What is the value of sinh((pii)/2) ?

The value of sinh((pii)/2) is isin(pi/2)

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sinh((pii)/2)

sinh (\frac{πi}{2})

i sin (\frac{π}{2})

Solution steps

sinh (\frac{πi}{2})

Use the Hyperbolic identity:

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

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

Simplify

\frac{e ^{\frac{iπ}{2}} - e ^{- \frac{iπ}{2}}}{2} : \frac{- cos ( - \frac{π}{2} ) + cos ( \frac{π}{2} )}{2} + i \frac{- sin ( - \frac{π}{2} ) + sin ( \frac{π}{2} )}{2}

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

e^{\frac{iπ}{2}} - e^{- \frac{iπ}{2}} = cos (\frac{π}{2}) + i sin (\frac{π}{2}) - (cos (- \frac{π}{2}) + i sin (- \frac{π}{2}))

e^{\frac{iπ}{2}} - e^{- \frac{iπ}{2}}

Apply imaginary number rule:

e^{ia} = cos (a) + i sin (a)

= cos (\frac{π}{2}) + i sin (\frac{π}{2}) - e^{- \frac{iπ}{2}}

Apply imaginary number rule:

e^{ia} = cos (a) + i sin (a)

= cos (\frac{π}{2}) + i sin (\frac{π}{2}) - (cos (- \frac{π}{2}) + i sin (- \frac{π}{2}))

= \frac{cos ( \frac{π}{2} ) + i sin ( \frac{π}{2} ) - ( cos ( - \frac{π}{2} ) + i sin ( - \frac{π}{2} ) )}{2}

Expand

cos (\frac{π}{2}) + sin (\frac{π}{2}) i - (cos (- \frac{π}{2}) + sin (- \frac{π}{2}) i) : cos (\frac{π}{2}) + sin (\frac{π}{2}) i - cos (- \frac{π}{2}) - sin (- \frac{π}{2}) i

cos (\frac{π}{2}) + sin (\frac{π}{2}) i - (cos (- \frac{π}{2}) + sin (- \frac{π}{2}) i)

= cos (\frac{π}{2}) + i sin (\frac{π}{2}) - (cos (- \frac{π}{2}) + i sin (- \frac{π}{2}))

- (cos (- \frac{π}{2}) + sin (- \frac{π}{2}) i) : - cos (- \frac{π}{2}) - sin (- \frac{π}{2}) i

- (cos (- \frac{π}{2}) + sin (- \frac{π}{2}) i)

Distribute parentheses

= - (cos (- \frac{π}{2})) - (sin (- \frac{π}{2}) i)

Apply minus-plus rules

+ (- a) = - a

= - cos (- \frac{π}{2}) - sin (- \frac{π}{2}) i

= cos (\frac{π}{2}) + sin (\frac{π}{2}) i - cos (- \frac{π}{2}) - sin (- \frac{π}{2}) i

= \frac{cos ( \frac{π}{2} ) + i sin ( \frac{π}{2} ) - cos ( - \frac{π}{2} ) - i sin ( - \frac{π}{2} )}{2}

Rewrite

\frac{cos ( \frac{π}{2} ) + sin ( \frac{π}{2} ) i - cos ( - \frac{π}{2} ) - sin ( - \frac{π}{2} ) i}{2}

in standard complex form:

\frac{cos ( \frac{π}{2} ) - cos ( - \frac{π}{2} )}{2} + \frac{sin ( \frac{π}{2} ) - sin ( - \frac{π}{2} )}{2} i

\frac{cos ( \frac{π}{2} ) + sin ( \frac{π}{2} ) i - cos ( - \frac{π}{2} ) - sin ( - \frac{π}{2} ) i}{2}

Apply the fraction rule:

