Inverse Trigonometric Functions | 反三角函数

📚 Inverse Trigonometric Functions | 反三角函数

Inverse trigonometric functions allow us to find an angle when given a trigonometric ratio. They are essential in calculus, geometry, and solving equations involving trigonometric expressions. In this article, we will explore their definitions, domains, ranges, graphs, derivatives, integrals, and key identities, all aligned with the AQA A-Level Mathematics specification.

反三角函数帮助我们根据一个三角函数值反过来求角度。它们在微积分、几何以及解含三角表达式的方程中至关重要。本文将围绕 AQA A-Level 数学考纲，系统讲解反三角函数的定义、定义域、值域、图像、导数、积分和重要恒等式。

1. Definition and Notation | 定义与符号

For a function to have an inverse, it must be one-to-one. The standard sine, cosine, and tangent functions are periodic and therefore not one-to-one on their entire domains. To define inverse functions, we restrict each trigonometric function to a suitable interval where it is strictly monotonic.

一个函数要有反函数，它必须是一一对应的。标准的正弦、余弦和正切函数是周期函数，因此在其整个定义域上不是一一对应的。为了定义反函数，我们需要将每个三角函数限制在一个合适的单调区间上。

y = sin⁻¹ x (also written arcsin x) means sin y = x, with -π/2 ≤ y ≤ π/2.
y = sin⁻¹ x（也写作 arcsin x）表示 sin y = x，其中 -π/2 ≤ y ≤ π/2。
y = cos⁻¹ x (arccos x) means cos y = x, with 0 ≤ y ≤ π.
y = cos⁻¹ x（也写作 arccos x）表示 cos y = x，其中 0 ≤ y ≤ π。
y = tan⁻¹ x (arctan x) means tan y = x, with -π/2 < y < π/2.
y = tan⁻¹ x（也写作 arctan x）表示 tan y = x，其中 -π/2 < y < π/2。

It is important to note that sin⁻¹ x is not the same as (sin x)⁻¹ = 1 / sin x. The −1 superscript denotes the inverse function, not a reciprocal.

需要特别注意的是，sin⁻¹ x 并不等于 (sin x)⁻¹ = 1 / sin x。这里的 −1 上标表示反函数，而不是倒数。

2. Domains and Ranges | 定义域与值域

Because the trigonometric functions are restricted, the inverse functions have specific input and output intervals. These are essential for solving problems and evaluating expressions without ambiguity.

由于三角函数被限制了定义域，反三角函数就有了特定的输入区间和输出区间。这些区间对于求解问题和无歧义地计算表达式至关重要。

Function	Domain (x)	Range (y)
y = sin⁻¹ x	−1 ≤ x ≤ 1	−π/2 ≤ y ≤ π/2
y = cos⁻¹ x	−1 ≤ x ≤ 1	0 ≤ y ≤ π
y = tan⁻¹ x	all real numbers	−π/2 < y < π/2

For example, sin⁻¹(1/2) = π/6, but sin⁻¹(1/2) cannot be 5π/6 because 5π/6 is outside the range of arcsin. Similarly, cos⁻¹(−1/2) = 2π/3, not −2π/3.

例如，sin⁻¹(1/2) = π/6，但 sin⁻¹(1/2) 不可能是 5π/6，因为 5π/6 不在 arcsin 的值域内。同理，cos⁻¹(−1/2) = 2π/3，而不是 −2π/3。

3. Graphs of Inverse Trigonometric Functions | 反三角函数的图像

The graph of an inverse function is the reflection of the restricted original function in the line y = x. Understanding these graphs helps visualise symmetry and important values.

反函数的图像是原函数在限制定义域后关于直线 y = x 对称的镜像。理解这些图像有助于直观把握对称性和关键取值。

y = sin⁻¹ x is increasing, passes through (−1, −π/2), (0, 0), (1, π/2), and is symmetric about the origin.
y = sin⁻¹ x 是递增的，经过 (−1, −π/2)、(0, 0)、(1, π/2)，并且关于原点对称。
y = cos⁻¹ x is decreasing, passes through (−1, π), (0, π/2), (1, 0). It is symmetric about the point (0, π/2).
y = cos⁻¹ x 是递减的，经过 (−1, π)、(0, π/2)、(1, 0)。它关于点 (0, π/2) 对称。
y = tan⁻¹ x has horizontal asymptotes y = π/2 and y = −π/2, and passes through (0, 0).
y = tan⁻¹ x 有水平渐近线 y = π/2 和 y = −π/2，并经过 (0, 0)。

When transforming graphs, changes such as y = 2 sin⁻¹(3x) can be analysed using standard transformation rules. For example, the domain of sin⁻¹(3x) becomes −1/3 ≤ x ≤ 1/3.

