# arctriangle

Canonical: https://www.mirstar.net/2025/09/20/arctriangle/
Author: onlymatt
Published: 2025-09-20T15:07:00.000Z
Updated: 2026-10-21T11:21:55.746Z
Categories: Mathematics, Calculus
Tags: Inverse Trigonometric Functions

# 反三角函数公式汇总
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### **一、基本恒等式**
1. **余角关系**  
   $$
   \begin{aligned}
   arcsin(x) + arccos(x) &= \frac{\pi}{2} \\
   arctan(x) + arccot(x) &= \frac{\pi}{2} \\
   arcsec(x) + arccsc(x) &= \frac{\pi}{2}
   \end{aligned}
   $$

2. **负数关系**  
   $$
   \begin{aligned}
   arcsin(-x) &= -arcsin(x), \\
   arccos(-x) &= \pi - arccos(x), \\
   arctan(-x) &= -arctan(x), \\
   arccot(-x) &= \pi - arccot(x)
   \end{aligned}
   $$

3. **倒数关系**  
   $$
   \begin{aligned}
   arcsin\left(\frac{1}{x}\right) &= arccsc(x), \\
   arccos\left(\frac{1}{x}\right) &= arcsec(x), \\
   arctan\left(\frac{1}{x}\right) &= arccot(x) \quad (x > 0)
   \end{aligned}
   $$

---

### **二、导数公式**
$$
\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}x} arcsin(x) &= \frac{1}{\sqrt{1 - x^2}}, \\
\frac{\mathrm{d}}{\mathrm{d}x} arccos(x) &= -\frac{1}{\sqrt{1 - x^2}}, \\
\frac{\mathrm{d}}{\mathrm{d}x} arctan(x) &= \frac{1}{1 + x^2}, \\
\frac{\mathrm{d}}{\mathrm{d}x} arccot(x) &= -\frac{1}{1 + x^2}, \\
\frac{\mathrm{d}}{\mathrm{d}x} arcsec(x) &= \frac{1}{|x|\sqrt{x^2 - 1}}, \\
\frac{\mathrm{d}}{\mathrm{d}x} arccsc(x) &= -\frac{1}{|x|\sqrt{x^2 - 1}}.
\end{aligned}
$$

---

### **三、积分公式**
$$
\begin{aligned}
\int arcsin(x) \, \mathrm{d}x &= x arcsin(x) + \sqrt{1 - x^2} + C, \\
\int arccos(x) \, \mathrm{d}x &= x arccos(x) - \sqrt{1 - x^2} + C, \\
\int arctan(x) \, \mathrm{d}x &= x arctan(x) - \frac{1}{2} \ln(1 + x^2) + C, \\
\int arccot(x) \, \mathrm{d}x &= x arccot(x) + \frac{1}{2} \ln(1 + x^2) + C.
\end{aligned}
$$

---

### **四、运算性质**
1. **加减公式**  
   $$
   arctan(x) + arctan(y) = arctan\left(\frac{x + y}{1 - xy}\right) \quad (xy < 1)
   $$

2. **复合函数关系**  
   $$
   \sin(arcsin(x)) = x \quad (|x| \leq 1), \quad
   \cos(arccos(x)) = x \quad (|x| \leq 1), \quad
   \tan(arctan(x)) = x \quad (x \in \mathbb{R})
   $$

---

### **五、泰勒展开**
1. **反正切函数**  
   $$
   arctan(x) = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \quad (|x| \leq 1)
   $$

2. **反正弦函数**  
   $$
   arcsin(x) = x + \frac{1}{6}x^3 + \frac{3}{40}x^5 + \cdots \quad (|x| < 1)
   $$

---

### **六、特殊值**
$$
\begin{array}{|c|c|c|}
\hline
x & arcsin(x) & arccos(x) & arctan(x) \\
\hline
0 & 0 & \frac{\pi}{2} & 0 \\
\frac{\sqrt{2}}{2} & \frac{\pi}{4} & \frac{\pi}{4} & \frac{\pi}{4} \\
1 & \frac{\pi}{2} & 0 & \frac{\pi}{4} \\
\hline
\end{array}
$$

---

### **七、反三角函数的转换**
$$
arctan(x) = arcsin\left(\frac{x}{\sqrt{1 + x^2}}\right), \\\\\
arccot(x) = arctan\left(\frac{1}{x}\right) \quad (x > 0)
$$

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