A-Level Maths: Graphs of Secant, Cosecant and Cotangent | A-Level 数学：正割、余割与余切的函数图像

📚 A-Level Maths: Graphs of Secant, Cosecant and Cotangent | A-Level 数学：正割、余割与余切的函数图像

In A-Level Mathematics, the three reciprocal trigonometric functions — secant (sec x), cosecant (cosec x) and cotangent (cot x) — play a vital role in solving trigonometric equations, proving identities and analysing periodic behaviour. Understanding their graphs is essential for mastering topics such as transformations, asymptotes and solving inequalities.

在 A-Level 数学中，三个倒数三角函数——正割（sec x）、余割（cosec x）和余切（cot x）——在解三角方程、证明恒等式以及分析周期行为方面发挥着至关重要的作用。理解它们的函数图像对于掌握变换、渐近线和不等式求解等知识点至关重要。

1. Definitions and Reciprocal Identities | 定义与倒数关系

Each of the three functions is defined as the reciprocal of a primary trigonometric function, provided the denominator is not zero.

这三个函数中的每一个都被定义为某个基本三角函数的倒数，前提是分母不为零。

sec x = 1 / cos x, cosec x = 1 / sin x, cot x = 1 / tan x = cos x / sin x

It is crucial to remember that cot x is also the reciprocal of tan x, but it can equally be expressed as the ratio of cos x to sin x. This duality is often exploited in identity proofs.

务必记住，cot x 不仅是 tan x 的倒数，也可以表示为 cos x 与 sin x 之比。这种双重身份在恒等式证明中经常被利用。

2. Domain Restrictions from Denominators | 由分母决定的自变量范围

Because division by zero is undefined, the domain of each reciprocal function excludes the zeros of its denominator.

由于除以零是没有定义的，每个倒数函数的定义域都排除其分母为零的点。

sec x = 1 / cos x: defined when cos x ≠ 0, i.e. x ≠ π/2 + nπ (where n is any integer).

sec x = 1 / cos x：当 cos x ≠ 0 时有定义，即 x ≠ π/2 + nπ（n 为任意整数）。
cosec x = 1 / sin x: defined when sin x ≠ 0, i.e. x ≠ nπ.

cosec x = 1 / sin x：当 sin x ≠ 0 时有定义，即 x ≠ nπ。
cot x = cos x / sin x: defined when sin x ≠ 0, i.e. x ≠ nπ.

cot x = cos x / sin x：当 sin x ≠ 0 时有定义，即 x ≠ nπ。

These excluded points appear as vertical asymptotes on the respective graphs. Memorising these restrictions is the first step towards sketching any reciprocal trigonometric graph correctly.

这些被排除的点在对应图像上表现为垂直渐近线。记住这些限制条件，是正确绘制任何倒数三角函数图像的第一步。

3. Graph of y = sec x | y = sec x 的图像

The graph of y = sec x is closely related to that of y = cos x. Where cos x = 0, the secant graph has a vertical asymptote; where cos x = ±1, sec x reaches its extreme values of ±1.

y = sec x 的图像与 y = cos x 的图像密切相关。当 cos x = 0 时，正割图像有垂直渐近线；当 cos x = ±1 时，sec x 取得极值 ±1。

Key features of y = sec x:

y = sec x 的主要特征：

Vertical asymptotes at x = π/2 + nπ.

垂直渐近线位于 x = π/2 + nπ。
U-shaped branches opening upwards above y = 1, and downwards below y = -1.

U 形分支在 y = 1 上方开口向上，在 y = -1 下方开口向下。
The curve never lies between -1 and 1 on the y-axis.

曲线在 y 轴方向上永远不会落在 -1 与 1 之间。
The graph passes through (0, 1), (2π, 1), and (π, -1).

图像经过点 (0, 1)、(2π, 1) 和 (π, -1)。

4. Graph of y = cosec x | y = cosec x 的图像

Similarly, y = cosec x is the reciprocal of y = sin x. The zeros of sin x become vertical asymptotes for cosec x.

类似地，y = cosec x 是 y = sin x 的倒数。sin x 的零点变成 cosec x 的垂直渐近线。

Key features of y = cosec x:

y = cosec x 的主要特征：

Vertical asymptotes at x = nπ.

垂直渐近线位于 x = nπ。
Turning points at x = π/2 + nπ, where sin x = ±1, giving cosec x = ±1.

极值点位于 x = π/2 + nπ，此时 sin x = ±1，cosec x = ±1。
Period of 2π, same as sin x.

周期为 2π，与 sin x 相同。
The branches are U-shaped, alternating above y = 1 and below y = -1.

分支为 U 形，交替出现在 y = 1 上方和 y = -1 下方。

Note that the shapes of secant and cosecant graphs are very similar; the key difference lies in the horizontal shift of their asymptotes by π/2.

注意正割和余割图像的形状非常相似；关键区别在于它们的渐近线在水平方向相差 π/2。

5. Graph of y = cot x | y = cot x 的图像

The cotangent function is fundamentally different from sec and cosec because it is a reciprocal of tan, which itself has vertical asymptotes, and its period is π, not 2π.