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

\frac{cos ( \frac{π}{2} ) + sin ( \frac{π}{2} ) i - cos ( - \frac{π}{2} ) - sin ( - \frac{π}{2} ) i}{2} = \frac{cos ( \frac{π}{2} )}{2} + \frac{sin ( \frac{π}{2} ) i}{2} - \frac{cos ( - \frac{π}{2} )}{2} - \frac{sin ( - \frac{π}{2} ) i}{2}

= \frac{cos ( \frac{π}{2} )}{2} + \frac{i sin ( \frac{π}{2} )}{2} - \frac{cos ( - \frac{π}{2} )}{2} - \frac{i sin ( - \frac{π}{2} )}{2}

Group like terms

= \frac{cos ( \frac{π}{2} )}{2} + \frac{i sin ( \frac{π}{2} )}{2} - \frac{cos ( - \frac{π}{2} )}{2} - \frac{i sin ( - \frac{π}{2} )}{2}

Group the real part and the imaginary part of the complex number

= (\frac{cos ( \frac{π}{2} )}{2} - \frac{cos ( - \frac{π}{2} )}{2}) + (\frac{sin ( \frac{π}{2} )}{2} - \frac{sin ( - \frac{π}{2} )}{2}) i

\frac{sin ( \frac{π}{2} )}{2} - \frac{sin ( - \frac{π}{2} )}{2} = \frac{sin ( \frac{π}{2} ) - sin ( - \frac{π}{2} )}{2}

\frac{sin ( \frac{π}{2} )}{2} - \frac{sin ( - \frac{π}{2} )}{2}

Apply rule

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

= \frac{sin ( \frac{π}{2} ) - sin ( - \frac{π}{2} )}{2}

= (\frac{cos ( \frac{π}{2} )}{2} - \frac{cos ( - \frac{π}{2} )}{2}) + \frac{sin ( \frac{π}{2} ) - sin ( - \frac{π}{2} )}{2} i

\frac{cos ( \frac{π}{2} )}{2} - \frac{cos ( - \frac{π}{2} )}{2} = \frac{cos ( \frac{π}{2} ) - cos ( - \frac{π}{2} )}{2}

\frac{cos ( \frac{π}{2} )}{2} - \frac{cos ( - \frac{π}{2} )}{2}

Apply rule

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

= \frac{cos ( \frac{π}{2} ) - cos ( - \frac{π}{2} )}{2}

= \frac{cos ( \frac{π}{2} ) - cos ( - \frac{π}{2} )}{2} + \frac{sin ( \frac{π}{2} ) - sin ( - \frac{π}{2} )}{2} i

= \frac{cos ( \frac{π}{2} ) - cos ( - \frac{π}{2} )}{2} + \frac{sin ( \frac{π}{2} ) - sin ( - \frac{π}{2} )}{2} i

= \frac{- cos ( - \frac{π}{2} ) + cos ( \frac{π}{2} )}{2} + i \frac{- sin ( - \frac{π}{2} ) + sin ( \frac{π}{2} )}{2}

Use the following property:

sin (- x) = - sin (x)

sin (- \frac{π}{2}) = - sin (\frac{π}{2})

= \frac{- cos ( - \frac{π}{2} ) + cos ( \frac{π}{2} )}{2} + i \frac{- ( - sin ( \frac{π}{2} ) ) + sin ( \frac{π}{2} )}{2}

Use the following property:

cos (- x) = cos (x)

cos (- \frac{π}{2}) = cos (\frac{π}{2})

= \frac{- cos ( \frac{π}{2} ) + cos ( \frac{π}{2} )}{2} + i \frac{- ( - sin ( \frac{π}{2} ) ) + sin ( \frac{π}{2} )}{2}

Simplify

= i sin (\frac{π}{2})

sin (\frac{29 π}{4})

cos ((\frac{3 π}{4}) + (\frac{5 π}{3}))

3 cos (8 5^{\circ})

arccos (\frac{- 12}{13})

\frac{150}{sin ( 4 3 ^{\circ} )}