 U 形分支组成，被垂直渐近线隔开。在区间 (-π/2, π/2) 上，cos x 从 1 递减到 0 再回到 1，因此 sec x 在 (0, 1) 处有一个极小点，然后在渐近线 x = -π/2 和 x = π/2 处升向 +∞。在区间 (π/2, 3π/2) 上，cos x 为负值，从略小于 0 到 -1 再返回，所以 sec x 从 -∞ 开始，在 (π, -1) 处达到极大点，再回到 -∞。这个模式每 2π 重复一次。该函数是偶函数，其图像关于 y 轴对称。
(π, -1), (3π/2 undefined), (2π, 1). The branches in the positive region resemble an upward‑opening U with minimum at y=1, while branches in the negative region are upside‑down U with maximum at y=-1.
一个周期（0 到 2π）内的关键点：(0, 1)，(π/2 无定义)，(π, -1)，(3π/2 无定义)，(2π, 1)。正区域中的分支像一个在 y=1 处最小值的向上开口 U 形，而负区域的分支则是在 y=-1 处最大值的倒 U 形。
6. Graph of y = cosec x | y = cosec x 的图像
s similar to that of sec x, but shifted horizontally by π/2. It has vertical asymptotes at
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