Inverse Trigonometric Functions: Definitions and Graphs | 反三角函数的定义与图像性质

📚 Inverse Trigonometric Functions: Definitions and Graphs | 反三角函数的定义与图像性质

Inverse trigonometric functions are the inverses of the trigonometric functions when restricted to carefully chosen intervals. They allow us to work backwards from a known ratio to the angle that produced it, and they are essential in calculus, geometry, and physics.

反三角函数是三角函数在特定限制区间上的反函数。它们使我们能够从已知的比值反推出对应的角度，在微积分、几何和物理中都非常重要。

1. Why Inverse Trigonometric Functions? | 为什么要研究反三角函数？

Trigonometric functions such as sin x, cos x and tan x are periodic, so they are not one-to-one over their entire domains. This means that an equation like sin θ = 0.5 has infinitely many solutions unless we restrict the domain.

正弦函数 sin x、余弦函数 cos x 和正切函数 tan x 具有周期性，因此在整个定义域上并不一一对应。这意味着，除非限制定义域，否则像 sin θ = 0.5 这样的方程会有无穷多个解。

To define an inverse, we restrict the original function to an interval where it is strictly monotonic (always increasing or always decreasing). On that interval the function becomes one-to-one, so its inverse exists.

为了定义反函数，我们将原函数限制在一个严格单调（始终递增或始终递减）的区间上。在这个区间上函数是一一对应的，因此其反函数存在。

2. Restricting the Domain | 定义域的限制

Conventionally, we choose the following principal intervals: for sine, [−π/2, π/2]; for cosine, [0, π]; for tangent, (−π/2, π/2). These intervals contain the origin where possible, and each function is monotonic and covers its full range.

通常我们选择如下主区间：对正弦函数取 [−π/2, π/2]；对余弦函数取 [0, π]；对正切函数取 (−π/2, π/2)。这些区间尽可能包含原点，并且函数在该区间内单调且覆盖其全部值域。

The inverse functions built on these intervals are denoted arcsin, arccos and arctan, respectively. The notation sin⁻¹ is also common, but it can be confused with the reciprocal 1/sin x, so many textbooks prefer the arc-prefix notation.

在这些区间上建立的反函数分别记作 arcsin、arccos 和 arctan。虽然 sin⁻¹ 也很常见，但它容易与倒数 1/sin x 混淆，因此许多教材更倾向于使用 arc 前缀的记法。

3. Definition of arcsin | arcsin 的定义

The inverse sine function is defined as follows:

反正弦函数的定义如下：

y = arcsin x ⇔ x = sin y, y ∈ [−π/2, π/2]

Its domain is [−1, 1] and its range is [−π/2, π/2]. The graph is strictly increasing and passes through the origin. Key points are (−1, −π/2), (0, 0) and (1, π/2).

其定义域为 [−1, 1]，值域为 [−π/2, π/2]。图像严格递增且经过原点。关键点为 (−1, −π/2)、(0, 0) 和 (1, π/2)。

The graph of arcsin is a reflection of the restricted sine curve across the line y = x. Because the restricted sine is odd, arcsin is also odd: arcsin(−x) = −arcsin x.

arcsin 的图像是限制后的正弦曲线关于直线 y = x 的反射。由于限制后的正弦函数是奇函数，arcsin 也是奇函数：arcsin(−x) = −arcsin x。

4. Definition of arccos | arccos 的定义

The inverse cosine function is defined as:

反余弦函数的定义如下：

y = arccos x ⇔ x = cos y, y ∈ [0, π]

Its domain is [−1, 1] and its range is [0, π]. The graph is strictly decreasing, with key points (−1, π), (0, π/2) and (1, 0).

其定义域为 [−1, 1]，值域为 [0, π]。图像严格递减，关键点为 (−1, π)、(0, π/2) 和 (1, 0)。

Notice that arccos is neither even nor odd. The graph is symmetric in the sense that arccos(−x) = π − arccos x, which reflects the property cos(π − θ) = −cos θ.

