What is the value of sinh(pi/4) ?

The value of sinh(pi/4) is (e^{pi/2}-1)/(2e^{pi/4)}

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sinh(pi/4)

sinh (\frac{π}{4})

\frac{e ^{\frac{π}{2}} - 1}{2 e ^{\frac{π}{4}}}

Decimal Notation

0.86867 \dots

Solution steps

sinh (\frac{π}{4})

Use the Hyperbolic identity:

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

= \frac{e ^{\frac{π}{4}} - e ^{- \frac{π}{4}}}{2}

\frac{e ^{\frac{π}{4}} - e ^{- \frac{π}{4}}}{2} = \frac{e ^{\frac{π}{2}} - 1}{2 e ^{\frac{π}{4}}}

\frac{e ^{\frac{π}{4}} - e ^{- \frac{π}{4}}}{2}

Apply exponent rule:

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

= \frac{e ^{\frac{π}{4}} - \frac{1}{e ^{\frac{π}{4}}}}{2}

Join

e^{\frac{π}{4}} - \frac{1}{e ^{\frac{π}{4}}} : \frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{4}}}

e^{\frac{π}{4}} - \frac{1}{e ^{\frac{π}{4}}}

Convert element to fraction:

e^{\frac{π}{4}} = \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}}}{e ^{\frac{π}{4}}}

= \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}}}{e ^{\frac{π}{4}}} - \frac{1}{e ^{\frac{π}{4}}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{\frac{π}{4}} e ^{\frac{π}{4}} - 1}{e ^{\frac{π}{4}}}

e^{\frac{π}{4}} e^{\frac{π}{4}} - 1 = e^{2 \cdot \frac{π}{4}} - 1

e^{\frac{π}{4}} e^{\frac{π}{4}} - 1

e^{\frac{π}{4}} e^{\frac{π}{4}} = e^{2 \cdot \frac{π}{4}}

e^{\frac{π}{4}} e^{\frac{π}{4}}

Apply exponent rule:

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

e^{\frac{π}{4}} e^{\frac{π}{4}} = e^{\frac{π}{4} + \frac{π}{4}}

= e^{\frac{π}{4} + \frac{π}{4}}

Add similar elements:

\frac{π}{4} + \frac{π}{4} = 2 \cdot \frac{π}{4}

= e^{2 \cdot \frac{π}{4}}

= e^{2 \cdot \frac{π}{4}} - 1

= \frac{e ^{2 \cdot \frac{π}{4}} - 1}{e ^{\frac{π}{4}}}

Multiply

2 \cdot \frac{π}{4} : \frac{π}{2}

2 \cdot \frac{π}{4}

Multiply fractions:

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

= \frac{π 2}{4}

Cancel the common factor:

2

= \frac{π}{2}

= \frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{4}}}

= \frac{\frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{4}}}}{2}

Apply the fraction rule:

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

= \frac{e ^{\frac{π}{2}} - 1}{e ^{\frac{π}{4}} \cdot 2}

= \frac{e ^{\frac{π}{2}} - 1}{2 e ^{\frac{π}{4}}}

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What is the value of sinh(pi/4) ?
The value of sinh(pi/4) is (e^{pi/2}-1)/(2e^{pi/4)}