余切函数与正割和余割有本质上的区别，因为它是 tan 的倒数，而 tan 本身就有垂直渐近线，且它的周期是 π，不是 2π。

Key features of y = cot x:

y = cot x 的主要特征：

Vertical asymptotes at x = nπ.

垂直渐近线位于 x = nπ。
The graph crosses the x-axis at x = π/2 + nπ, where tan x = 0 equivalent to cot x = 0.

图像在 x = π/2 + nπ 处穿过 x 轴，此时 cot x = 0。
The graph is a decreasing function between successive asymptotes.

在相邻两条渐近线之间，图像是递减的。
cot x is an odd function, so the graph is symmetric about the origin.

cot x 是奇函数，所以图像关于原点对称。

Unlike sec and cosec, the cotangent graph covers the entire real range (-∞, ∞) and has no gaps in its y-values.

与 sec 和 cosec 不同，余切图像覆盖整个实数范围 (-∞, ∞)，在 y 值上没有空缺。

6. Periodicity and Symmetry | 周期性与对称性

All three functions are periodic, but their periods differ. Understanding the period allows efficient sketching over any interval.

这三个函数都是周期函数，但它们的周期不同。理解周期可以在任意区间内高效地绘制图像。

Function	Period	Symmetry	函数	周期	对称性
sec x	2π	Even: sec(-x) = sec x	正割 x	2π	偶函数：sec(-x) = sec x
cosec x	2π	Odd: cosec(-x) = -cosec x	余割 x	2π	奇函数：cosec(-x) = -cosec x
cot x	π	Odd: cot(-x) = -cot x	余切 x	π	奇函数：cot(-x) = -cot x

The even symmetry of sec x means its graph is a reflection of itself across the y-axis, whereas the odd symmetry of cosec x and cot x produces rotational symmetry of 180° about the origin.

sec x 的偶对称性意味着它的图像关于 y 轴镜像对称；而 cosec x 和 cot x 的奇对称性则产生了关于原点旋转 180° 的对称性。

7. Domain and Range Comparison | 定义域与值域对比

A common exam question asks students to state the domain and range of reciprocal trigonometric functions. The table below summarises these properties.

常见的考试题目要求学生写出倒数三角函数的定义域和值域。下表总结了这些性质。

Function	Domain (principal)	Range	函数	定义域（主区间）	值域
sec x	x ≠ π/2 + nπ	y ≤ -1 or y ≥ 1	正割 x	x ≠ π/2 + nπ	y ≤ -1 或 y ≥ 1
cosec x	x ≠ nπ	y ≤ -1 or y ≥ 1	余割 x	x ≠ nπ	y ≤ -1 或 y ≥ 1
cot x	x ≠ nπ	y ∈ ℝ (all reals)	余切 x	x ≠ nπ	y ∈ ℝ（一切实数）

A frequent source of error is writing the range of cot x as [-1, 1] — this is completely incorrect, since cot x can take any real value, including values greater than 100 or less than -100.

一个常见的错误是把 cot x 的值域写成 [-1, 1]——这是完全错误的，因为 cot x 可以取任意实数值，包括大于 100 或小于 -100 的值。

8. Special Angle Values | 特殊角的函数值

In examination problems, you are often required to evaluate these functions at standard angles without a calculator. The following exact values are essential.

在考试题目中，你经常需要在没有计算器的情况下计算这些函数在标准角处的值。以下精确值至关重要。

Angle θ	θ = 0	θ = π/6	θ = π/4	θ = π/3	θ = π/2	角度 θ
sec θ	1	2/√3	√2	2	undefined	正割 θ
cosec θ	undefined	2	√2	2/√3	1	余割 θ
cot θ	undefined	√3	1	1/√3	0	余切 θ

For example, at θ = π/3, since cos(π/3) = 1/2, we immediately obtain sec(π/3) = 2. Similarly, sin(π/3) = √3/2 gives cosec(π/3) = 2/√3.

例如，在 θ = π/3 处，因为 cos(π/3) = 1/2，我们立即得到 sec(π/3) = 2。类似地，sin(π/3) = √3/2 给出 cosec(π/3) = 2/√3。

9. Transformations of Graphs | 图像变换

Just like sin x and cos x, the reciprocal trigonometric graphs can be translated, stretched and reflected. The standard transformation rules apply directly.

与 sin x 和 cos x 一样，倒数三角函数的图像可以进行平移、伸缩和反射。标准的变换规则直接适用。

For y = a sec(bx − c) + d:

对于 y = a sec(bx − c) + d：

The amplitude factor a stretches the graph vertically: if |a| > 1, the branches move further away from the x-axis.

振幅因子 a 垂直拉伸图像：如果 |a| > 1，分支离 x 轴更远。
The coefficient b compresses the graph horizontally: period = 2π/|b|.

系数 b 水平压缩图像：周期 = 2π/|b|。
The phase shift is c/b to the right, shifting all asymptotes accordingly.

相位偏移为向右移动 c/b，所有渐近线也相应移动。
The vertical displacement d shifts the entire graph up or down.

