What is the value of tanh(1.4) ?

The value of tanh(1.4) is (e^{14/5}-1)/(e^{14/5)+1}

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tanh(1.4)

Solution

tanh (1.4)

Solution

\frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{14}{5}} + 1}

Decimal Notation

0.88535 \dots

Solution steps

tanh (1.4)

= tanh (\frac{7}{5})

Rewrite using trig identities:

\frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{14}{5}} + 1}

tanh (\frac{7}{5})

Use the Hyperbolic identity:

tanh (x) = \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}

= \frac{e ^{\frac{7}{5}} - e ^{- \frac{7}{5}}}{e ^{\frac{7}{5}} + e ^{- \frac{7}{5}}}

\frac{e ^{\frac{7}{5}} - e ^{- \frac{7}{5}}}{e ^{\frac{7}{5}} + e ^{- \frac{7}{5}}} = \frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{14}{5}} + 1}

\frac{e ^{\frac{7}{5}} - e ^{- \frac{7}{5}}}{e ^{\frac{7}{5}} + e ^{- \frac{7}{5}}}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

e^{- \frac{7}{5}} = \frac{1}{e ^{\frac{7}{5}}}

= \frac{e ^{\frac{7}{5}} - \frac{1}{e ^{\frac{7}{5}}}}{e ^{\frac{7}{5}} + \frac{1}{e ^{\frac{7}{5}}}}

Join

e^{\frac{7}{5}} + \frac{1}{e ^{\frac{7}{5}}} : \frac{e ^{\frac{14}{5}} + 1}{e ^{\frac{7}{5}}}

e^{\frac{7}{5}} + \frac{1}{e ^{\frac{7}{5}}}

Convert element to fraction:

e^{\frac{7}{5}} = \frac{e ^{\frac{7}{5}} e ^{\frac{7}{5}}}{e ^{\frac{7}{5}}}

= \frac{e ^{\frac{7}{5}} e ^{\frac{7}{5}}}{e ^{\frac{7}{5}}} + \frac{1}{e ^{\frac{7}{5}}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{\frac{7}{5}} e ^{\frac{7}{5}} + 1}{e ^{\frac{7}{5}}}

e^{\frac{7}{5}} e^{\frac{7}{5}} + 1 = e^{2 \cdot \frac{7}{5}} + 1

e^{\frac{7}{5}} e^{\frac{7}{5}} + 1

e^{\frac{7}{5}} e^{\frac{7}{5}} = e^{2 \cdot \frac{7}{5}}

e^{\frac{7}{5}} e^{\frac{7}{5}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{\frac{7}{5}} e^{\frac{7}{5}} = e^{\frac{7}{5} + \frac{7}{5}}

= e^{\frac{7}{5} + \frac{7}{5}}

Add similar elements:

\frac{7}{5} + \frac{7}{5} = 2 \cdot \frac{7}{5}

= e^{2 \cdot \frac{7}{5}}

= e^{2 \cdot \frac{7}{5}} + 1

= \frac{e ^{2 \cdot \frac{7}{5}} + 1}{e ^{\frac{7}{5}}}

Multiply

2 \cdot \frac{7}{5} : \frac{14}{5}

2 \cdot \frac{7}{5}

Multiply fractions:

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

= \frac{7 \cdot 2}{5}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14}{5}

= \frac{e ^{\frac{14}{5}} + 1}{e ^{\frac{7}{5}}}

= \frac{e ^{\frac{7}{5}} - \frac{1}{e ^{\frac{7}{5}}}}{\frac{e ^{\frac{14}{5}} + 1}{e ^{\frac{7}{5}}}}

Join

e^{\frac{7}{5}} - \frac{1}{e ^{\frac{7}{5}}} : \frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{7}{5}}}

e^{\frac{7}{5}} - \frac{1}{e ^{\frac{7}{5}}}

Convert element to fraction:

e^{\frac{7}{5}} = \frac{e ^{\frac{7}{5}} e ^{\frac{7}{5}}}{e ^{\frac{7}{5}}}

= \frac{e ^{\frac{7}{5}} e ^{\frac{7}{5}}}{e ^{\frac{7}{5}}} - \frac{1}{e ^{\frac{7}{5}}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{\frac{7}{5}} e ^{\frac{7}{5}} - 1}{e ^{\frac{7}{5}}}

e^{\frac{7}{5}} e^{\frac{7}{5}} - 1 = e^{2 \cdot \frac{7}{5}} - 1

e^{\frac{7}{5}} e^{\frac{7}{5}} - 1

e^{\frac{7}{5}} e^{\frac{7}{5}} = e^{2 \cdot \frac{7}{5}}

e^{\frac{7}{5}} e^{\frac{7}{5}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{\frac{7}{5}} e^{\frac{7}{5}} = e^{\frac{7}{5} + \frac{7}{5}}

= e^{\frac{7}{5} + \frac{7}{5}}

Add similar elements:

\frac{7}{5} + \frac{7}{5} = 2 \cdot \frac{7}{5}

= e^{2 \cdot \frac{7}{5}}

= e^{2 \cdot \frac{7}{5}} - 1

= \frac{e ^{2 \cdot \frac{7}{5}} - 1}{e ^{\frac{7}{5}}}

Multiply

2 \cdot \frac{7}{5} : \frac{14}{5}

2 \cdot \frac{7}{5}

Multiply fractions:

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

= \frac{7 \cdot 2}{5}

Multiply the numbers:

7 \cdot 2 = 14

= \frac{14}{5}

= \frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{7}{5}}}

= \frac{\frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{7}{5}}}}{\frac{e ^{\frac{14}{5}} + 1}{e ^{\frac{7}{5}}}}

Divide fractions:

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

= \frac{( e ^{\frac{14}{5}} - 1 ) e ^{\frac{7}{5}}}{e ^{\frac{7}{5}} ( e ^{\frac{14}{5}} + 1 )}

Cancel the common factor:

e^{\frac{7}{5}}

= \frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{14}{5}} + 1}

= \frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{14}{5}} + 1}

= \frac{e ^{\frac{14}{5}} - 1}{e ^{\frac{14}{5}} + 1}

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Frequently Asked Questions (FAQ)

What is the value of tanh(1.4) ?
The value of tanh(1.4) is (e^{14/5}-1)/(e^{14/5)+1}