Inverse Trigonometric Functions: Concepts and Properties | 反三角函数的概念与性质

📚 Inverse Trigonometric Functions: Concepts and Properties | 反三角函数的概念与性质

Inverse trigonometric functions are one of the key topics in IB Mathematics, bridging trigonometry, algebra and calculus. They allow us to recover angles from given ratios, solve trigonometric equations and evaluate a family of important integrals.

反三角函数是IB数学中的核心专题之一，它将三角学、代数和微积分联系起来。借助反三角函数，我们可以由给定的比值求角、解三角方程，并计算一类重要的积分。

1. Why Inverse Trigonometric Functions? | 为什么要学习反三角函数？

Trigonometric functions such as sin, cos and tan take an angle as input and give a ratio as output. Conversely, we often need to find the angle that produces a known ratio. For example, if sin θ = ½, what is θ? The answer requires an inverse operation.

诸如sin、cos和tan这样的三角函数以角度为输入，输出一个比值。反之，我们常常需要找出产生已知比值的角度。例如，若sin θ = ½，那么θ是多少？回答这个问题就需要逆运算。

Since sine, cosine and tangent are periodic, they are not one-to-one over their entire domain. To define inverses, we must restrict the domain of each original function to an interval where it is strictly monotonic.

由于正弦、余弦和正切是周期函数，它们在整个定义域上不是一一对应的。为了定义反函数，我们必须将原函数的定义域限制在一个严格单调的区间上。

2. Definitions and Notation | 定义与记法

The standard notation for inverse trigonometric functions is arcsin x, arccos x and arctan x. Some textbooks also use sin⁻¹ x, cos⁻¹ x and tan⁻¹ x, but this can be confused with reciprocal functions.

反三角函数的标准记法是arcsin x、arccos x和arctan x。有些教材也使用sin⁻¹ x、cos⁻¹ x和tan⁻¹ x，但这些记法容易与倒数函数混淆。

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

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

Similarly, arccos x is the inverse of cos x on [0, π], and arctan x is the inverse of tan x on (-π/2, π/2).

类似地，arccos x是cos x在[0, π]上的反函数，arctan x是tan x在(-π/2, π/2)上的反函数。

3. Domain and Range | 定义域与值域

Each inverse trigonometric function has a restricted domain and range. These restrictions are essential for ensuring that the inverse is a well-defined function.

每个反三角函数都有确定的定义域和值域。这些限制是保证反函数为合法函数的关键。

Function / 函数	Domain / 定义域	Range / 值域
y = arcsin x	x ∈ [-1, 1]	y ∈ [-π/2, π/2]
y = arccos x	x ∈ [-1, 1]	y ∈ [0, π]
y = arctan x	x ∈ ℝ	y ∈ (-π/2, π/2)

Notice that arctan x can accept any real number, but its output always lies strictly between -π/2 and π/2. This is very useful when working with limits and horizontal asymptotes.

注意arctan x可以接受任意实数，但其输出始终严格位于-π/2与π/2之间。这在处理极限和水平渐近线时非常有用。

4. Graphs and Symmetry | 图像与对称性

The graph of each inverse trigonometric function is the reflection of the restricted original function across the line y = x. Since the restricted sine, cosine and tangent are monotonic, their reflections are also monotonic.

每个反三角函数的图像都是原函数在限制区间上的图像关于直线y = x的反射。由于限制后的正弦、余弦和正切是单调的，其反射图像也是单调的。

arcsin x is increasing and symmetric about the origin, meaning arcsin(-x) = -arcsin x. arccos x is decreasing and has no odd or even symmetry. arctan x is increasing, symmetric about the origin, and approaches -π/2 as x → -∞ and π/2 as x → +∞.

arcsin x是增函数，且关于原点对称，即arcsin(-x) = -arcsin x。arccos x是减函数，且没有奇偶对称性。arctan x是增函数，关于原点对称，并且当x → -∞时趋于-π/2，当x → +∞时趋于π/2。

5. Key Identities | 重要恒等式

The following identities are frequently needed in IB exams. They simplify expressions and allow conversion between inverse trigonometric functions.

以下恒等式在IB考试中经常用到。它们可以化简表达式，并允许在不同反三角函数之间进行转换。

arcsin x + arccos x = π/2 for all x ∈ [-1, 1]

对一切x ∈ [-1, 1]，arcsin x + arccos x = π/2。
arctan x + arctan(1/x) = π/2 for x > 0

当x > 0时，arctan x + arctan(1/x) = π/2。
arcsin(-x) = -arcsin x and arctan(-x) = -arctan x

arcsin(-x) = -arcsin x，arctan(-x) = -arctan x。
arccos(-x) = π – arccos x

arccos(-x) = π – arccos x。

These identities are derived from the symmetric properties of the restricted domains. For example, the first identity follows from the fact that the angles arcsin x and arccos x must sum to a right angle in a right-angled triangle.

这些恒等式可由限制定义域的对称性推出。例如，第一个恒等式说明arcsin x与arccos x这两个角在一个直角三角形中互为余角，因此和为直角。

6. Composing Functions and Simplification | 复合函数与化简

When composing trigonometric functions with inverse trigonometric functions, the result depends on the domain and range. The most basic rules are:

将三角函数与反三角函数复合时，结果取决于定义域和值域。最基本的规则是：

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, arcsin(sin θ) is not always θ. It equals θ only when θ ∈ [-π/2, π/2]. For other values of θ, the output is the unique angle in [-π/2, π/2] that has the same sine value. Similar restrictions apply to arccos(cos θ) and arctan(tan θ).

