What is the value of tanh(pi/4) ?

The value of tanh(pi/4) is ((e^{pi/2}-1)^2)/(e^pi-1)

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Popular >

tanh(pi/4)

Solution

tanh (\frac{π}{4})

Solution

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

Decimal Notation

0.65579 \dots

Solution steps

tanh (\frac{π}{4})

Use the Hyperbolic identity:

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

= \frac{e ^{\frac{π}{4}} - e ^{- \frac{π}{4}}}{e ^{\frac{π}{4}} + e ^{- \frac{π}{4}}}

\frac{e ^{\frac{π}{4}} - e ^{- \frac{π}{4}}}{e ^{\frac{π}{4}} + e ^{- \frac{π}{4}}} = \frac{( e ^{\frac{π}{2}} - 1 ) ^{2}}{e ^{π} - 1}

\frac{e ^{\frac{π}{4}} - e ^{- \frac{π}{4}}}{e ^{\frac{π}{4}} + e ^{- \frac{π}{4}}}

Apply exponent rule:

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

e^{- \frac{π}{4}} = \frac{1}{e ^{\frac{π}{4}}}

= \frac{e ^{\frac{π}{4}} - \frac{1}{e ^{\frac{π}{4}}}}{e ^{\frac{π}{4}} + \frac{1}{e ^{\frac{π}{4}}}}

Join

e^{\frac{π}{4}} + \frac{1}{e ^{\frac{π}{4}}} : \frac{e ^{\frac{π}{2}} + 1}{e ^{\frac{π}{4}}}

e^{\frac{π}{4}} + \frac{1}{e ^{\frac{π}{4}}}

Convert element to fraction:

e^{\frac{π}{4}} = \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}}}{e ^{\frac{π}{4}}}

= \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}}}{e ^{\frac{π}{4}}} + \frac{1}{e ^{\frac{π}{4}}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}} + 1}{e ^{\frac{π}{4}}}

e^{\frac{π}{4}} e^{\frac{π}{4}} + 1 = e^{2 \cdot \frac{π}{4}} + 1

e^{\frac{π}{4}} e^{\frac{π}{4}} + 1

e^{\frac{π}{4}} e^{\frac{π}{4}} = e^{2 \cdot \frac{π}{4}}

e^{\frac{π}{4}} e^{\frac{π}{4}}

Apply exponent rule:

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

e^{\frac{π}{4}} e^{\frac{π}{4}} = e^{\frac{π}{4} + \frac{π}{4}}

= e^{\frac{π}{4} + \frac{π}{4}}

Add similar elements:

\frac{π}{4} + \frac{π}{4} = 2 \cdot \frac{π}{4}

= e^{2 \cdot \frac{π}{4}}

= e^{2 \cdot \frac{π}{4}} + 1

= \frac{e ^{2 \cdot \frac{π}{4}} + 1}{e ^{\frac{π}{4}}}

Multiply

2 \cdot \frac{π}{4} : \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

= \frac{e ^{\frac{π}{2}} + 1}{e ^{\frac{π}{4}}}

= \frac{e ^{\frac{π}{4}} - \frac{1}{e ^{\frac{π}{4}}}}{\frac{e ^{\frac{π}{2}} + 1}{e ^{\frac{π}{4}}}}

Join

e^{\frac{π}{4}} - \frac{1}{e ^{\frac{π}{4}}} : \frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{4}}}

e^{\frac{π}{4}} - \frac{1}{e ^{\frac{π}{4}}}

Convert element to fraction:

e^{\frac{π}{4}} = \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}}}{e ^{\frac{π}{4}}}

= \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}}}{e ^{\frac{π}{4}}} - \frac{1}{e ^{\frac{π}{4}}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}} - 1}{e ^{\frac{π}{4}}}

e^{\frac{π}{4}} e^{\frac{π}{4}} - 1 = e^{2 \cdot \frac{π}{4}} - 1

e^{\frac{π}{4}} e^{\frac{π}{4}} - 1

e^{\frac{π}{4}} e^{\frac{π}{4}} = e^{2 \cdot \frac{π}{4}}

e^{\frac{π}{4}} e^{\frac{π}{4}}

Apply exponent rule:

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

e^{\frac{π}{4}} e^{\frac{π}{4}} = e^{\frac{π}{4} + \frac{π}{4}}

= e^{\frac{π}{4} + \frac{π}{4}}

Add similar elements:

\frac{π}{4} + \frac{π}{4} = 2 \cdot \frac{π}{4}

= e^{2 \cdot \frac{π}{4}}

= e^{2 \cdot \frac{π}{4}} - 1

= \frac{e ^{2 \cdot \frac{π}{4}} - 1}{e ^{\frac{π}{4}}}

Multiply

2 \cdot \frac{π}{4} : \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

= \frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{4}}}

= \frac{\frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{4}}}}{\frac{e ^{\frac{π}{2}} + 1}{e ^{\frac{π}{4}}}}

Divide fractions:

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

= \frac{( e ^{\frac{π}{2}} - 1 ) e ^{\frac{π}{4}}}{e ^{\frac{π}{4}} ( e ^{\frac{π}{2}} + 1 )}

Cancel the common factor:

e^{\frac{π}{4}}

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

Rationalize

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

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

Multiply by the conjugate

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

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

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

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

Apply exponent rule:

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

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

= (e^{\frac{π}{2}} - 1)^{1 + 1}

Add the numbers:

1 + 1 = 2

= (e^{\frac{π}{2}} - 1)^{2}

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

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

Apply Difference of Two Squares Formula:

(a + b) (a - b) = a^{2} - b^{2}

a = e^{\frac{π}{2}}, b = 1

= (e^{\frac{π}{2}})^{2} - 1^{2}

Simplify

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

(e^{\frac{π}{2}})^{2} - 1^{2}

Apply rule

1^{a} = 1

1^{2} = 1

= (e^{\frac{π}{2}})^{2} - 1

(e^{\frac{π}{2}})^{2} = e^{π}

(e^{\frac{π}{2}})^{2}

Apply exponent rule:

(a^{b})^{c} = a^{b c}

= e^{\frac{π}{2} \cdot 2}

\frac{π}{2} \cdot 2 = π

\frac{π}{2} \cdot 2

Multiply fractions:

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

= \frac{π 2}{2}

Cancel the common factor:

2

= π

= e^{π}

= e^{π} - 1

= e^{π} - 1

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

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

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

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Frequently Asked Questions (FAQ)

What is the value of tanh(pi/4) ?
The value of tanh(pi/4) is ((e^{pi/2}-1)^2)/(e^pi-1)