📚 Graphs of Sec, Cosec and Cot Functions | sec、cosec、cot 函数图像
Understanding the graphs of the reciprocal trigonometric functions is essential for solving equations, sketching transformations, and analysing periodic behaviour in A-level Mathematics.
理解余割、正割和余切这三个倒数三角函数的图像，是A-level数学中解方程、画变换图像和分析周期行为的重要基础。
1. Definitions and Basic Relationships | 定义与基本关系
The secant, cosecant and cotangent functions are defined as reciprocals of the three primary trigonometric functions.
正割函数、余割函数和余切函数分别是三个基本三角函数的倒数。
- Secant: sec x = 1 / cos x, defined for all x where cos x ≠ 0.
- 正割函数: sec x = 1 / cos x，在 cos x ≠ 0 的所有 x 处有定义。
- Cosecant: cosec x = 1 / sin x, defined for all x where sin x ≠ 0.
- 余割函数: cosec x = 1 / sin x，在 sin x ≠ 0 的所有 x 处有定义。
- Cotangent: cot x = 1 / tan x = cos x / sin x, defined for all x where sin x ≠ 0.
- 余切函数: cot x = 1 / tan x = cos x / sin x，在 sin x ≠ 0 的所有 x 处有定义。
It is also useful to remember the Pythagorean identities that connect these functions: 1 + tan² x = sec² x and 1 + cot² x = cosec² x.
还需要记住联系这些函数的勾股恒等式：1 + tan² x = sec² x 和 1 + cot² x = cosec² x。
2. The Graph of y = sec x | y = sec x 的图像
The graph of y = sec x is best understood by starting from y = cos x and taking reciprocals of the y-coordinates.
画 y = sec x 的图像最好从 y = cos x 出发，对每个 y 坐标取倒数。
Key features of y = sec x:
y = sec x 的关键特征：
- Period: 2π, since sec(x + 2π) = sec x.
- 周期：2π，因为 sec(x + 2π) = sec x。
- Vertical asymptotes: at x = π/2 + nπ, where cos x = 0.
- 垂直渐近线：在 x = π/2 + nπ 处，此时 cos x = 0。
- Range: (-∞, -1] ∪ [1, ∞). The curve never lies between -1 and 1.
- 值域：(-∞, -1] ∪ [1, ∞)。曲线永远不会落在 -1 和 1 之间。
- Maximum/minimum: At x = 2nπ, sec x = 1 (minimum turning point in each period); at x = (2n+1)π, sec x = -1 (maximum turning point).
- 极大/极小值：当 x = 2nπ 时，sec x = 1（每个周期内的极小值点）；当 x = (2n+1)π 时，sec x = -1（极大值点）。
For example, when x = 0, cos 0 = 1, so sec 0 = 1. When x = π/3, cos(π/3) = 1/2, so sec(π/3) = 2. As x approaches π/2 from the left, cos x approaches 0 from positive values, so sec x tends to +∞.
例如，当 x = 0 时，cos 0 = 1，所以 sec 0 = 1。当 x = π/3 时，cos(π/3) = 1/2，所以 sec(π/3) = 2。当 x 从左侧接近 π/2 时，cos x 从正值接近 0，所以 sec x 趋向 +∞。
3. The Graph of y = cosec x | y = cosec x 的图像
The graph of y = cosec x is obtained from y = sin x in the same reciprocal manner.
y = cosec x 的图像由 y = sin x 以相同的倒数方式得到。
Key features of y = cosec x:
y = cosec x 的关键特征：
- Period: 2π.
- 周期：2π。
- Vertical asymptotes: at x = nπ, where sin x = 0.
- 垂直渐近线：在 x = nπ 处，此时 sin x = 0。
- Range: (-∞, -1] ∪ [1, ∞).
- 值域：(-∞, -1] ∪ [1, ∞)。
- Turning points: At x = π/2 + 2nπ, cosec x = 1; at x = 3π/2 + 2nπ, cosec x = -1.
- 极值点：当 x = π/2 + 2nπ 时，cosec x = 1；当 x = 3π/2 + 2nπ 时，cosec x = -1。
Notice that because sin x is zero at x = 0, ±π, ±2π, etc., the cosecant graph has asymptotes at these points and consists of separate U-shaped branches alternating above and below the x-axis.
注意因为 sin x 在 x = 0、±π、±2π 等处为零，余割图像在这些点有渐近线，并由在 x 轴上下交替的分离的 U 形分支组成。
4. The Graph of y = cot x | y = cot x 的图像
The cotangent function is different in nature: it is the reciprocal of tan x, but it can also be expressed as cos x / sin x.
余切函数性质不同：它是 tan x 的倒数，但也可以表示为 cos x / sin x。
Key features of y = cot x:
y = cot x 的关键特征：
- Period: π, because cot(x + π) = cot x.
- 周期：π，因为 cot(x + π) = cot x。
- Vertical asymptotes: at x = nπ, where sin x = 0.
- 垂直渐近线：在 x = nπ 处，此时 sin x = 0。
- Range: all real numbers, (-∞, ∞).
