📚 Inverse and Reciprocal Trigonometric Functions | 反三角函数与倒数三角函数
In IB Mathematics, the study of inverse and reciprocal trigonometric functions extends your understanding of circular functions into new domains: from defining secant, cosecant, and cotangent, to working with arcsin, arccos, and arctan. Mastering the differences, graphs, identities, derivatives, and integrals of these functions is essential for success in both Analysis & Approaches and Applications & Interpretation. This article covers all key concepts with clear bilingual explanations to support your revision.
在 IB 数学中，反三角函数与倒数三角函数的学习将从定义 sec、csc 和 cot，到运用 arcsin、arccos 和 arctan，拓展你对圆函数的理解。掌握这些函数的区别、图像、恒等式、导数和积分对于在分析与方法和应用与解释两门课程中取得成功至关重要。本文通过清晰的中英双语讲解，涵盖所有核心概念，助力你的复习。
1. Distinguishing Inverse from Reciprocal | 区分反函数与倒数函数
Many students confuse inverse trigonometric functions (arcsin, arccos, arctan) with reciprocal trigonometric functions (cosecant, secant, cotangent). The inverse function undoes the original trigonometric operation, whereas the reciprocal is simply 1 divided by the function. For instance, arcsin(0.5) = π/6, but csc(π/6) = 1/sin(π/6) = 2.
许多学生混淆反三角函数（arcsin、arccos、arctan）与倒数三角函数（cosecant、secant、cotangent）。反函数是原三角函数的逆运算，而倒数只是 1 除以该函数。例如，arcsin(0.5) = π/6，但 csc(π/6) = 1/sin(π/6) = 2。
Notation is a common pitfall: sin⁻¹(x) denotes the inverse sine, not (sin x)⁻¹. To avoid ambiguity, IB often uses arcsin. The reciprocal of sin x is written as csc x or (sin x)⁻¹, and it is completely different from sin⁻¹(x). Always check the context.
记号是一个常见陷阱：sin⁻¹(x) 表示反正弦，而不是 (sin x)⁻¹。为避免歧义，IB 常使用 arcsin。sin x 的倒数写作 csc x 或 (sin x)⁻¹，它与 sin⁻¹(x) 完全不同。请务必根据上下文判断。
2. Reciprocal Trigonometric Functions | 倒数三角函数
The three reciprocal functions are secant (sec), cosecant (csc), and cotangent (cot). They are defined by sec θ = 1/cos θ, csc θ = 1/sin θ, and cot θ = 1/tan θ = cos θ/sin θ. These functions appear frequently in identities and calculus.
三个倒数函数分别是 sec（正割）、csc（余割）和 cot（余切）。它们的定义为：sec θ = 1/cos θ，csc θ = 1/sin θ，cot θ = 1/tan θ = cos θ/sin θ。这些函数经常出现在恒等式和微积分中。
Because they involve division, each reciprocal function is undefined where the original function equals zero. For sec θ, this means cos θ = 0, giving vertical asymptotes at θ = π/2 + nπ. For csc θ, asymptotes occur at θ = nπ, and for cot θ at θ = nπ as well.
因为它们包含除法，每个倒数函数在原函数为零的点处无定义。对于 sec θ，当 cos θ = 0 时，即在 θ = π/2 + nπ 处存在垂直渐近线。对于 csc θ，渐近线出现在 θ = nπ；cot θ 同样在 θ = nπ 处有渐近线。
3. Graphs of Reciprocal Functions | 倒数函数的图像
The graph of y = sec x has U-shaped branches extending to ±∞, with values satisfying |sec x| ≥ 1. It shares vertical asymptotes with cos x where cos x = 0. The secant curve does not intersect the x-axis and has a period of 2π.
y = sec x 的图像呈现 U 形分支，延伸至 ±∞，且满足 |sec x| ≥ 1。它在 cos x = 0 处与 cos x 共享垂直渐近线。正割曲线不与 x 轴相交，周期为 2π。
The graph of y = csc x also consists of alternating upward and downward branches, with asymptotes at integer multiples of π. Its range is (−∞, −1] ∪ [1, ∞). Meanwhile, y = cot x crosses the x-axis at π/2 + nπ and has a decreasing shape within each period, with asymptotes at x = nπ.
y = csc x 的图像同样由交替的向上和向下分支组成，渐近线在 π 的整数倍处。其值域为 (−∞, −1] ∪ [1, ∞)。另一方面，y = cot x 在 π/2 + nπ 处穿过 x 轴，在每个周期内呈递减趋势，渐近线位于 x = nπ。
Recognising these graphs helps in understanding domain restrictions and solving inequalities involving reciprocal trigonometric expressions.
识别这些图像有助于理解定义域限制以及解含有倒数三角函数的不等式。
4. Inverse Trigonometric Functions | 反三角函数
The inverse sine function, arcsin x or sin⁻¹ x, outputs the angle in [−π/2, π/2] whose sine is x. Since y = sin x is not injective on its full domain, we restrict it to [−π/2, π/2] to define a proper inverse.