垂直位移 d 将整个图像向上或向下移动。

When sketching transformed graphs, always start from the original sec, cosec or cot graph, then apply vertical stretch first, then horizontal shift, then vertical shift.

在绘制变换后的图像时，务必从原始的正割、余割或余切图像开始，然后先进行垂直伸缩，再进行水平平移，最后进行垂直平移。

10. Sketching sec x and cosec x from sin x and cos x | 从 sin x 和 cos x 绘制 sec x 与 cosec x

An efficient exam technique is to first sketch the corresponding primary function, then use the “reciprocal” relationship to obtain the target graph.

一个高效的考试技巧是先绘制对应的基本函数图像，然后利用“倒数”关系得出目标图像。

Step-by-step method for y = sec x:

绘制 y = sec x 的逐步方法：

Step 1: Sketch y = cos x over the required interval, marking zeros clearly.

第 1 步：在所需区间内绘制 y = cos x，并清楚标出零点。
Step 2: Draw vertical dashed lines at each zero of cos x — these are the asymptotes.

第 2 步：在 cos x 的每个零点处画垂直虚线——这些就是渐近线。
Step 3: Mark the points where cos x = ±1; these become the turning points of sec x.

第 3 步：标出 cos x = ±1 的点；这些点成为 sec x 的极值点。
Step 4: Draw U-shaped curves approaching the asymptotes, passing through the turning points.

第 4 步：绘制逼近渐近线并经过极值点的 U 形曲线。

The identical method applies for cosec x, using y = sin x as the starting graph. This approach ensures that no asymptote or turning point is missed.

完全相同的方法适用于 cosec x，以 y = sin x 作为起始图像。这种方法确保不会遗漏任何渐近线或极值点。

11. Sketching cot x from tan x | 从 tan x 绘制 cot x

For cot x, an alternative approach is to sketch y = tan x first, then take reciprocals.

对于 cot x，另一种方法是先绘制 y = tan x，然后取倒数。

Where tan x = 0 (at x = nπ), cot x has vertical asymptotes.

在 tan x = 0 处（x = nπ），cot x 有垂直渐近线。
Where tan x is undefined (at x = π/2 + nπ), cot x = 0, giving x-axis crossings.

在 tan x 无定义处（x = π/2 + nπ），cot x = 0，即与 x 轴相交。
The graph of cot x decreases throughout each interval between consecutive asymptotes.

cot x 的图像在相邻渐近线之间的每个区间内始终递减。
At x = π/4, tan(π/4) = 1, so cot(π/4) = 1.

在 x = π/4 处，tan(π/4) = 1，所以 cot(π/4) = 1。

Notice an important visual difference: while tan x increases between its asymptotes, cot x decreases. Remembering this distinction prevents a very common sketching error.

请注意一个重要的视觉差异：tan x 在其渐近线之间递增，而 cot x 递减。记住这一区别可以避免一个非常常见的绘图错误。

12. Common Exam Pitfalls and Revision Tips | 常见考试陷阱与复习建议

Many students lose marks unnecessarily on reciprocal trigonometric graph questions. Here are the most common pitfalls and strategies to avoid them.

许多学生在倒数三角函数图像问题上不必要地丢分。以下是最常见的陷阱以及避免它们的策略。

Pitfall 1: Forgetting that sec x and cosec x never take values in the interval (-1, 1). Always check that no part of your graph crosses into this forbidden zone.

陷阱 1：忘记 sec x 和 cosec x 永远不会取 (-1, 1) 区间内的值。始终检查你的图像是否有任何部分进入了这个禁区。
Pitfall 2: Misplacing asymptotes. sec x asymptotes at π/2 + nπ; cosec x and cot x asymptotes at nπ.

陷阱 2：放错渐近线位置。sec x 的渐近线在 π/2 + nπ；cosec x 和 cot x 的渐近线在 nπ。
Pitfall 3: Using the wrong period for cot x. Its period is π, not 2π.

陷阱 3：对 cot x 使用错误周期。它的周期是 π，而不是 2π。
Pitfall 4: In transformation problems, forgetting that the period of cosec(bx) is 2π/|b|, while the period of cot(bx) is π/|b|.

陷阱 4：在变换问题中，忘记 cosec(bx) 的周期是 2π/|b|，而 cot(bx) 的周期是 π/|b|。
Pitfall 5: In identity proofs, incorrectly writing 1 + cot²x instead of 1 + cot²x = cosec²x. Always remember: sec²x = 1 + tan²x and cosec²x = 1 + cot²x.

陷阱 5：在恒等式证明中，错误地写出 1 + cot²x 而不是正确的关系 1 + cot²x = cosec²x。始终记住：sec²x = 1 + tan²x，cosec²x = 1 + cot²x。

By internalising the domain restrictions, memorising exact values, and practising the sketch method described above, you can confidently solve any question on these three essential functions.

通过内化定义域的限制、熟记精确值并练习上述绘图方法，你可以自信地解决关于这三个重要函数的任何问题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

A-Level Maths: Graphs of Secant, Cosecant and Cotangent | A-Level 数学：正割、余割与余切的函数图像