What is the value of sinh(ln(3/2)) ?

The value of sinh(ln(3/2)) is 5/12

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sinh(ln(3/2))

Solution

sinh (ln (\frac{3}{2}))

Solution

\frac{5}{12}

Decimal Notation

0.41666 \dots

Solution steps

sinh (ln (\frac{3}{2}))

Use the Hyperbolic identity:

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

= \frac{e ^{l n (\frac{3}{2})} - e ^{- l n (\frac{3}{2})}}{2}

\frac{e ^{l n (\frac{3}{2})} - e ^{- l n (\frac{3}{2})}}{2} = \frac{5}{12}

\frac{e ^{l n (\frac{3}{2})} - e ^{- l n (\frac{3}{2})}}{2}

e^{l n (\frac{3}{2})} = \frac{3}{2}

e^{l n (\frac{3}{2})}

Apply log rule:

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

= \frac{3}{2}

e^{- l n (\frac{3}{2})} = \frac{2}{3}

e^{- l n (\frac{3}{2})}

Apply exponent rule:

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

= (e^{l n (\frac{3}{2})})^{- 1}

Apply log rule:

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

e^{l n (\frac{3}{2})} = \frac{3}{2}

= (\frac{3}{2})^{- 1}

Apply exponent rule:

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

= \frac{1}{\frac{3}{2}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{2}{3}

= \frac{\frac{3}{2} - \frac{2}{3}}{2}

Simplify

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

Join

\frac{3}{2} - \frac{2}{3} : \frac{5}{6}

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

Least Common Multiplier of

2, 3 : 6

2, 3

Least Common Multiplier (LCM)

Prime factorization of

2 : 2

2

2

is a prime number, therefore no factorization is possible

= 2

Prime factorization of

3 : 3

3

3

is a prime number, therefore no factorization is possible

= 3

Multiply each factor the greatest number of times it occurs in either

2

3

= 2 \cdot 3

Multiply the numbers:

2 \cdot 3 = 6

= 6

Adjust Fractions based on the LCM

Multiply each numerator by the same amount needed to multiply its

corresponding denominator to turn it into the LCM

6

For

\frac{3}{2} :

multiply the denominator and numerator by

3

\frac{3}{2} = \frac{3 \cdot 3}{2 \cdot 3} = \frac{9}{6}

For

\frac{2}{3} :

multiply the denominator and numerator by

2

\frac{2}{3} = \frac{2 \cdot 2}{3 \cdot 2} = \frac{4}{6}

= \frac{9}{6} - \frac{4}{6}

Since the denominators are equal, combine the fractions:

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

= \frac{9 - 4}{6}

Subtract the numbers:

9 - 4 = 5

= \frac{5}{6}

= \frac{\frac{5}{6}}{2}

Apply the fraction rule:

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

= \frac{5}{6 \cdot 2}

Multiply the numbers:

6 \cdot 2 = 12

= \frac{5}{12}

= \frac{5}{12}

= \frac{5}{12}

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Frequently Asked Questions (FAQ)

What is the value of sinh(ln(3/2)) ?
The value of sinh(ln(3/2)) is 5/12