What is the value of coth(3) ?

The value of coth(3) is (e^6+1)/(e^6-1)

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coth(3)

coth (3)

\frac{e ^{6} + 1}{e ^{6} - 1}

Decimal Notation

1.00496 \dots

Solution steps

coth (3)

Use the Hyperbolic identity:

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

= \frac{e ^{3} + e ^{- 3}}{e ^{3} - e ^{- 3}}

\frac{e ^{3} + e ^{- 3}}{e ^{3} - e ^{- 3}} = \frac{e ^{6} + 1}{e ^{6} - 1}

\frac{e ^{3} + e ^{- 3}}{e ^{3} - e ^{- 3}}

Apply exponent rule:

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

e^{- 3} = \frac{1}{e ^{3}}

= \frac{e ^{3} + \frac{1}{e ^{3}}}{e ^{3} - \frac{1}{e ^{3}}}

Join

e^{3} - \frac{1}{e ^{3}} : \frac{e ^{6} - 1}{e ^{3}}

e^{3} - \frac{1}{e ^{3}}

Convert element to fraction:

e^{3} = \frac{e ^{3} e ^{3}}{e ^{3}}

= \frac{e ^{3} e ^{3}}{e ^{3}} - \frac{1}{e ^{3}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{3} e ^{3} - 1}{e ^{3}}

e^{3} e^{3} - 1 = e^{6} - 1

e^{3} e^{3} - 1

e^{3} e^{3} = e^{6}

e^{3} e^{3}

Apply exponent rule:

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

e^{3} e^{3} = e^{3 + 3}

= e^{3 + 3}

Add the numbers:

3 + 3 = 6

= e^{6}

= e^{6} - 1

= \frac{e ^{6} - 1}{e ^{3}}

= \frac{e ^{3} + \frac{1}{e ^{3}}}{\frac{e ^{6} - 1}{e ^{3}}}

Join

e^{3} + \frac{1}{e ^{3}} : \frac{e ^{6} + 1}{e ^{3}}

e^{3} + \frac{1}{e ^{3}}

Convert element to fraction:

e^{3} = \frac{e ^{3} e ^{3}}{e ^{3}}

= \frac{e ^{3} e ^{3}}{e ^{3}} + \frac{1}{e ^{3}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{3} e ^{3} + 1}{e ^{3}}

e^{3} e^{3} + 1 = e^{6} + 1

e^{3} e^{3} + 1

e^{3} e^{3} = e^{6}

e^{3} e^{3}

Apply exponent rule:

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

e^{3} e^{3} = e^{3 + 3}

= e^{3 + 3}

Add the numbers:

3 + 3 = 6

= e^{6}

= e^{6} + 1

= \frac{e ^{6} + 1}{e ^{3}}

= \frac{\frac{e ^{6} + 1}{e ^{3}}}{\frac{e ^{6} - 1}{e ^{3}}}

Divide fractions:

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

= \frac{( e ^{6} + 1 ) e ^{3}}{e ^{3} ( e ^{6} - 1 )}

Cancel the common factor:

e^{3}

= \frac{e ^{6} + 1}{e ^{6} - 1}

= \frac{e ^{6} + 1}{e ^{6} - 1}

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