Inverse Trigonometric Functions | 反三角函数

📚 Inverse Trigonometric Functions | 反三角函数

Inverse trigonometric functions, often called arcus functions, reverse the action of sine, cosine and tangent. For the Edexcel A-Level Mathematics specification, arcsin x, arccos x and arctan x are essential tools in differentiation, integration and the solution of trigonometric equations. They appear year after year in Pure Mathematics papers, so a secure understanding of their domains, ranges and calculus is non-negotiable for top grades.

反三角函数（也称反圆函数）用于逆转正弦、余弦和正切的作用。在 Edexcel A-Level 数学考纲中，arcsin x、arccos x 与 arctan x 是微分、积分及解三角方程的必备工具。它们在纯数试卷中年年出现，因此扎实掌握它们的定义域、值域以及相关微积分结论，是冲击高分的必要条件。

1. Why Trig Functions Must Be Restricted | 为什么必须限制三角函数的定义域

A function can only have an inverse if it is one-to-one, meaning every output corresponds to exactly one input. Sine, cosine and tangent are periodic, so they take the same value infinitely many times across their natural domains. For instance, sin(0) = sin(π) = sin(2π) = 0, so without restrictions there would be no unique value for sin⁻¹(0).

一个函数只有满足一一对应时才有反函数，即每个函数值只对应唯一一个自变量。正弦、余弦和正切都是周期函数，在自然定义域上同一个函数值会出现无数次。例如 sin(0) = sin(π) = sin(2π) = 0，若不施加限制，sin⁻¹(0) 就没有唯一确定的值。

To fix this, we restrict each trigonometric function to a chosen principal interval on which it is strictly monotonic (always increasing or always decreasing). On such an interval the function is one-to-one, and the inverse can be defined cleanly. The inverse functions are usually written as arcsin x, arccos x and arctan x (the notation sin⁻¹x is also used).

为解决这一问题，我们将每个三角函数限制在一个选定的“主区间”上，使函数在该区间上严格单调（始终递增或始终递减）。在这样的区间上函数是一一对应的，反函数便可被干净地定义。反函数通常写作 arcsin x、arccos x 与 arctan x（有时也写作 sin⁻¹x）。

2. Defining arcsin x | 反正弦函数 arcsin x

If y = arcsin x, then x = sin y, where the principal value of y is chosen in the interval -π/2 ≤ y ≤ π/2. The domain of arcsin x is -1 ≤ x ≤ 1, and its range is -π/2 ≤ y ≤ π/2. On this interval sine is strictly increasing, so arcsin x is also strictly increasing.

若 y = arcsin x，则 x = sin y，其中 y 的主值选在区间 -π/2 ≤ y ≤ π/2 内。arcsin x 的定义域为 -1 ≤ x ≤ 1，值域为 -π/2 ≤ y ≤ π/2。由于正弦函数在该区间上严格递增，arcsin x 也严格递增。

Three values are especially worth memorising:

三个特殊值特别值得记住：

arcsin(0) = 0 because sin(0) = 0

arcsin(0) = 0，因为 sin(0) = 0
arcsin(1) = π/2 because sin(π/2) = 1

arcsin(1) = π/2，因为 sin(π/2) = 1
arcsin(-1) = -π/2 because sin(-π/2) = -1

arcsin(-1) = -π/2，因为 sin(-π/2) = -1

3. Defining arccos x | 反余弦函数 arccos x

If y = arccos x, then x = cos y, where the principal value of y is chosen in the interval 0 ≤ y ≤ π. The domain of arccos x is -1 ≤ x ≤ 1, and its range is 0 ≤ y ≤ π. On this interval cosine is strictly decreasing, so arccos x is strictly decreasing as well.

若 y = arccos x，则 x = cos y，其中 y 的主值选在区间 0 ≤ y ≤ π 内。arccos x 的定义域为 -1 ≤ x ≤ 1，值域为 0 ≤ y ≤ π。由于余弦函数在该区间上严格递减，arccos x 也严格递减。

Key values include:

关键值包括：

arccos(0) = π/2 because cos(π/2) = 0

arccos(0) = π/2，因为 cos(π/2) = 0
arccos(1) = 0 because cos(0) = 1

arccos(1) = 0，因为 cos(0) = 1
arccos(-1) = π because cos(π) = -1

arccos(-1) = π，因为 cos(π) = -1

4. Defining arctan x | 反正切函数 arctan x

If y = arctan x, then x = tan y, where the principal value of y is chosen in the open interval -π/2 < y < π/2. Unlike arcsin x and arccos x, the domain of arctan x is all real numbers, ℝ. The range is the open interval -π/2 < y < π/2.