注意，arccos 既不是奇函数也不是偶函数。图像具有 arccos(−x) = π − arccos x 的对称关系，这反映了 cos(π − θ) = −cos θ 的性质。

5. Definition of arctan | arctan 的定义

The inverse tangent function is defined as:

反正切函数的定义如下：

y = arctan x ⇔ x = tan y, y ∈ (−π/2, π/2)

Its domain is all real numbers (ℝ) and its range is (−π/2, π/2). The graph passes through the origin, is strictly increasing, and has two horizontal asymptotes: y = π/2 as x → +∞ and y = −π/2 as x → −∞.

其定义域为全体实数 (ℝ)，值域为 (−π/2, π/2)。图像经过原点，严格递增，并且有两条水平渐近线：当 x → +∞ 时 y = π/2，当 x → −∞ 时 y = −π/2。

Because tan is odd over the chosen interval, arctan is also odd: arctan(−x) = −arctan x. The graph approaches the asymptotes but never touches them.

由于在所取区间上 tan 是奇函数，arctan 也是奇函数：arctan(−x) = −arctan x。图像逐渐接近渐近线但永远不会触及它们。

6. Other Inverse Trigonometric Functions | 其他反三角函数

The reciprocal trigonometric functions also have inverses: arccot, arcsec and arccsc. They are less commonly tested but may appear in advanced courses.

余切、正割和余割也有对应的反函数：arccot、arcsec 和 arccsc。它们虽然不常考，但在进阶课程中可能会出现。

Conventional principal ranges are (0, π) for arccot, [0, π] with y ≠ π/2 for arcsec, and [−π/2, π/2] with y ≠ 0 for arccsc. These choices ensure that each function is one-to-one.

通常的主值区间为：arccot 取 (0, π)，arcsec 取 [0, π] 且 y ≠ π/2，arccsc 取 [−π/2, π/2] 且 y ≠ 0。这样的选择确保每个函数是一一对应的。

7. Summary Table of Domains and Ranges | 定义域与值域汇总表

Function	Domain	Range
arcsin x	[−1, 1]	[−π/2, π/2]
arccos x	[−1, 1]	[0, π]
arctan x	ℝ	(−π/2, π/2)
arccot x	ℝ	(0, π)
arcsec x	(−∞, −1] ∪ [1, ∞)	[0, π/2) ∪ (π/2, π]
arccsc x	(−∞, −1] ∪ [1, ∞)	[−π/2, 0) ∪ (0, π/2]

This table is an essential reference when solving equations or sketching graphs. In A-level courses, arcsin, arccos and arctan are the most important.

这张表格是解方程或画图时的重要参考。在 A-level 课程中，最常考的是 arcsin、arccos 和 arctan。

8. Key Graphical Features | 图像的主要特征

The graph of arcsin is always increasing, with slope steepest near x = 0 and approaching vertical near x = ±1. The graph is symmetric with respect to the origin.

arcsin 的图像总是递增，在 x = 0 附近斜率最大，在 x = ±1 附近趋于竖直。图像关于原点对称。

The graph of arccos is always decreasing, with a horizontal tangent at x = 0, and it is symmetric about the point (0, π/2).

arccos 的图像总是递减，在 x = 0 处有水平切线，并且关于点 (0, π/2) 中心对称。

The graph of arctan has no vertical asymptotes, but the horizontal asymptotes y = ±π/2 are boundaries that the curve approaches as x tends to ±∞. The slope at x = 0 is 1.

arctan 的图像没有垂直渐近线，但有两条水平渐近线 y = ±π/2，曲线在 x 趋于 ±∞ 时接近它们。在 x = 0 处的斜率为 1。

All inverse trigonometric graphs are the reflections of the corresponding restricted trigonometric graphs across the line y = x. This reflection explains why vertical asymptotes of the original become horizontal asymptotes of the inverse.

所有反三角函数的图像都是对应的受限三角函数图像关于直线 y = x 的反射。这一反射解释了为什么原函数的垂直渐近线变成了反函数的水平渐近线。

9. Symmetry and Identities | 对称性与恒等式

Several symmetry relations are particularly useful:

以下对称关系非常有用：

arcsin(−x) = −arcsin x

arcsin(−x) = −arcsin x
arctan(−x) = −arctan x

arctan(−x) = −arctan x
arccos(−x) = π − arccos x

arccos(−x) = π − arccos x
arccot(−x) = π − arccot x

arccot(−x) = π − arccot x

The most fundamental identities connecting the functions are:

最基本的联系恒等式如下：

arcsin x + arccos x = π/2

arctan x + arccot x = π/2

These identities hold for every x in the common domain and are useful when simplifying expressions involving the derivatives or integrals.

