What is the value of tanh(pi) ?

The value of tanh(pi) is (e^{2pi}-1)/(e^{2pi)+1}

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tanh(pi)

tanh (π)

\frac{e ^{2 π} - 1}{e ^{2 π} + 1}

Decimal Notation

0.99627 \dots

Solution steps

tanh (π)

Use the Hyperbolic identity:

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

= \frac{e ^{π} - e ^{- π}}{e ^{π} + e ^{- π}}

\frac{e ^{π} - e ^{- π}}{e ^{π} + e ^{- π}} = \frac{e ^{2 π} - 1}{e ^{2 π} + 1}

\frac{e ^{π} - e ^{- π}}{e ^{π} + e ^{- π}}

Apply exponent rule:

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

e^{- π} = \frac{1}{e ^{π}}

= \frac{e ^{π} - \frac{1}{e ^{π}}}{e ^{π} + \frac{1}{e ^{π}}}

Join

e^{π} + \frac{1}{e ^{π}} : \frac{e ^{2 π} + 1}{e ^{π}}

e^{π} + \frac{1}{e ^{π}}

Convert element to fraction:

e^{π} = \frac{e ^{π} e ^{π}}{e ^{π}}

= \frac{e ^{π} e ^{π}}{e ^{π}} + \frac{1}{e ^{π}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{π} e ^{π} + 1}{e ^{π}}

e^{π} e^{π} + 1 = e^{2 π} + 1

e^{π} e^{π} + 1

e^{π} e^{π} = e^{2 π}

e^{π} e^{π}

Apply exponent rule:

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

e^{π} e^{π} = e^{π + π}

= e^{π + π}

Add similar elements:

π + π = 2 π

= e^{2 π}

= e^{2 π} + 1

= \frac{e ^{2 π} + 1}{e ^{π}}

= \frac{e ^{π} - \frac{1}{e ^{π}}}{\frac{e ^{2 π} + 1}{e ^{π}}}

Join

e^{π} - \frac{1}{e ^{π}} : \frac{e ^{2 π} - 1}{e ^{π}}

e^{π} - \frac{1}{e ^{π}}

Convert element to fraction:

e^{π} = \frac{e ^{π} e ^{π}}{e ^{π}}

= \frac{e ^{π} e ^{π}}{e ^{π}} - \frac{1}{e ^{π}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{π} e ^{π} - 1}{e ^{π}}

e^{π} e^{π} - 1 = e^{2 π} - 1

e^{π} e^{π} - 1

e^{π} e^{π} = e^{2 π}

e^{π} e^{π}

Apply exponent rule:

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

e^{π} e^{π} = e^{π + π}

= e^{π + π}

Add similar elements:

π + π = 2 π

= e^{2 π}

= e^{2 π} - 1

= \frac{e ^{2 π} - 1}{e ^{π}}

= \frac{\frac{e ^{2 π} - 1}{e ^{π}}}{\frac{e ^{2 π} + 1}{e ^{π}}}

Divide fractions:

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

= \frac{( e ^{2 π} - 1 ) e ^{π}}{e ^{π} ( e ^{2 π} + 1 )}

Cancel the common factor:

e^{π}

= \frac{e ^{2 π} - 1}{e ^{2 π} + 1}

= \frac{e ^{2 π} - 1}{e ^{2 π} + 1}

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What is the value of tanh(pi) ?
The value of tanh(pi) is (e^{2pi}-1)/(e^{2pi)+1}