arctriangle

反三角函数公式汇总

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一、基本恒等式

1. 余角关系

   $$arcsin(x) + arccos(x) = \frac{\pi}{2}\\\\ arctan(x) + arccot(x) = \frac{\pi}{2}\\\\ arcsec(x) + arccsc(x) = \frac{\pi}{2}$$

2. 负数关系

   $$arcsin(-x) = -arcsin(x), \\\\ arccos(-x) = \pi - arccos(x), \\\\ arctan(-x) = -arctan(x), \\\\ arccot(-x) = \pi - arccot(x)$$

3. 倒数关系

   $$arcsin\left(\frac{1}{x}\right) = arccsc(x),\\\\ arccos\left(\frac{1}{x}\right) = arcsec(x), \\\\ arctan\left(\frac{1}{x}\right) = arccot(x) \quad (x > 0)$$

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二、导数公式

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

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三、积分公式

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

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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

   $$\sin(arcsin(x)) = x \quad (|x| \leq 1), \quad \cos(arccos(x)) = x \quad (|x| \leq 1), \quad \tan(arctan(x)) = x \quad (x \in \mathbb{R})$$

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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

   $$arcsin(x) = x + \frac{1}{6}x^3 + \frac{3}{40}x^5 + \cdots \quad (|x| < 1)$$

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六、特殊值

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

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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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