在对图像进行变换时，诸如 y = 2 sin⁻¹(3x) 的表达式可以按照标准变换规则分析。例如，sin⁻¹(3x) 的定义域变为 −1/3 ≤ x ≤ 1/3。

4. Evaluating Exact Values | 计算精确值

To evaluate expressions like sin⁻¹(cos θ) or tan⁻¹(tan θ), we must consider the range of the inverse function. The composite function only returns the principal value.

计算像 sin⁻¹(cos θ) 或 tan⁻¹(tan θ) 这样的表达式时，必须考虑反函数的值域。复合函数只返回主值。

Example: sin⁻¹(sin(3π/4)) = π/4, not 3π/4, because 3π/4 is outside the range of sin⁻¹.

例：sin⁻¹(sin(3π/4)) = π/4，而不是 3π/4，因为 3π/4 不在 sin⁻¹ 的值域内。

For exact evaluation, we often draw a right-angled triangle. For instance, if θ = sin⁻¹(3/5), then sin θ = 3/5, and we can find cos θ = 4/5, tan θ = 3/4.

在求精确值时，我们通常画一个直角三角形。例如，若 θ = sin⁻¹(3/5)，则 sin θ = 3/5，由此可得 cos θ = 4/5，tan θ = 3/4。

An important technique is to compose an inverse function with another trigonometric function. For example, to find cos(sin⁻¹ x), let y = sin⁻¹ x, so sin y = x. Then cos y = √(1 − x²), taking the positive root because y ∈ [−π/2, π/2].

一个重要技巧是将反函数与另一个三角函数复合。例如，求 cos(sin⁻¹ x) 时，令 y = sin⁻¹ x，则 sin y = x。于是 cos y = √(1 − x²)，取正根，因为 y ∈ [−π/2, π/2]。

5. Inverse Tangent Addition Formulas | 反正切叠加公式

The inverse tangent function has a useful addition formula, analogous to the tangent addition formula:

反正切函数有一个非常有用的叠加公式，类似正切的加法公式：

tan⁻¹ a + tan⁻¹ b = tan⁻¹((a + b) / (1 − ab)), provided ab < 1.

tan⁻¹ a + tan⁻¹ b = tan⁻¹((a + b) / (1 − ab))，其中 ab < 1。

If ab > 1, the result needs adjustment by adding or subtracting π to land in the principal range. For example, tan⁻¹ 1 + tan⁻¹ 1 = π/4 + π/4 = π/2, but tan⁻¹((1+1)/(1−1)) is undefined, so the formula must be used carefully.

若 ab > 1，结果需要加上或减去 π 来调整到主值区间。例如，tan⁻¹ 1 + tan⁻¹ 1 = π/4 + π/4 = π/2，但 tan⁻¹((1+1)/(1−1)) 无定义，因此使用该公式时要小心。

This formula is often used to solve equations such as tan⁻¹ x + tan⁻¹(2x) = π/4.

这个公式常用于解方程，例如 tan⁻¹ x + tan⁻¹(2x) = π/4。

6. Solving Equations with Inverse Functions | 利用反函数解方程

Inverse functions transform trigonometric equations into algebraic ones. For example, to solve 3 sin θ = 2 for θ in [0, 2π), first find the principal solution θ = sin⁻¹(2/3), then use symmetry to find the second solution θ = π − sin⁻¹(2/3).

反函数能将三角方程转化为代数方程。例如，在 [0, 2π) 上解 3 sin θ = 2，先找到主解 θ = sin⁻¹(2/3)，然后利用对称性找到第二个解 θ = π − sin⁻¹(2/3)。

For tan θ = k, the general solution is θ = tan⁻¹ k + nπ, where n is any integer. This compact form is used in modelling periodic phenomena.

对于 tan θ = k，通解为 θ = tan⁻¹ k + nπ，其中 n 为任意整数。这种简洁形式常用于描述周期现象。

When solving equations involving compositions such as sin(2θ) = 0.5, we set 2θ = sin⁻¹(0.5) + 2nπ or 2θ = π − sin⁻¹(0.5) + 2nπ, then divide by 2 to find all solutions in the required interval.