这些恒等式对公共定义域中的所有 x 都成立，在化简导数或积分表达式时非常有用。

10. Composition with Trigonometric Functions | 与三角函数的复合

Inside their domains, we have the cancellation laws:

在定义域内部，我们有以下抵消法则：

sin(arcsin x) = x for x ∈ [−1, 1]

当 x ∈ [−1, 1] 时，sin(arcsin x) = x
cos(arccos x) = x for x ∈ [−1, 1]

当 x ∈ [−1, 1] 时，cos(arccos x) = x
tan(arctan x) = x for all real x

对所有实数 x，tan(arctan x) = x

However, the reverse direction is not always true. For example, arcsin(sin 2π) = arcsin(0) = 0, not 2π. The identity arcsin(sin y) = y holds only when y ∈ [−π/2, π/2].

然而，反向复合并不总是成立。例如，arcsin(sin 2π) = arcsin(0) = 0，而不是 2π。恒等式 arcsin(sin y) = y 仅在 y ∈ [−π/2, π/2] 时成立。

Similarly, arccos(cos y) = y only for y ∈ [0, π], and arctan(tan y) = y only for y ∈ (−π/2, π/2). These restrictions are a direct consequence of the chosen principal intervals.

类似地，arccos(cos y) = y 仅在 y ∈ [0, π] 时成立，arctan(tan y) = y 仅在 y ∈ (−π/2, π/2) 时成立。这些限制直接来源于所选的主值区间。

11. Transformations of Inverse Trig Graphs | 反三角函数图像的变换

As with any function, inverse trigonometric graphs can be translated, stretched and reflected. For example, y = 2 arcsin(x/3) has domain [−3, 3] and range [−π, π], because the input is scaled by 3 and the output by 2.

与任何函数一样，反三角函数的图像也可以平移、伸缩和反射。例如，y = 2 arcsin(x/3) 的定义域为 [−3, 3]，值域为 [−π, π]，因为输入放大了 3 倍，输出放大了 2 倍。

A vertical shift, such as y = arctan x + π/4, moves the graph upward by π/4. A horizontal reflection, y = arccos(−x), is equivalent to replacing x by −x and produces the mirror image across the y-axis.

垂直平移，如 y = arctan x + π/4，将图像向上移动 π/4。水平反射，如 y = arccos(−x)，相当于把 x 替换为 −x，产生关于 y 轴的镜像。

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

Although this article focuses on definitions and graphs, the derivatives of these functions are closely related to their graphical properties and are standard knowledge in calculus:

虽然本文主要关注定义和图像，但这些函数的导数与其图像性质紧密相关，也是微积分中的标准内容：

d/dx arcsin x = 1 / √(1 − x²)

d/dx arccos x = −1 / √(1 − x²)

d/dx arctan x = 1 / (1 + x²)

Notice that the derivative of arccos is the negative of the derivative of arcsin, which is consistent with the graph of arccos being decreasing. The steep slopes near x = ±1 for arcsin correspond to the singular behaviour of 1/√(1 − x²).

注意，arccos 的导数是 arcsin 导数的相反数，这与 arccos 图像递减一致。arcsin 在 x = ±1 附近的陡峭斜率对应着 1/√(1 − x²) 的奇异行为。

These derivatives also lead to important integrals, such as ∫ 1/√(1 − x²) dx = arcsin x + C and ∫ 1/(1 + x²) dx = arctan x + C. Recognising the graph shapes helps you choose the correct inverse function when integrating.

这些导数还导出重要的积分公式，例如 ∫ 1/√(1 − x²) dx = arcsin x + C 和 ∫ 1/(1 + x²) dx = arctan x + C。识别图像形状有助于在积分时选择正确的反三角函数。

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Inverse Trigonometric Functions: Definitions and Graphs | 反三角函数的定义与图像性质