然而，arcsin(sin θ)并不总是等于θ。只有θ ∈ [-π/2, π/2]时才成立。对于其他θ，输出是[-π/2, π/2]中具有相同正弦值的唯一角度。arccos(cos θ)和arctan(tan θ)也有类似的限制。

Expressions such as sin(arccos x) can be simplified using right-triangle diagrams or trigonometric identities:

像sin(arccos x)这样的表达式可以通过构造直角三角形或使用三角恒等式化简：

sin(arccos x) = √(1 – x²)

This is valid for x ∈ [-1, 1], and the positive root is chosen because arccos x ∈ [0, π] where sine is non-negative.

该式在x ∈ [-1, 1]时成立，取正根是因为arccos x ∈ [0, π]，在此区间内正弦非负。

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

Differentiation of inverse trigonometric functions is a standard IB Calculus topic. The derivatives are obtained by implicit differentiation of the defining equations.

反三角函数求导是IB微积分中的标准内容。这些导数可以通过对定义方程进行隐函数求导得到。

d/dx [arcsin x] = 1/√(1 – x²), for -1 < x < 1

d/dx [arccos x] = -1/√(1 – x²), for -1 < x < 1

d/dx [arctan x] = 1/(1 + x²), for all real x

Notice that the derivative of arcsin x and arccos x differ only by a sign. The domain restrictions exclude the endpoints ±1, where the derivative is undefined because the tangent line becomes vertical.

注意arcsin x与arccos x的导数只相差一个符号。定义域排除了端点±1，因为在这些点处切线垂直，导数不存在。

These formulas are often combined with the chain rule, for example d/dx [arcsin(3x)] = 3/√(1 – 9x²).

这些公式常与链式法则结合使用，例如d/dx [arcsin(3x)] = 3/√(1 – 9x²)。

8. Integrals Involving Inverse Trigonometric Functions | 反三角函数的积分

Because differentiation of inverse trigonometric functions produces simple algebraic expressions, their antiderivatives appear naturally when integrating rational or radical functions.

由于反三角函数求导后产生简单的代数表达式，其原函数自然地出现在有理函数或根式函数的积分中。

∫ 1/√(1 – x²) dx = arcsin x + C

∫ 1/(1 + x²) dx = arctan x + C

A useful variant is ∫ 1/(a² + x²) dx = (1/a) arctan(x/a) + C. Similarly, ∫ 1/√(a² – x²) dx = arcsin(x/a) + C for a > 0.

常用的变式是∫ 1/(a² + x²) dx = (1/a) arctan(x/a) + C。类似地，当a > 0时，∫ 1/√(a² – x²) dx = arcsin(x/a) + C。

In IB exams, you may need to rewrite an integrand using algebraic manipulation before applying these standard forms. Completing the square is a common technique.

在IB考试中，你可能需要先通过代数变形改写被积函数，再套用这些标准形式。配方法是一种常用技巧。

9. Common Mistakes and Misconceptions | 常见错误与易混淆点

Students often confuse inverse trigonometric functions with reciprocal trigonometric functions. Remember that arcsin x is not the same as (sin x)⁻¹ = csc x.

学生常常把反三角函数与倒数三角函数混淆。请记住，arcsin x不等同于(sin x)⁻¹ = csc x。

Mistake: Assuming arcsin(sin θ) = θ for all θ. This is only true when θ lies in the restricted interval.

错误：认为arcsin(sin θ) = θ对所有θ都成立。实际上只有θ在限制区间内时才成立。
Mistake: Forgetting the domain restriction when solving equations, e.g. writing arcsin(2) = some real number. Since 2 ∉ [-1, 1], arcsin(2) is undefined.

错误：解方程时忘记定义域限制，例如写出arcsin(2)等于某个实数。由于2 ∉ [-1, 1]，arcsin(2)无意义。
Mistake: Using the wrong branch for arccos when evaluating arccos(cos θ). The result must lie in [0, π], not simply θ.

错误：计算arccos(cos θ)时选错分支。结果必须位于[0, π]内，而不一定等于θ。
Mistake: Dropping the absolute value or sign when simplifying √(1 – cos²θ). The sign should match the interval of the original angle.

错误：化简√(1 – cos²θ)时忽略绝对值或符号。符号应与原角所在的区间一致。

10. Problem-Solving Strategies and IB Exam Tips | 解题策略与IB考点提示

In IB Paper 1, inverse trigonometric functions often appear in differentiation, integration or algebraic simplification. In Paper 2, they may be used to model periodic phenomena or solve trigonometric equations with restricted domains.

在IB Paper 1中，反三角函数常出现在求导、积分或代数化简中。在Paper 2中，它们可能用于模拟周期现象或在限制定义域内求解三角方程。

Start by identifying the restricted range. When solving an equation involving arcsin, arccos or arctan, verify that the input lies in the function’s domain. Use the identities to convert expressions into simpler forms.

首先确定限制值域。在解含有arcsin、arccos或arctan的方程时，验证输入是否在函数定义域内。利用恒等式将表达式转化为更简单的形式。

For calculus problems, memorize the derivative and integral formulas exactly. When using the chain rule, remember to multiply by the derivative of the inner function. For integration, match the integrand to the standard form by completing the square or using substitution.

对于微积分问题，要准确记忆导数和积分公式。使用链式法则时，记得乘以内层函数的导数。对于积分，通过配方或换元使被积函数匹配标准形式。

Finally, practice sketching the graphs of arcsin x, arccos x and arctan x. Visualising their horizontal asymptotes and symmetry will help you answer range questions quickly and correctly.

最后，多练习画arcsin x、arccos x和arctan x的图像。直观理解它们的水平渐近线和对称性，可以帮助你快速准确地回答值域问题。

Published by TutorHao | Mathematics Revision Series | aleveler.com

更多咨询请联系16621398022（同微信）

Inverse Trigonometric Functions: Concepts and Properties | 反三角函数的概念与性质