What is the value of 20(sinh(30/20)-sinh(-30/20)) ?

The value of 20(sinh(30/20)-sinh(-30/20)) is (20(e^3-1))/(e^{3/2)}

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20(sinh(30/20)-sinh(-30/20))

Solution

20 (sinh (\frac{30}{20}) - sinh (- \frac{30}{20}))

Solution

\frac{20 ( e ^{3} - 1 )}{e ^{\frac{3}{2}}}

Decimal Notation

85.17117 \dots

Solution steps

20 (sinh (\frac{30}{20}) - sinh (- \frac{30}{20}))

Simplify:

\frac{30}{20} = \frac{3}{2}

\frac{30}{20}

Cancel the common factor:

10

= \frac{3}{2}

Simplify:

- \frac{30}{20} = - \frac{3}{2}

- \frac{30}{20}

Cancel the common factor:

10

= - \frac{3}{2}

= 20 (sinh (\frac{3}{2}) - sinh (- \frac{3}{2}))

Use the following property:

sinh (- x) = - sinh (x)

sinh (- \frac{3}{2}) = - sinh (\frac{3}{2})

= 20 (sinh (\frac{3}{2}) - (- sinh (\frac{3}{2})))

Simplify

= 40 sinh (\frac{3}{2})

Rewrite using trig identities:

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

sinh (\frac{3}{2})

Use the Hyperbolic identity:

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

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

\frac{e ^{\frac{3}{2}} - e ^{- \frac{3}{2}}}{2} = \frac{e ^{3} - 1}{2 e ^{\frac{3}{2}}}

\frac{e ^{\frac{3}{2}} - e ^{- \frac{3}{2}}}{2}

Apply exponent rule:

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

= \frac{e ^{\frac{3}{2}} - \frac{1}{e ^{\frac{3}{2}}}}{2}

Join

e^{\frac{3}{2}} - \frac{1}{e ^{\frac{3}{2}}} : \frac{e ^{3} - 1}{e ^{\frac{3}{2}}}

e^{\frac{3}{2}} - \frac{1}{e ^{\frac{3}{2}}}

Convert element to fraction:

e^{\frac{3}{2}} = \frac{e ^{\frac{3}{2}} e ^{\frac{3}{2}}}{e ^{\frac{3}{2}}}

= \frac{e ^{\frac{3}{2}} e ^{\frac{3}{2}}}{e ^{\frac{3}{2}}} - \frac{1}{e ^{\frac{3}{2}}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{\frac{3}{2}} e ^{\frac{3}{2}} - 1}{e ^{\frac{3}{2}}}

e^{\frac{3}{2}} e^{\frac{3}{2}} - 1 = e^{2 \cdot \frac{3}{2}} - 1

e^{\frac{3}{2}} e^{\frac{3}{2}} - 1

e^{\frac{3}{2}} e^{\frac{3}{2}} = e^{2 \cdot \frac{3}{2}}

e^{\frac{3}{2}} e^{\frac{3}{2}}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{\frac{3}{2}} e^{\frac{3}{2}} = e^{\frac{3}{2} + \frac{3}{2}}

= e^{\frac{3}{2} + \frac{3}{2}}

Add similar elements:

\frac{3}{2} + \frac{3}{2} = 2 \cdot \frac{3}{2}

= e^{2 \cdot \frac{3}{2}}

= e^{2 \cdot \frac{3}{2}} - 1

= \frac{e ^{2 \cdot \frac{3}{2}} - 1}{e ^{\frac{3}{2}}}

Multiply

2 \cdot \frac{3}{2} : 3

2 \cdot \frac{3}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 3

= \frac{e ^{3} - 1}{e ^{\frac{3}{2}}}

= \frac{\frac{e ^{3} - 1}{e ^{\frac{3}{2}}}}{2}

Apply the fraction rule:

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

= \frac{e ^{3} - 1}{e ^{\frac{3}{2}} \cdot 2}

= \frac{e ^{3} - 1}{2 e ^{\frac{3}{2}}}

= 40 \cdot \frac{e ^{3} - 1}{2 e ^{\frac{3}{2}}}

Simplify

40 \cdot \frac{e ^{3} - 1}{2 e ^{\frac{3}{2}}} : \frac{20 ( e ^{3} - 1 )}{e ^{\frac{3}{2}}}

40 \cdot \frac{e ^{3} - 1}{2 e ^{\frac{3}{2}}}

Multiply fractions:

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

= \frac{( e ^{3} - 1 ) \cdot 40}{2 e ^{\frac{3}{2}}}

Divide the numbers:

\frac{40}{2} = 20

= \frac{20 ( e ^{3} - 1 )}{e ^{\frac{3}{2}}}

= \frac{20 ( e ^{3} - 1 )}{e ^{\frac{3}{2}}}

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Frequently Asked Questions (FAQ)

What is the value of 20(sinh(30/20)-sinh(-30/20)) ?
The value of 20(sinh(30/20)-sinh(-30/20)) is (20(e^3-1))/(e^{3/2)}