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受欢迎的三角函数 >

arctanh(0.99)

解答

arctanh (0.99)

解答

\frac{ln ( 199 )}{2}

+1

十进制

2.64665 \dots

求解步骤

arctanh (0.99)

= arctanh (\frac{99}{100})

使用三角恒等式改写:

\frac{ln ( 199 )}{2}

arctanh (\frac{99}{100})

使用双曲函数恒等式:

arctanh (x) = \frac{ln ( \frac{1 + x}{1 - x} )}{2}

= \frac{ln ( \frac{1 + \frac{99}{100}}{1 - \frac{99}{100}} )}{2}

\frac{ln ( \frac{1 + \frac{99}{100}}{1 - \frac{99}{100}} )}{2} = \frac{ln ( 199 )}{2}

\frac{ln ( \frac{1 + \frac{99}{100}}{1 - \frac{99}{100}} )}{2}

\frac{1 + \frac{99}{100}}{1 - \frac{99}{100}} = 199

\frac{1 + \frac{99}{100}}{1 - \frac{99}{100}}

化简

1 - \frac{99}{100} : \frac{1}{100}

1 - \frac{99}{100}

将项转换为分式:

1 = \frac{1 \cdot 100}{100}

= \frac{1 \cdot 100}{100} - \frac{99}{100}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 100 - 99}{100}

1 \cdot 100 - 99 = 1

1 \cdot 100 - 99

数字相乘:

1 \cdot 100 = 100

= 100 - 99

数字相减:

100 - 99 = 1

= 1

= \frac{1}{100}

= \frac{1 + \frac{99}{100}}{\frac{1}{100}}

化简

1 + \frac{99}{100} : \frac{199}{100}

1 + \frac{99}{100}

将项转换为分式:

1 = \frac{1 \cdot 100}{100}

= \frac{1 \cdot 100}{100} + \frac{99}{100}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 100 + 99}{100}

1 \cdot 100 + 99 = 199

1 \cdot 100 + 99

数字相乘:

1 \cdot 100 = 100

= 100 + 99

数字相加:

100 + 99 = 199

= 199

= \frac{199}{100}

= \frac{\frac{199}{100}}{\frac{1}{100}}

分式相除:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{199 \cdot 100}{100 \cdot 1}

整理后得

= \frac{199 \cdot 100}{100}

约分:

100

= 199

= \frac{ln ( 199 )}{2}

= \frac{ln ( 199 )}{2}

= \frac{ln ( 199 )}{2}

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