📚 Inverse Trigonometric Functions | 反三角函数
Inverse trigonometric functions, also known as arcus functions, are essential tools in calculus and algebra. They allow us to determine an angle given the value of a trigonometric ratio. In the IB Mathematics: Analysis and Approaches HL course, a solid understanding of arcsin, arccos, and arctan is required, along with their domains, ranges, graphs, derivatives, and integrals. This article provides a comprehensive review with exam-focused explanations and examples.
反三角函数（亦称反圆函数）是微积分和代数中的重要工具。它们使我们能够根据三角比的值求出对应的角。在IB数学分析与方法HL课程中，需要对反正弦、反余弦和反正切有扎实的理解，并掌握它们的定义域、值域、图形、导数和积分。本文提供全面的复习，包括紧扣考点的解释与实例。
1. What Are Inverse Trigonometric Functions? | 什么是反三角函数？
iting θ = arcsin(x) or θ = sin⁻¹(x). Because sine, cosine, and tangent are periodic, their "inverses" ions unless we restrict their domains. This restriction gives rise to the principal values – the agreed ranges that make each inverse relation a proper function.
反三角函数将标准正弦、余弦和正切函数的作用反转。例如，若 sin(θ) = x，我们可通过写作 θ = arcsin(x) 或 θ = sin⁻¹(x) 来求出 θ。由于正弦、余弦和正切都是周期函数，除非限定其定义域，否则它们的“反函数”并不是真正的函数。这种限定就产生了主值概念——即约定的值域，使得每个反关系成为一个真正的函数。
tion sin⁻¹(x) must never be confused with the reciprocal 1/sin(x). This is a very common pitfall, and examiners frequently test this understanding.
在IB HL数学中，你应当能熟练地处理 y = sin⁻¹(x)、y = cos⁻¹(x) 和 y = tan⁻¹(x)。切记，符号 sin⁻¹(x) 绝不能与倒数 1/sin(x) 混淆。这是一个极为常见的陷阱，考官经常考查这一点。
2. Definitions and Principal Ranges | 定义与主值范围
Each inverse trigonometric function is defined by a specific trigonometric equation together with a restricted interval that ensures it is one-to-one. The formal definitions are:
每个反三角函数都由一个特定的三角方程及一个限定的区间来定义，以保证其是一一映射。正式定义如下：
arcsin: y = sin⁻¹(x) ⇔ sin y = x, where y ∈ [−π/2, π/2]
反正弦：y = sin⁻¹(x) ⇔ sin y = x，其中 y ∈ [−π/2, π/2]
arccos: y = cos⁻¹(x) ⇔ cos y = x, where y ∈ [0, π]
反余弦：y = cos⁻¹(x) ⇔ cos y = x，其中 y ∈ [0, π]
arctan: y = tan⁻¹(x) ⇔ tan y = x, where y ∈ (−π/2, π/2)
反正切：y = tan⁻¹(x) ⇔ tan y = x，其中 y ∈ (−π/2, π/2)
: its range never actually reaches ±π/2, though it approaches these values asymptotically.
这些限制后的值域被称为主值范围。它们保证定义域内的每个输入 x 都恰好对应一个输出 y。注意反正切的值域是开区间：虽然它无限趋近 ±π/2，但永远不会等于这两个值。
3. Domain and Range in Detail | 定义域与值域详解
Memorising the domains and ranges is essential for graphing, solving equations, and performing compositions.
熟记定义域和值域对于绘图、解方程以及进行函数的复合运算至关重要。
n (−∞, ∞) because the tangent function can take any real number. Its range is (−π/2, π/2).
arcsin x 和 arccos x 的定义域是 [−1, 1]，因为 sin 和 cos 的输出被限制在 −1 到 1 之间。arcsin x 的值域是 [−π/2, π/2]；arccos x 的值域是 [0, π]。而 arctan x 的定义域是 (−∞, ∞)，因为正切函数可以取任意实数值；它的值域是 (−π/2, π/2)。
should be able to instantly recall these restrictions. For example, a question might ask: "What is the domain of f(x) = arcsin(2x − 3)?" You must solve −1 ≤ 2x − 3 ≤ 1 to obtain 1 ≤ x ≤ 2.
为在考试中取得成功，你应当能立即想起这些限制。例如，题目可能会问：“函数 f(x) = arcsin(2x − 3) 的定义域是什么？”你必须解 −1 ≤ 2x − 3 ≤ 1，得出 1 ≤ x ≤ 2。
4. Graphs and Key Features | 图形与关键特征
ections of restricted sections of the original trigonometric graphs across the line y = x.
反三角函数的图形是将原三角函数在限制区间内的图像沿直线 y = x 进行反射得到的。
that is increasing throughout its domain, with horizontal asymptotes y = −π/2 and y = π/2.
y = arcsin x 的图像单调递增，经过原点，并在 x = ±1 处有垂直切线。y = arccos x 的图像单调递减，从点 (1, 0) 开始，到点 (−1, π) 结束。y = arctan x 的图像是一条贯穿其定义域的 S 形递增曲线，水平渐近线为 y = −π/2 与 y = π/2。
Knowing these shapes helps you justify why, for instance, arcsin x is an odd function (symmetric about the origin), while arccos x is neither odd nor even.
掌握这些图形形状有助于你解释为什么 arcsin x 是奇函数（关于原点对称），而 arccos x 既非奇函数也非偶函数。
5. Symmetry and Fundamental Identities | 对称性与基本恒等式
Several identities must be at your fingertips. They arise from the symmetry of the unit circle and the complementary nature of sine and cosine.
有几个恒等式你必须能够熟练运用。它们源于单位圆的
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