- 值域：一切实数，(-∞, ∞)。
- Zeros: at x = π/2 + nπ, where cos x = 0.
- 零点：在 x = π/2 + nπ 处，此时 cos x = 0。
- Behaviour: The graph decreases from +∞ to -∞ over each interval (0, π), passing through (π/4, 1) and (3π/4, -1).
- 行为特征：在每个区间 (0, π) 内，图像从 +∞ 递减到 -∞，经过点 (π/4, 1) 和 (3π/4, -1)。
Unlike sec and cosec, cot x has no turning points and its graph is continuous between asymptotes.
与 sec 和 cosec 不同，cot x 没有极值点，并且其图像在渐近线之间是连续的。
5. Comparing Periods and Ranges | 周期与值域对比
The table below summarises the essential features of the three functions.
下表总结了这三个函数的基本特征。
| Function | Period | Asymptotes | Range |
| sec x | 2π | x = π/2 + nπ | (-∞, -1] ∪ [1, ∞) |
| cosec x | 2π | x = nπ | (-∞, -1] ∪ [1, ∞) |
| cot x | π | x = nπ | (-∞, ∞) |
Note that sec x and cosec x share the same period and range, but their asymptotes and turning points are shifted by π/2. Meanwhile cot x has a shorter period and a full real-number range.
注意 sec x 和 cosec x 有相同的周期和值域，但它们的渐近线和极值点相差 π/2。而 cot x 的周期更短且值域为全体实数。
6. Sketching Transformed Graphs | 画变换后的图像
Transformations of sec, cosec and cot functions follow the same rules as other functions: f(x + a), f(x) + a, a f(x), and f(ax) cause translations, vertical stretches, and horizontal stretches/compressions.
sec、cosec 和 cot 函数的变换遵循与一般函数相同的规则：f(x + a)、f(x) + a、a f(x) 和 f(ax) 分别产生平移、纵向伸缩和横向伸缩。
For example, to sketch y = 2 sec x – 1:
例如，要画 y = 2 sec x – 1：
- Start with y = sec x.
- 先画 y = sec x。
- Stretch vertically by factor 2: the points at y = 1 and y = -1 move to y = 2 and y = -2.
- 纵向拉伸 2 倍：y = 1 和 y = -1 的点移到 y = 2 和 y = -2。
- Translate down by 1: the turning points now occur at y = 1 and y = -3.
- 向下平移 1 个单位：极值点现在位于 y = 1 和 y = -3。
- Asymptotes remain at x = π/2 + nπ, because the transformation does not affect x-coordinates.
- 渐近线仍在 x = π/2 + nπ，因为该变换不影响 x 坐标。
7. Solving Equations Using Graphs | 用图像解方程
Graphs of reciprocal trig functions are often used to solve equations such as sec x = 2 or cot x = √3 within a given interval.
倒数三角函数的图像常用于解形如 sec x = 2 或 cot x = √3 在给定区间内的方程。
Example: Solve sec x = 2 for 0 ≤ x ≤ 2π.
例：解方程 sec x = 2，其中 0 ≤ x ≤ 2π。
Since sec x = 2 means cos x = 1/2, we solve the simpler equation. In the interval [0, 2π], cos x = 1/2 has solutions x = π/3 and x = 5π/3.
因为 sec x = 2 意味着 cos x = 1/2，我们解这个更简单的方程。在区间 [0, 2π] 内，cos x = 1/2 的解为 x = π/3 和 x = 5π/3。
In general, crossing a horizontal line y = k with the graph of a reciprocal function corresponds to the same x-values as solving the corresponding primary equation, with careful attention to domain restrictions.
一般来说，用水平线 y = k 与倒数函数图像相交所得的 x 值，与解相应基本方程所得一致，但需注意定义域限制。
8. Asymptotes and the Limiting Behaviour | 渐近线与极限行为
Understanding asymptotes is crucial for correct sketching. For sec x, asymptotes occur where cos x = 0; for cosec x and cot x, they occur where sin x = 0.
理解渐近线对正确画图至关重要。对于 sec x，渐近线出现在 cos x = 0 处；对于 cosec x 和 cot x，它们出现在 sin x = 0 处。
Around an asymptote, the function value tends to either +∞ or -∞ depending on the sign of the denominator:
在渐近线附近，函数值趋向 +∞ 或 -∞，具体取决于分母的符号：
- For sec x, as x → π/2⁻, cos x → 0⁺, so sec x → +∞; as x → π/2⁺, sec x → -∞.
- 对于 sec x，当 x → π/2⁻ 时，cos x → 0⁺，所以 sec x → +∞；当 x → π/2⁺ 时，sec x → -∞。
- For cosec x, as x → 0⁺, sin x → 0⁺, so cosec x → +∞; as x → 0⁻, cosec x → -∞.
- 对于 cosec x，当 x → 0⁺ 时，sin x → 0⁺，所以 cosec x → +∞；当 x → 0⁻ 时，cosec x → -∞。
- For cot x, as x → 0⁺, cot x → +∞; as x → 0⁻, cot x → -∞.
- 对于 cot x，当 x → 0⁺
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