📚 Inverse Trigonometric Functions | 反三角函数
In AQA A-Level Mathematics, inverse trigonometric functions appear in pure topics such as graphs, differentiation, integration and solving equations. They are not reciprocal functions: each one is defined by restricting a trigonometric function to a domain on which it is one-to-one, allowing a unique inverse to exist.
在 AQA A-Level 数学中，反三角函数出现在图像、求导、积分和解方程等纯数学主题中。它们并不是倒数函数：每个反三角函数都是通过限制三角函数的定义域，使其在该区间内一对一，从而得到唯一的反函数。
1. The Need for Inverse Trigonometric Functions | 为什么需要反三角函数
domains. For example, sin x takes the value 0.5 infinitely many times. A statement such as 'sin x = 0.5' infinitely many solutions unless we restrict the output values we are prepared to accept.
正弦、余弦和正切都是周期函数，因此在自然定义域内是多对一的。例如，sin x 在无穷多个点上取值为 0.5，所以像 'sin x = 0.5' 这样的表达式有无穷多个解，除非我们限制允许接受的输出值范围。
To make inverse functions possible, we choose a principal range for each inverse. In AQA questions, arcsin x, arccos x and arctan x always refer to these principal values.
为了让反函数成为可能，我们为每个反三角函数选择一个主值范围。在 AQA 考试题中，arcsin x、arccos x 和 arctan x 始终表示这些主值。
arcsin x is NOT the same as 1/sin x
arcsin x 与 1/sin x 不同
2. Defining arcsin, arccos and arctan | 定义 arcsin、arccos 和 arctan
The three principal inverse trigonometric functions are defined by stating both the equation and the allowed interval for the angle.
三个主值反三角函数通过同时给出方程和允许的角度区间来定义。
arcsin x = θ ⇔ sin θ = x, −π/2 ≤ θ ≤ π/2, −1 ≤ x ≤ 1
arcsin x = θ ⇔ sin θ = x，其中 −π/2 ≤ θ ≤ π/2，且 −1 ≤ x ≤ 1
arccos x = θ ⇔ cos θ = x, 0 ≤ θ ≤ π, −1 ≤ x ≤ 1
arccos x = θ ⇔ cos θ = x，其中 0 ≤ θ ≤ π，且 −1 ≤ x ≤ 1
arctan x = θ ⇔ tan θ = x, −π/2 < θ < π/2, x ∈ ℝ
arctan x = θ ⇔ tan θ = x，其中 −π/2 < θ < π/2，且 x ∈ ℝ
value range. The inequalities are strict for arctan because tangent is undefined at ±π/2.
在每个定义中，θ 所在的区间称为主值范围。对于 arctan，不等式是严格的，因为正切在 ±π/2 处无定义。
3. Domains and Ranges | 定义域与值域
You must memorise the domain and range of each inverse trigonometric function. Many AQA questions test whether an input is valid before any calculation can be carried out.
你必须牢记每个反三角函数的定义域和值域。许多 AQA 题目会先检验输入是否有效，然后才能进行计算。
| Function | Domain | Range |
|---|---|---|
| arcsin x | −1 ≤ x ≤ 1 | −π/2 ≤ arcsin x ≤ π/2 |
| arccos x | −1 ≤ x ≤ 1 | 0 ≤ arccos x ≤ π |
| arctan x | x ∈ ℝ | −π/2 < arctan x < π/2 |
In practice, outputs for arcsin and arctan are always acute, right or negative acute angles, while outputs for arccos are angles from 0 to π.
实际上，arcsin 和 arctan 的输出总是锐角、直角或负锐角，而 arccos 的输出是从 0 到 π 的角。
= −π/2 while arccos(−1) = π. These values show that the ranges are deliberately different.
例如，arcsin 1 = π/2，arccos 1 = 0；但 arcsin(−1) = −π/2，而 arccos(−1) = π。这些取值说明两者的值域是刻意不同的。
4. Graphs of the Inverse Functions | 反三角函数的图像
QA sketch questions, you need the general shape, key endpoints, intercepts and asymptotes.
