📚 Inverse Trigonometric Functions | 反三角函数
Inverse trigonometric functions reverse the sine, cosine and tangent operations. Because trigonometric functions are periodic, their inverses only exist after restricting their domains. In AQA A Level Mathematics you need to know their definitions, principal values, graphs, derivatives, integrals and applications.
反三角函数是正弦、余弦和正切运算的逆运算。由于三角函数具有周期性，只有限制定义域后它们才有反函数。在 AQA A Level 数学中，你需要掌握反三角函数的定义、主值、图像、导数、积分以及应用。
1. What Are Inverse Trigonometric Functions? | 什么是反三角函数？
inverse operation is written x = sin⁻¹ y, also called arcsin y. The notation sin⁻¹ x means 'the angle whose sine is x'; it does not mean the reciprocal (sin x)⁻¹.
如果 y = sin x，那么逆运算写作 x = sin⁻¹ y，也称为 arcsin y。符号 sin⁻¹ x 表示“正弦为 x 的角”，而不是倒数 (sin x)⁻¹。
Similar definitions apply to cos⁻¹ x and tan⁻¹ x. They answer questions such as: which angle in the principal range has cosine 0.5?
类似地，cos⁻¹ x 和 tan⁻¹ x 分别表示余弦和正切的逆运算。它们回答诸如“主值范围内哪个角的余弦为 0.5”这样的问题。
2. Domains, Ranges and Principal Values | 定义域、值域与主值
To make the trigonometric functions one-to-one, we restrict their domains. The inverse functions then have the following domains and principal ranges.
为了使三角函数一一对应，我们需要限制它们的定义域。反三角函数因此具有以下定义域和主值范围。
| Function | Domain | Principal range |
| sin⁻¹ x | −1 ≤ x ≤ 1 | −π/2 ≤ sin⁻¹ x ≤ π/2 |
| cos⁻¹ x | −1 ≤ x ≤ 1 | 0 ≤ cos⁻¹ x ≤ π |
| tan⁻¹ x | all real x | −π/2 < tan⁻¹ x < π/2 |
You must learn these ranges by heart. They guarantee that each input gives exactly one angle, called the principal value.
你必须熟记这些范围。它们保证每个输入都对应唯一一个角，称为主值。
3. Graphs of arcsin x, arccos x and arctan x | 反正弦、反余弦与反正切函数的图像
stricted to [−π/2, π/2] in the line y = x. It passes through the origin and is increasing.
y = sin⁻¹ x 的图像是 y = sin x 在 [−π/2, π/2] 上关于直线 y = x 的对称图像。它经过原点，且单调递增。
the reflection of y = cos x restricted to [0, π]. It is decreasing, from (1,0) to (−1,π).
y = cos⁻¹ x 是 y = cos x 在 [0, π] 上关于直线 y = x 的对称图像。它单调递减，从 (1,0) 到 (−1,π)。
= tan⁻¹ x is increasing for all real x and has horizontal asymptotes y = π/2 and y = −π/2.
y = tan⁻¹ x 在整个实数集上单调递增，并有水平渐近线 y = π/2 和 y = −π/2。
symmetries: arcsin(−x) = −arcsin x, arccos(−x) = π − arccos x, and arctan(−x) = −arctan x.
图像还显示出对称性：arcsin(−x) = −arcsin x，arccos(−x) = π − arccos x，arctan(−x) = −arctan x。
4. Key Identities for Inverse Trig Functions | 反三角函数的重要恒等式
in(sin⁻¹ x) = x, cos(cos⁻¹ x) = x, tan(tan⁻¹ x) = x, provided x is in the relevant domain.
最直接的恒等式来自定义：sin(sin⁻¹ x) = x，cos(cos⁻¹ x) = x，tan(tan⁻¹ x) = x，前提是 x 在相应的定义域内。
ies in [−π/2, π/2]. For other angles, you must adjust by an integer multiple of π or use symmetry.
但 sin⁻¹(sin x) = x 只有当 x 在 [−π/2, π/2] 内才成立。对于其他角，你必须通过 π 的整数倍或对称性进行调整。
Pythagorean identities give very useful results: cos(sin⁻¹ x) = √1 − x², sin(cos⁻¹ x) = √1 − x² for x in [−1,1], and tan(sin⁻¹ x) = x / √1 − x².
勾股恒等式给出非常有用的结果：当 x ∈ [−1,1] 时，cos(sin⁻¹ x) = √1 − x²，sin(cos⁻¹ x) = √1 − x²，且 tan(sin⁻¹ x) = x / √1 − x²。
Another identity is sin⁻¹ x + cos⁻¹ x = π/2 for all x in [−1,1].
另一个恒等式是对于所有 x ∈ [−1,1]，sin⁻¹ x + cos⁻¹ x = π/2。
5. Differentiating Inverse Trigonometric Functions | 反三角函数的求导
The standard derivatives are:
标准导数公式为：
d/dx (sin⁻¹ x) = 1 / √1 − x²
d/dx (cos⁻¹ x) = −1 / √1 − x²
d/dx (tan⁻¹ x) = 1 / (1 + x²)
For y = sin⁻¹ x, write sin y = x. Im
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