📚 Inverse Trigonometric Functions | 反三角函数
Inverse trigonometric functions, often written as arcsin, arccos and arctan (or sin−1, cos−1, tan−1), play a central role in A-Level Edexcel Mathematics. They allow us to recover the angle when we know the value of a trigonometric ratio. Because sine, cosine and tangent are periodic and not one-to-one on their natural domains, we must first restrict their domains before we can talk about their inverses. This article explores the definitions, domains, ranges, graphs, derivatives, integrals and key identities of these functions, helping you build the confidence needed for exam success.
反三角函数通常记作 arcsin、arccos 和 arctan（或 sin−1、cos−1、tan−1），在A-Level Edexcel数学中扮演着核心角色。它们让我们在已知一个三角比值时能够还原出角度。由于正弦、余弦和正切在其自然定义域上是周期函数且不是一一映射，我们必须先限制定义域才能讨论它们的反函数。本文探讨这些函数的定义、定义域、值域、图像、导数、积分以及重要恒等式，帮助你在考试中树立信心。
1. Why We Need Restricted Domains | 为什么需要限制定义域
ontal line y = 0 cuts the graph infinitely many times. An inverse function can only exist if we restrict each function to an interval where it is monotonic and covers its full range exactly once. For sin x, the standard restricted domain is [−π/2, π/2]; on this interval the function increases from −1 to 1, each value is taken exactly once, and the range is [−1, 1].
三角函数 sin x、cos x 和 tan x 在整个实数集上不是单射（一一映射）。例如 sin 0 = 0 且 sin π = 0，因此水平线 y = 0 会与图像相交无数次。只有当我们将每个函数限制在一个单调且恰好完整覆盖其值域的区间上时，反函数才可能存在。对于 sin x，标准的限制定义域是 [−π/2, π/2]；在该区间上函数从 −1 增加到 1，每一个值恰好取一次，值域为 [−1, 1]。
t soars from −∞ to ∞ and passes the horizontal line test. These choices produce the principal branches that define the inverse trigonometric functions we use throughout calculus and equation solving.
类似地，cos x 被限制在 [0, π] 上，在此期间函数从 1 递减到 −1。正切函数则被限制在开区间 (−π/2, π/2) 上，此时曲线从 −∞ 飞升至 ∞ 并且通过水平线检验。这些选择给出了定义反三角函数的主值分支，我们在微积分和解方程中持续使用它们。
2. Defining arcsin(x) | 反正弦函数的定义
The arcsine function, denoted arcsin x or sin−1 he principal range [−π/2, π/2]. This means the output of arcsine is always an angle between −π/2 and π/2 inclusive. The notation sin−1 x is not a reciprocal; it stands for the inverse function. The domain of arcsin x is [−1, 1], because the sine of any angle in the principal range cannot exceed 1 or drop below −1.
反正弦函数，记作 arcsin x 或 sin−1 x，定义为 y = arcsin x 当且仅当 sin y = x 且 y 落在主值范围 [−π/2, π/2] 内。这意味着反正弦的输出总是介于 −π/2 和 π/2 之间的角度。符号 sin−1 x 不是倒数，它代表反函数。反正弦的定义域为 [−1, 1]，因为主值范围内任何角度的正弦不可能超出 1 或低于 −1。
For example, arcsin(1/2) asks: 'which angle y in [−π/2, π/2] has sin y = 1/2?' se sin(−π/2) = −1. The function is odd, so arcsin(−x) = −arcsin x for all x in the domain.
例如，arcsin(1/2) 在问：“在 [−π/2, π/2] 中哪个角度 y 满足 sin y = 1/2？”答案是 π/6。同理，arcsin(−1) = −π/2，因为 sin(−π/2) = −1。该函数是奇函数，因此对于定义域内的任意 x，有 arcsin(−x) = −arcsin x。
3. Defining arccos(x) | 反余弦函数的定义
The arccosine function, arccos x or cos−1 x, is defined by y = arccos x ⇔ while decreasing monotonically. The domain of arccos x is [−1, 1] and the range is [0, π].
反余弦函数，arccos x 或 cos−1 x，定义为 y = arccos x ⇔ cos y = x，其中 y 被限制在区间 [0, π] 上。这一主值范围的选择保证了 cos y 在单调递减的过程中恰好取遍 −1 到 1 之间的每一个值。反余弦的定义域是 [−1, 1]，值域是 [0, π]。
π − π/3 = 2π/3. The smallest output is 0 (when x = 1) and the largest is π (when x = −1).
与反正弦不同，反余弦不是奇函数，它满足 arccos(−x) = π − arccos x。例如，arccos(1/2) = π/3，而 arccos(−1/2) = π − π/3 = 2π/3。最小输出值是 0（当 x = 1），最大输出值是 π（当 x = −1）。
4. Defining arctan(x) | 反正切函数的定义
The arctangent function, arctan x or tan−1 x, is defined by y = arctan x ⇔ l real numbers precisely once on this interval, the domain of arctan x is all real numbers ℝ proached only as limits; they are not actually attained, giving two horizontal asymptotes.
反正切函数，arctan x 或 tan−1 x，定义为 y = arctan x ⇔ tan y = x，其中 y 取自开区间 (−π/2, π/2)。由于正切函数在该区间上恰好遍历所有实数，反正切的定义域是整个实数集 ℝ，值域是 (−π/2, π/2)。值 −π/2 和 π/2 只能作为极限无限接近，实际上并不会取到，从而形成两条水平渐近线。
Arctan is an odd function: arctan(−x) = −arctan x. For example, arctan(1) = π/4, arctan(√3) = π/3, and arctan(0) = 0. The function grows very slowly and approaches π/2 as x → ∞.
反正切是奇函数：arctan(−x) = −arctan x。例如，arctan(1) = π/4，arctan(√3) = π/3，arctan(0) = 0。该函数增长非常缓慢，当 x → ∞ 时趋于 π/2。
5. Summary Table of Domains and Ranges | 定义域与值域总结
Memorising the precise domain and range of each inverse trigonometric function is essential for answering exam questions accurately. The table below collects the key facts.
记住每个反三角函数的精确定义域和值域对于准确回答考题至关重要。下表汇集了关键信息。
| Function | Domain | Range (principal values) |
|---|---|---|
| y = arcsin x (sin−1 x) | [−1, 1] | [−π/2, π/2] |
| y = arccos x (cos−1 x) | [−1, 1] | [0, π] |
| y = arctan x (tan−1 x) | ℝ | (−π/2, π/2) |
Notice that arcsin and arccos share the same domain [−1, 1], but their ranges are different. For arctan, any real input is allowed. Whenever you solve an equation involving these functions, always check that the values lie within the principal ranges and that the arguments are within the domains.
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