反三角函数的图像是原限制函数关于直线 y = x 的对称图形。在 AQA 作图中，你需要掌握大致形状、关键端点、截距和渐近线。
y = arcsin x: passes through (0,0), endpoints (−1, −π/2) and (1, π/2)
y = arcsin x：经过 (0,0)，端点 (−1, −π/2) 和 (1, π/2)
y = arccos x: endpoints (−1, π) and (1, 0), crosses y-axis at (0, π/2)
y = arccos x：端点 (−1, π) 和 (1, 0)，与 y 轴交于 (0, π/2)
y = arctan x: passes through (0,0), asymptotes y = ±π/2
y = arctan x：经过 (0,0)，渐近线 y = ±π/2
strictly decreasing. The function y = arctan x is strictly increasing on all real numbers.
函数 y = arcsin x 在其定义域内严格递增，而 y = arccos x 严格递减。函数 y = arctan x 在所有实数上严格递增。
, arccos is not odd; instead it has the property arccos(−x) = π − arccos x for −1 ≤ x ≤ 1.
注意，arcsin 和 arctan 是奇函数：arcsin(−x) = −arcsin x，arctan(−x) = −arctan x。然而，arccos 不是奇函数；对于 −1 ≤ x ≤ 1，它具有性质 arccos(−x) = π − arccos x。
5. Key Exact Values | 重要精确值
AQA questions often expect exact answers in radians. The following table summarises the most common exact values for arcsine and arccosine.
AQA 题目通常要求以弧度给出精确答案。下表总结了最常用的反正弦和反余弦精确值。
| x | arcsin x | arccos x |
|---|---|---|
| −1 | −π/2 | π |
| −√3/2 | −π/3 | 5π/6 |
| −1/√2 | −π/4 | 3π/4 |
| −1/2 | −π/6 | 2π/3 |
| 0 | 0 | π/2 |
| 1/2 | π/6 | π/3 |
| 1/√2 | π/4 | π/4 |
| √3/2 | π/3 | π/6 |
| 1 | π/2 | 0 |
For arctan, the most useful exact values are arctan 0 = 0, arctan(1/√3) = π/6, arctan 1 = π/4 and arctan(√3) = π/3.
对于 arctan，最常用的精确值是 arctan 0 = 0，arctan(1/√3) = π/6，arctan 1 = π/4 以及 arctan(√3) = π/3。
Remember that negative inputs produce negative outputs for arcsin and arctan, but arccos(−x) is always in the second quadrant and is greater than π/2.
请记住，对于 arcsin 和 arctan，负输入产生负输出；但 arccos(−x) 始终位于第二象限，且大于 π/2。
6. Symmetry and Basic Identities | 对称性与基本恒等式
Inverse trigonometric functions satisfy several identities that appear in AQA proofs and simplifications. The most important identity is the complementary relationship between arcsin and arccos.
反三角函数满足若干恒等式，这些恒等式常出现在 AQA 的证明和化简题中。其中最重要的是 arcsin 与 arccos 之间的互补关系。
arcsin x + arccos x = π/2 for all −1 ≤ x ≤ 1
对于所有 −1 ≤ x ≤ 1，arcsin x + arccos x = π/2
This identity can be proved by considering a right-angled triangle or by differentiating the left-hand side and showing it is constant.
这个恒等式可以通过构造直角三角形来证明，也可以对左边求导并证明其导数为零，从而说明它是常数。
Other useful relationships include:
其他有用的关系式包括：
sin(arccos x) = cos(arcsin x) = √1 − x², for −1 ≤ x ≤ 1
sin(arccos x) = cos(arcsin x) = √1 − x²，其中 −1 ≤ x ≤ 1
tan(arcsin x) = x / √1 − x², for −1 < x < 1
tan(arcsin x) = x / √1 − x²，其中 −1 < x < 1
x > 0, and it equals −π/2 for x < 0. This is another standard result used in AQA multiple-choice and proof questions.
此外，当 x > 0 时，arctan x + arctan(1/x) = π/2；当 x < 0 时，其值为 −π/2。这是 AQA 选择题和证明题中的另一个标准结论。
7. Differentiating Inverse Trigonometric Functions | 反三角函数的求导
und by implicit differentiation. Let y = arcsin x. Then sin y = x and cos y dy/dx = 1, so:
arcsin x 的导数可以通过隐函数求导得到。设 y = arcsin x，则 sin y = x，且 cos y dy/dx = 1，因此：
d/dx (arcsin x) = 1 / √1 − x², for −1 < x <
Published by TutorHao | A-Level Mathematics Revision Series | aleveler.com 更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图