当解复合方程如 sin(2θ) = 0.5 时，我们令 2θ = sin⁻¹(0.5) + 2nπ 或 2θ = π − sin⁻¹(0.5) + 2nπ，然后除以 2 得到指定区间内的所有解。

7. Derivatives of Inverse Trigonometric Functions | 反三角函数的导数

The derivatives of inverse trigonometric functions are standard results that must be memorised. They are derived using implicit differentiation.

反三角函数的导数是必须牢记的标准结果。它们可以通过隐函数求导推导出来。

d/dx (sin⁻¹ x) = 1 / √(1 − x²), for −1 < x < 1

d/dx (cos⁻¹ x) = −1 / √(1 − x²), for −1 < x < 1

d/dx (tan⁻¹ x) = 1 / (1 + x²), for all real x

Notice that d/dx (sin⁻¹ x) + d/dx (cos⁻¹ x) = 0, which reflects the identity sin⁻¹ x + cos⁻¹ x = π/2.

注意 d/dx (sin⁻¹ x) + d/dx (cos⁻¹ x) = 0，这正是恒等式 sin⁻¹ x + cos⁻¹ x = π/2 的体现。

For composite functions, use the chain rule. For example, d/dx [sin⁻¹(2x)] = 2 / √(1 − 4x²).

对于复合函数，使用链式法则。例如，d/dx [sin⁻¹(2x)] = 2 / √(1 − 4x²)。

8. Deriving the Derivatives | 推导导数公式

To derive d/dx (sin⁻¹ x), let y = sin⁻¹ x. Then sin y = x. Differentiating both sides with respect to x gives cos y · dy/dx = 1, so dy/dx = 1 / cos y. Since cos y ≥ 0 for y ∈ [−π/2, π/2], we have cos y = √(1 − sin² y) = √(1 − x²). Hence the derivative is 1 / √(1 − x²).

推导 d/dx (sin⁻¹ x) 时，令 y = sin⁻¹ x，则 sin y = x。两边对 x 求导得 cos y · dy/dx = 1，所以 dy/dx = 1 / cos y。由于 y ∈ [−π/2, π/2] 时 cos y ≥ 0，所以 cos y = √(1 − sin² y) = √(1 − x²)。因此导数为 1 / √(1 − x²)。

Similarly, for y = tan⁻¹ x, we write tan y = x. Differentiate: sec² y · dy/dx = 1, so dy/dx = 1 / sec² y = 1 / (1 + tan² y) = 1 / (1 + x²).

同理，对于 y = tan⁻¹ x，写出 tan y = x。求导得 sec² y · dy/dx = 1，所以 dy/dx = 1 / sec² y = 1 / (1 + tan² y) = 1 / (1 + x²)。

9. Integrals Involving Inverse Trigonometric Functions | 含反三角函数的积分

Derivatives of inverse functions give immediate standard integrals. These are particularly useful for integrating rational functions with quadratic denominators.

反函数的导数直接给出标准积分。这些积分在处理含二次分母的有理函数时特别有用。

∫ 1 / √(a² − x²) dx = sin⁻¹(x/a) + C

∫ 1 / (a² + x²) dx = (1/a) tan⁻¹(x/a) + C

For example, ∫ 1 / (4 + x²) dx = (1/2) tan⁻¹(x/2) + C. And ∫ 1 / √(9 − x²) dx = sin⁻¹(x/3) + C.

例如，∫ 1 / (4 + x²) dx = (1/2) tan⁻¹(x/2) + C。以及 ∫ 1 / √(9 − x²) dx = sin⁻¹(x/3) + C。

Sometimes a substitution is needed to match the form. For instance, ∫ 1 / √(4 − 9x²) dx can be written as (1/3) ∫ 1 / √((4/9) − x²) dx = (1/3) sin⁻¹(3x/2) + C.

有时需要变量代换以匹配标准形式。例如，∫ 1 / √(4 − 9x²) dx 可改写为 (1/3) ∫ 1 / √((4/9) − x²) dx = (1/3) sin⁻¹(3x/2) + C。

10. Key Identities and Relationships | 关键恒等式与关系

Several identities link inverse trigonometric functions. They often simplify expressions and are tested in exam questions.

