What is the value of coth(ln(6)) ?

The value of coth(ln(6)) is 37/35

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coth(ln(6))

Solution

coth (ln (6))

Solution

\frac{37}{35}

Decimal Notation

1.05714 \dots

Solution steps

coth (ln (6))

Use the Hyperbolic identity:

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

= \frac{e ^{l n (6)} + e ^{- l n (6)}}{e ^{l n (6)} - e ^{- l n (6)}}

\frac{e ^{l n (6)} + e ^{- l n (6)}}{e ^{l n (6)} - e ^{- l n (6)}} = \frac{37}{35}

\frac{e ^{l n (6)} + e ^{- l n (6)}}{e ^{l n (6)} - e ^{- l n (6)}}

e^{l n (6)} = 6

e^{l n (6)}

Apply log rule:

a^{l o g_{a} (b)} = b

= 6

e^{- l n (6)} = 6^{- 1}

e^{- l n (6)}

Apply exponent rule:

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

= (e^{l n (6)})^{- 1}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (6)} = 6

= 6^{- 1}

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

e^{l n (6)} = 6

e^{l n (6)}

Apply log rule:

a^{l o g_{a} (b)} = b

= 6

e^{- l n (6)} = 6^{- 1}

e^{- l n (6)}

Apply exponent rule:

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

= (e^{l n (6)})^{- 1}

Apply log rule:

a^{l o g_{a} (b)} = b

e^{l n (6)} = 6

= 6^{- 1}

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

Simplify

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

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

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

Join

6 - \frac{1}{6} : \frac{35}{6}

6 - \frac{1}{6}

Convert element to fraction:

6 = \frac{6 \cdot 6}{6}

= \frac{6 \cdot 6}{6} - \frac{1}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{6 \cdot 6 - 1}{6}

6 \cdot 6 - 1 = 35

6 \cdot 6 - 1

Multiply the numbers:

6 \cdot 6 = 36

= 36 - 1

Subtract the numbers:

36 - 1 = 35

= 35

= \frac{35}{6}

= \frac{6 + \frac{1}{6}}{\frac{35}{6}}

Join

6 + \frac{1}{6} : \frac{37}{6}

6 + \frac{1}{6}

Convert element to fraction:

6 = \frac{6 \cdot 6}{6}

= \frac{6 \cdot 6}{6} + \frac{1}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{6 \cdot 6 + 1}{6}

6 \cdot 6 + 1 = 37

6 \cdot 6 + 1

Multiply the numbers:

6 \cdot 6 = 36

= 36 + 1

Add the numbers:

36 + 1 = 37

= 37

= \frac{37}{6}

= \frac{\frac{37}{6}}{\frac{35}{6}}

Divide fractions:

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

= \frac{37 \cdot 6}{6 \cdot 35}

Cancel the common factor:

6

= \frac{37}{35}

= \frac{37}{35}

= \frac{37}{35}

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Frequently Asked Questions (FAQ)

What is the value of coth(ln(6)) ?
The value of coth(ln(6)) is 37/35