IB Mathematics: Domain and Graphs of Inverse Trigonometric Functions | IB数学：反三角函数的定义域与图像

📚 IB Mathematics: Domain and Graphs of Inverse Trigonometric Functions | IB数学：反三角函数的定义域与图像

Inverse trigonometric functions are a core topic in IB Mathematics: Analysis and Approaches HL, and they often appear in questions on transformations, equations, calculus and paper-style problem solving. Because the original sine, cosine and tangent functions are periodic, they are not one-to-one over their natural domains. To define an inverse, we must first restrict each trigonometric function to a carefully chosen interval. That chosen interval produces the principal branch of the inverse, and it determines the exact domain and range that students must memorise.

反三角函数是IB数学分析与方法HL中的核心内容，也经常出现在图像变换、方程求解、微积分以及试卷压轴题中。由于原始的正弦、余弦和正切函数具有周期性，它们在其自然定义域上不是一一对应的。为了定义反函数，我们必须先把每个三角函数限制到一个经过认真选择的区间上。这个被选中的区间产生反函数的主支，也决定了学生必须准确记住的定义域和值域。

1. Why Do We Need Inverse Trigonometric Functions? | 为什么需要反三角函数？

For a function to have an inverse, it must be one-to-one. A horizontal line intersects the graph of y = sin x infinitely many times over the real line, so sine as a complete function has no inverse. The same is true for cosine and tangent.

一个函数要有反函数，必须是一一对应的。在整个实数范围内，水平线与 y = sin x 的图像相交无限多次，因此完整的正弦函数没有反函数。余弦和正切也是如此。

We solve this by restricting the original function to a small interval where it is monotonic. On that interval, the function is one-to-one and therefore invertible. The inverse so obtained is called the principal inverse trigonometric function.

解决方法是把原函数限制在一个较小且单调的区间上。在这个区间内，函数是一一对应的，因此可以求反函数。这样得到的反函数称为反三角函数的主支。

The inverse of a restricted function does not simply mean “reciprocal”. For example, sin⁻¹ x is an inverse function, not 1/sin x. This distinction is essential in IB exams.

反函数并不意味着“倒数”。例如，sin⁻¹ x 是反函数，而不是 1/sin x。这个区别在IB考试中至关重要。

2. The Inverse of Sine: y = arcsin x | 正弦函数的反函数：y = arcsin x

Restrict the sine function to the interval [-π/2, π/2]. On this interval, sine is increasing and every value in [-1,1] is reached exactly once. The inverse of this restricted function is

将正弦函数限制在区间 [-π/2, π/2] 上。在这个区间内，正弦函数单调递增，[-1,1] 中的每个值都恰好被取到一次。这个限制函数的反函数是

y = arcsin x = sin⁻¹ x, x ∈ [-1,1], y ∈ [-π/2, π/2]

• The domain of arcsin x is [-1,1].

  arcsin x 的定义域是 [-1,1]。

• The range of arcsin x is [-π/2, π/2].

  arcsin x 的值域是 [-π/2, π/2]。

• The graph is increasing, contains (-1, -π/2), (0,0) and (1, π/2), and is symmetric about the origin.

  图像单调递增，经过 (-1, -π/2)、(0,0) 和 (1, π/2)，并且关于原点对称。

The graph of y = arcsin x is the reflection of the restricted sine graph across the line y = x. The endpoint (-1, -π/2) is included because both -1 and π/2 belong to the domain/range.

y = arcsin x 的图像是受限制的正弦图像关于直线 y = x 的反射。端点 (-1, -π/2) 是实心点，因为 -1 和 -π/2 都在定义域和值域内。

3. The Inverse of Cosine: y = arccos x | 余弦函数的反函数：y = arccos x

For cosine, we choose the restricted interval [0, π]. On this interval cosine is continuous and decreasing, and it takes every value from 1 down to -1 exactly once. Its inverse is

对于余弦函数，我们选择限制区间 [0, π]。在这个区间上，余弦函数连续且单调递减，并且从 1 到 -1 的每个值都恰好被取到一次。其余函数是

y = arccos x = cos⁻¹ x, x ∈ [-1,1], y ∈ [0, π]

• The domain of arccos x is [-1,1].

  arccos x 的定义域是 [-1,1]。

• The range of arccos x is [0, π].

  arccos x 的值域是 [0, π]。

• The graph is decreasing, connecting (1,0) to (-1, π).

  图像单调递减，连接点 (1,0) 和 (-1, π)。

Unlike arcsin, arccos is neither even nor odd. The graph has no symmetry about the origin or the y-axis.

与 arcsin 不同，arccos 既不是奇函数也不是偶函数。其图像关于原点或 y 轴均不对称。

4. The Inverse of Tangent: y = arctan x | 正切函数的反函数：y = arctan x

Tangent is restricted to the open interval (-π/2, π/2). On this interval it is increasing and its range is all real numbers. Therefore the inverse has a larger domain than arcsin or arccos.

正切函数被限制在开区间 (-π/2, π/2) 上。在这个区间上它单调递增，并且值域是整个实数集。因此它的反函数比 arcsin 或 arccos 有更大的定义域。

y = arctan x = tan⁻¹ x, x ∈ ℝ, y ∈ (-π/2, π/2)

• The domain of arctan x is all real numbers, ℝ.

  arctan x 的定义域是所有实数 ℝ。

• The range of arctan x is (-π/2, π/2).

  arctan x 的值域是 (-π/2, π/2)。

• The graph is increasing and has horizontal asymptotes y = π/2 and y = -π/2.

  图像单调递增，并且有水平渐近线 y = π/2 和 y = -π/2。

The vertical asymptotes of tan x become horizontal asymptotes for arctan x. As x approaches +∞, arctan x approaches π/2 but never reaches it.

tan x 的垂直渐近线变成了 arctan x 的水平渐近线。当 x 趋向 +∞ 时，arctan x 趋向 π/2，但永远达不到它。

5. Quick Reference Table | 速查表

The table below summarises the essential facts for the three inverse trigonometric functions that appear most frequently in IB questions.

下表总结了在IB题目中出现

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