一些恒等式将反三角函数联系起来。它们常用于化简表达式，也是考试中的常见考点。

sin⁻¹ x + cos⁻¹ x = π/2 for all x ∈ [−1, 1].
对所有 x ∈ [−1, 1]，sin⁻¹ x + cos⁻¹ x = π/2。
tan⁻¹ x + tan⁻¹(1/x) = π/2 for x > 0, and −π/2 for x < 0.
当 x > 0 时，tan⁻¹ x + tan⁻¹(1/x) = π/2；当 x < 0 时为 −π/2。
sin(cos⁻¹ x) = √(1 − x²), cos(sin⁻¹ x) = √(1 − x²).
sin(cos⁻¹ x) = √(1 − x²)，cos(sin⁻¹ x) = √(1 − x²)。
tan(sin⁻¹ x) = x / √(1 − x²), for −1 < x < 1.
tan(sin⁻¹ x) = x / √(1 − x²)，其中 −1 < x < 1。

These identities are derived from the definitions and the Pythagorean identity. They are useful in integration and in proving other results.

这些恒等式可由定义和勾股恒等式推导。它们在积分和证明其他结论时非常有用。

11. Differentiation of Composite Inverse Trigonometric Functions | 复合反三角函数的微分

In AQA A-Level, you may need to differentiate functions like sin⁻¹(f(x)) or tan⁻¹(f(x)). The chain rule is applied along with the known derivatives.

在 AQA A-Level 中，你可能需要对 sin⁻¹(f(x)) 或 tan⁻¹(f(x)) 这类函数求导。此时需要使用链式法则配合已知导数公式。

d/dx [sin⁻¹(f(x))] = f'(x) / √(1 − f(x)²)

d/dx [tan⁻¹(f(x))] = f'(x) / (1 + f(x)²)

Example: find d/dx [tan⁻¹(√x)]. Here f(x) = √x, f'(x) = 1/(2√x). Therefore the derivative is (1/(2√x)) / (1 + x) = 1 / (2√x (1 + x)).

例：求 d/dx [tan⁻¹(√x)]。这里 f(x) = √x，f'(x) = 1/(2√x)。因此导数为 (1/(2√x)) / (1 + x) = 1 / (2√x (1 + x))。

Sometimes a substitution such as x = tan θ simplifies an expression before differentiation. For example, to differentiate sin⁻¹(2x/(1+x²)), recognising the double-angle identity 2 tan θ / (1 + tan² θ) = sin 2θ can lead to a simpler derivative.

有时先做代换如 x = tan θ 可以简化表达式再求导。例如，对 sin⁻¹(2x/(1+x²)) 求导时，识别二倍角公式 2 tan θ / (1 + tan² θ) = sin 2θ 可以得到更简单的导数形式。

12. Common Exam Pitfalls | 常见考试易错点

Students often lose marks on inverse trigonometric questions due to subtle mistakes. Here are the most common pitfalls and how to avoid them.

学生在反三角函数题目上经常因为一些细微错误丢分。以下是最常见的易错点以及如何避免。

Confusing inverse with reciprocal: sin⁻¹ x is not 1/sin x. Always read the notation carefully.
混淆反函数与倒数： sin⁻¹ x 不等于 1/sin x。读题时务必仔细。
Forgetting the restricted range: When writing arcsin or arccos results, check that the angle lies in the correct interval.
忘记主值区间： 写出 arcsin 或 arccos 的结果时，检查角度是否在正确的区间内。
Wrong sign in derivatives: d/dx (cos⁻¹ x) is negative. Missing the negative sign is a common error.
导数符号写错： d/dx (cos⁻¹ x) 是负的。漏掉负号是常见错误。
Using the addition formula without conditions: tan⁻¹ a + tan⁻¹ b may need an extra π term.
使用叠加公式时忽略条件： tan⁻¹ a + tan⁻¹ b 可能需要额外加上 π。
Ignoring domain restrictions in integrals: Given ∫ 1/√(a² − x²) dx, remember that |x| < a.
积分中忽略定义域限制： 对于 ∫ 1/√(a² − x²) dx，注意 |x| < a。

Always write the range of an inverse function in a solution step, especially when solving equations or evaluating composite expressions. This habit prevents many careless errors.

在解题过程中，尤其是在解方程或计算复合表达式时，始终写出反函数的值域。这个好习惯能避免许多粗心错误。

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Inverse Trigonometric Functions | 反三角函数