反正弦函数 arcsin x 或 sin⁻¹ x 输出 [−π/2, π/2] 内的角度，其正弦值为 x。由于 y = sin x 在其整个定义域上不是一一对应的，我们将其限制在 [−π/2, π/2] 上来定义一个合适的反函数。
The inverse cosine function, arccos x, uses the restricted domain [0, π] for cos x, so arccos x ∈ [0, π]. The inverse tangent, arctan x, takes domain ℝ and maps to (−π/2, π/2). These principal value branches guarantee that each inverse function is unique.
反余弦函数 arccos x 使用 cos x 的限制定义域 [0, π]，因此 arccos x ∈ [0, π]。反正切 arctan x 的定义域为 ℝ，值域为 (−π/2, π/2)。这些主值分支确保了每个反函数的唯一性。
While arcsec, arccsc, and arccot exist, IB courses typically focus on arcsin, arccos, and arctan. The principles of restricting domains apply similarly.
虽然存在 arcsec、arccsc 和 arccot，但 IB 课程通常重点关注 arcsin、arccos 和 arctan。限制定义域的原则同样适用于它们。
5. Graphs of Inverse Functions | 反函数的图像
The graph of y = arcsin x is obtained by reflecting the restricted sine curve y = sin x, x ∈ [−π/2, π/2], across the line y = x. The result is a strictly increasing function that passes through the origin and has endpoints (−1, −π/2) and (1, π/2).
y = arcsin x 的图像是通过将限制后的正弦曲线 y = sin x，x ∈ [−π/2, π/2]，关于直线 y = x 反射得到的。结果是一个严格递增的函数，经过原点，端点为 (−1, −π/2) 和 (1, π/2)。
The graph of y = arccos x is strictly decreasing, connecting (1, 0) to (−1, π). Arctan x has a characteristic S-shape, bounded by horizontal asymptotes at y = π/2 and y = −π/2, approaching them as x → +∞ and x → −∞.
y = arccos x 的图像是严格递减的，连接 (1, 0) 和 (−1, π)。arctan x 具有典型的 S 形，被水平渐近线 y = π/2 和 y = −π/2 所限制，当 x → +∞ 和 x → −∞ 时分别趋近它们。
Recognising the symmetry between a function and its inverse is critical when solving equations like arcsin(2x) = k or when analysing composite functions.
理解原函数与其反函数之间的对称性，对于求解 arcsin(2x) = k 这类方程或分析复合函数至关重要。
6. Domain and Range | 定义域与值域
The following table summarises the domains and ranges of the principal inverse functions and the three reciprocal functions. Memorising these is essential for correct evaluation and equation solving.
下表总结了主值反函数和三个倒数函数的定义域与值域。熟记这些对于正确求值和解方程必不可少。
| Function | Domain | Range |
|---|---|---|
| arcsin x (sin⁻¹ x) | [−1, 1] | [−π/2, π/2] |
| arccos x (cos⁻¹ x) | [−1, 1] | [0, π] |
| arctan x (tan⁻¹ x) | ℝ | (−π/2, π/2) |
| sec x | x ≠ π/2 + nπ | (−∞, −1] ∪ [1, ∞) |
| csc x | x ≠ nπ | (−∞, −1] ∪ [1, ∞) |
| cot x | x ≠ nπ | ℝ |
For inverse functions, the domain is determined by the range of the original trigonometric function on the restricted interval. Reciprocal functions inherit domain restrictions from where sine, cosine, or tangent are zero.
对于反函数，定义域取决于原三角函数在限制区间上的值域。倒数函数则从正弦、余弦或正切为零的点继承定义域限制。
7. Key Trigonometric Identities | 关键恒等式
Reciprocal identities link each pair: sin θ · csc θ = 1, cos θ · sec θ = 1, tan θ · cot θ = 1. Pythagorean-derived identities include 1 + tan² θ = sec² θ and 1 + cot² θ = csc² θ. These are particularly useful when simplifying expressions or proving other identities.
倒数恒等式连接每一对：sin θ · csc θ = 1，cos θ · sec θ = 1，tan θ · cot θ = 1。由毕达哥拉斯恒等式导出的恒等式包括 1 + tan² θ = sec² θ 和 1 + cot² θ = csc² θ。这些在化简表达式或证明其他恒等式时特别有用。
For inverse functions, the complementary angle identities hold: arcsin x + arccos x = π/2 for x ∈ [−1, 1], and arctan x + arccot x = π/2. Moreover, sin(arcsin x) = x and arcsin(sin x) = x only when x lies within the principal range [−π/2, π/2].
对于反函数，余角恒等式成立：对于 x ∈ [−1, 1]，arcsin x + arccos x = π/2；arctan x + arccot x = π/2。此外，sin(arcsin x) = x 恒成立，而 arcsin(sin x) = x 仅在 x 位于主值区间 [−π/2, π/2] 时成立。
Double-angle formulas with reciprocal functions often require conversion to sines and cosines first. For instance, sec 2θ =
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