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人気のある三角関数 >

arctanh(0.99)

解

arctanh (0.99)

解

\frac{ln ( 199 )}{2}

+1

十進法表記

2.64665 \dots

解答ステップ

arctanh (0.99)

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

三角関数の公式を使用して書き換える:

\frac{ln ( 199 )}{2}

arctanh (\frac{99}{100})

双曲線の公式を使用する:

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}}

結合

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

1 - \frac{99}{100}

元を分数に変換する:

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

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

分母が等しいので, 分数を組み合わせる:

\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

数を乗じる:

1 \cdot 100 = 100

= 100 - 99

数を引く:

100 - 99 = 1

= 1

= \frac{1}{100}

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

結合

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

1 + \frac{99}{100}

元を分数に変換する:

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

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

分母が等しいので, 分数を組み合わせる:

\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

数を乗じる:

1 \cdot 100 = 100

= 100 + 99

数を足す:

100 + 99 = 199

= 199

= \frac{199}{100}

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

分数を割る:

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

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

改良

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

共通因数を約分する:

100

= 199

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

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

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

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