What is the value of coth(ln(5)) ?

The value of coth(ln(5)) is 13/12

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Popular Trigonometry >

coth(ln(5))

Solution

coth (ln (5))

Solution

\frac{13}{12}

Decimal

1.08333 \dots

Solution steps

coth (ln (5))

Use the Hyperbolic identity:

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

= \frac{e ^{l n (5)} + e ^{- l n (5)}}{e ^{l n (5)} - e ^{- l n (5)}}

\frac{e ^{l n (5)} + e ^{- l n (5)}}{e ^{l n (5)} - e ^{- l n (5)}} = \frac{13}{12}

\frac{e ^{l n (5)} + e ^{- l n (5)}}{e ^{l n (5)} - e ^{- l n (5)}}

e^{l n (5)} = 5

e^{l n (5)}

Apply log rule:

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

= 5

e^{- l n (5)} = 5^{- 1}

e^{- l n (5)}

Apply exponent rule:

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

= (e^{l n (5)})^{- 1}

Apply log rule:

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

e^{l n (5)} = 5

= 5^{- 1}

= \frac{e ^{l n (5)} + e ^{- l n (5)}}{5 - 5 ^{- 1}}

e^{l n (5)} = 5

e^{l n (5)}

Apply log rule:

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

= 5

e^{- l n (5)} = 5^{- 1}

e^{- l n (5)}

Apply exponent rule:

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

= (e^{l n (5)})^{- 1}

Apply log rule:

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

e^{l n (5)} = 5

= 5^{- 1}

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

Simplify

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

Apply exponent rule:

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

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

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

Apply exponent rule:

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

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

= \frac{5 + \frac{1}{5}}{5 - \frac{1}{5}}

Join

5 - \frac{1}{5} : \frac{24}{5}

5 - \frac{1}{5}

Convert element to fraction:

5 = \frac{5 \cdot 5}{5}

= \frac{5 \cdot 5}{5} - \frac{1}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{5 \cdot 5 - 1}{5}

5 \cdot 5 - 1 = 24

5 \cdot 5 - 1

Multiply the numbers:

5 \cdot 5 = 25

= 25 - 1

Subtract the numbers:

25 - 1 = 24

= 24

= \frac{24}{5}

= \frac{5 + \frac{1}{5}}{\frac{24}{5}}

Join

5 + \frac{1}{5} : \frac{26}{5}

5 + \frac{1}{5}

Convert element to fraction:

5 = \frac{5 \cdot 5}{5}

= \frac{5 \cdot 5}{5} + \frac{1}{5}

Since the denominators are equal, combine the fractions:

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

= \frac{5 \cdot 5 + 1}{5}

5 \cdot 5 + 1 = 26

5 \cdot 5 + 1

Multiply the numbers:

5 \cdot 5 = 25

= 25 + 1

Add the numbers:

25 + 1 = 26

= 26

= \frac{26}{5}

= \frac{\frac{26}{5}}{\frac{24}{5}}

Divide fractions:

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

= \frac{26 \cdot 5}{5 \cdot 24}

Cancel the common factor:

5

= \frac{26}{24}

Cancel the common factor:

2

= \frac{13}{12}

= \frac{13}{12}

= \frac{13}{12}

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Frequently Asked Questions (FAQ)

What is the value of coth(ln(5)) ?
The value of coth(ln(5)) is 13/12