What is the value of sinh(2ln(5)) ?

The value of sinh(2ln(5)) is 312/25

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sinh(2ln(5))

sinh (2 ln (5))

\frac{312}{25}

Decimal

12.48

Solution steps

sinh (2 ln (5))

Use the Hyperbolic identity:

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

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

\frac{e ^{2 l n (5)} - e ^{- 2 l n (5)}}{2} = \frac{312}{25}

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

e^{2 l n (5)} = 5^{2}

e^{2 l n (5)}

Apply exponent rule:

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

= (e^{l n (5)})^{2}

Apply log rule:

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

e^{l n (5)} = 5

= 5^{2}

e^{- 2 l n (5)} = 5^{- 2}

e^{- 2 l n (5)}

Apply exponent rule:

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

= (e^{l n (5)})^{- 2}

Apply log rule:

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

e^{l n (5)} = 5

= 5^{- 2}

= \frac{5 ^{2} - 5 ^{- 2}}{2}

Simplify

\frac{5 ^{2} - 5 ^{- 2}}{2}

5^{2} = 25

= \frac{25 - 5 ^{- 2}}{2}

5^{- 2} = \frac{1}{25}

5^{- 2}

Apply exponent rule:

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

= \frac{1}{5 ^{2}}

5^{2} = 25

= \frac{1}{25}

= \frac{25 - \frac{1}{25}}{2}

Join

25 - \frac{1}{25} : \frac{624}{25}

25 - \frac{1}{25}

Convert element to fraction:

25 = \frac{25 \cdot 25}{25}

= \frac{25 \cdot 25}{25} - \frac{1}{25}

Since the denominators are equal, combine the fractions:

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

= \frac{25 \cdot 25 - 1}{25}

25 \cdot 25 - 1 = 624

25 \cdot 25 - 1

Multiply the numbers:

25 \cdot 25 = 625

= 625 - 1

Subtract the numbers:

625 - 1 = 624

= 624

= \frac{624}{25}

= \frac{\frac{624}{25}}{2}

Apply the fraction rule:

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

= \frac{624}{25 \cdot 2}

Multiply the numbers:

25 \cdot 2 = 50

= \frac{624}{50}

Cancel the common factor:

2

= \frac{312}{25}

= \frac{312}{25}

= \frac{312}{25}