若 y = arctan x，则 x = tan y，其中 y 的主值选在开区间 -π/2 < y < π/2 内。与 arcsin x 和 arccos x 不同，arctan x 的定义域是全体实数 ℝ。其值域为开区间 -π/2 < y < π/2。

Because tan y approaches +∞ as y → (π/2)⁻ and -∞ as y → (-π/2)⁺, the graph of arctan x has two horizontal asymptotes, y = π/2 and y = -π/2. The function is strictly increasing and passes through the origin.

由于当 y → (π/2)⁻ 时 tan y 趋于 +∞，当 y → (-π/2)⁺ 时 tan y 趋于 -∞，arctan x 的图像有两条水平渐近线 y = π/2 与 y = -π/2。该函数严格递增且经过原点。

5. Summary Table of Domains and Ranges | 定义域与值域总结表

The three principal inverse trigonometric functions are compared below. You should be able to reproduce this table from memory in an exam.

下表对比三个主要反三角函数。你应当能够在考试中凭记忆写出这张表。

Function 函数	Domain 定义域	Principal Range 主值范围	Monotonicity 单调性
arcsin x	-1 ≤ x ≤ 1	-π/2 ≤ y ≤ π/2	Increasing 递增
arccos x	-1 ≤ x ≤ 1	0 ≤ y ≤ π	Decreasing 递减
arctan x	x ∈ ℝ	-π/2 < y < π/2	Increasing 递增

A common exam trap is writing the range of arcsin x as [0, π], which actually belongs to arccos x. Always match the range to the function explicitly.

一个常见考场陷阱是把 arcsin x 的值域写成 [0, π]，那其实是 arccos x 的值域。务必把值域与函数一一对应起来。

6. Graphs and Key Features | 图像与关键特征

The graph of y = arcsin x is obtained by reflecting the graph of y = sin x restricted to [-π/2, π/2] in the line y = x. Likewise, y = arccos x is the reflection of restricted cosine, and y = arctan x is the reflection of restricted tangent.

y = arcsin x 的图像是将限制在 [-π/2, π/2] 上的 y = sin x 关于直线 y = x 反射得到。同理，y = arccos x 是限制余弦的反射，y = arctan x 是限制正切的反射。

y = arcsin x passes through (-1, -π/2), (0, 0) and (1, π/2), with vertical-looking tangents at the endpoints.

y = arcsin x 经过 (-1, -π/2)、(0, 0) 和 (1, π/2)，在端点处切线近乎竖直。
y = arccos x passes through (-1, π), (0, π/2) and (1, 0), falling from left to right.

y = arccos x 经过 (-1, π)、(0, π/2) 和 (1, 0)，从左到右递减。
y = arctan x passes through the origin and approaches the asymptotes y = π/2 and y = -π/2 without crossing them.

y = arctan x 经过原点，并无限接近渐近线 y = π/2 与 y = -π/2 但永不相交。

7. The Identity arcsin x + arccos x = π/2 | 恒等式 arcsin x + arccos x = π/2

For every x in [-1, 1], the relationship arcsin x + arccos x = π/2 always holds. This is a direct analogue of the complementary angle identity sin θ = cos(π/2 – θ), written in inverse form.

对任意 x ∈ [-1, 1]，恒有 arcsin x + arccos x = π/2。这是余角恒等式 sin θ = cos(π/2 – θ) 的反函数形式，二者完全对应。

A quick proof: let θ = arcsin x. Then sin θ = x, so cos(π/2 – θ) = sin θ = x. Since θ ∈ [-π/2, π/2], we have π/2 – θ ∈ [0, π], which is exactly the principal range of arccos x. Hence arccos x = π/2 – θ, and arcsin x + arccos x = π/2.

简要证明：令 θ = arcsin x，则 sin θ = x，故 cos(π/2 – θ) = sin θ = x。由于 θ ∈ [-π/2, π/2]，可得 π/2 – θ ∈ [0, π]，恰好落在 arccos x 的主值范围内。因此 arccos x = π/2 – θ，即 arcsin x + arccos x = π/2。

Similarly, for x > 0, arctan x + arctan(1/x) = π/2. This follows from the symmetry of the tangent function about θ = π/4 and is useful in integration problems.Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com

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Inverse Trigonometric Functions | 反三角函数