coth(ln(2))

The hyperbolic cotangent is read as

\coth (u) = \frac{\cosh (u)}{\sinh (u)}

, on each interval where

\sinh (u) \neq 0

1) Let's calculate:

\frac{\cosh (\ln (2))}{\sinh (\ln (2))}

Replace cosh by its definition:

\cosh (x) = \frac{e^{x} + e^{- x}}{2}

Replace sinh by its definition:

\sinh (x) = \frac{e^{x} - e^{- x}}{2}

ⅰ

\sinh (\ln (2))

\frac{e^{\ln (2)} - e^{- \ln (2)}}{2}

\frac{\frac{e^{\ln (2)} + e^{- \ln (2)}}{2}}{\frac{e^{\ln (2)} - e^{- \ln (2)}}{2}}

ⓘ For real

x

\ln (\exp (x)) = x

; wherever

\ln (x)

is defined,

\exp (\ln (x)) = x

e^{\ln (2)}

2

e^{- \ln (2)}

e^{\ln (\frac{1}{2})}

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

ⓘ For real

x

\ln (\exp (x)) = x

; wherever

\ln (x)

is defined,

\exp (\ln (x)) = x

e^{\ln (\frac{1}{2})}

\frac{1}{2}

■To add fractions, put them over the same denominator, then add the numerators:

2 + \frac{1}{2}

\frac{2 \times 2 + 1}{2}

2 - \frac{1}{2}

\frac{2 \times 2 - 1}{2}

\frac{\frac{\frac{2 \times 2 + 1}{2}}{2}}{\frac{\frac{2 \times 2 - 1}{2}}{2}}

ⓘ Perform multiplications before additions and subtractions:

2 \times 2

4

\frac{\frac{\frac{4 + 1}{2}}{2}}{\frac{\frac{4 - 1}{2}}{2}}

ⓘ The numerator and the denominator are computed before the division:

4 + 1

5

4 - 1

3

\frac{\frac{\frac{5}{2}}{2}}{\frac{\frac{3}{2}}{2}}

ⓘ Dividing a fraction by

2

is multiplying its denominator by

2

\frac{\frac{5}{2}}{2}

\frac{5}{2 \times 2}

\frac{\frac{3}{2}}{2}

\frac{3}{2 \times 2}

■Dividing is multiplying by the reciprocal:

\frac{\frac{5}{2 \times 2}}{\frac{3}{2 \times 2}}

\frac{5}{2 \times 2} \times \frac{2 \times 2}{3}

\frac{5}{2 \times 2} \times \frac{2 \times 2}{3}

ⅰ

2 \times 2

4

\frac{5}{4} \times \frac{4}{3}

ⓘ The product of fractions is found by multiplying their numerators together and their denominators together:

\frac{5}{4} \times \frac{4}{3}

\frac{5 \times 4}{4 \times 3}

\frac{20}{12}

\frac{20}{12}

ⓘ We simplify the fraction by dividing the numerator and the denominator by

4

, their greatest common divisor:

\frac{20}{12}

\frac{5 \times 4}{3 \times 4}

\frac{5}{3}

▶Result:

\frac{5}{3}

2) The numeric value of:

\frac{5}{3}

equals:

▷

\approx 1.6667

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