What is the value of sinh(pi/6 i) ?

The value of sinh(pi/6 i) is isin(pi/6)

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

sinh(pi/6 i)

sinh (\frac{π}{6} i)

i sin (\frac{π}{6})

Solution steps

sinh (\frac{π}{6} i)

Simplify:

\frac{π}{6} i = \frac{πi}{6}

\frac{π}{6} i

Multiply fractions:

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

= \frac{πi}{6}

= sinh (\frac{πi}{6})

Rewrite using trig identities:

i sin (\frac{π}{6})

sinh (\frac{πi}{6})

Use the Hyperbolic identity:

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

= \frac{e ^{\frac{πi}{6}} - e ^{- \frac{πi}{6}}}{2}

Simplify

\frac{e ^{\frac{πi}{6}} - e ^{- \frac{πi}{6}}}{2} : \frac{- cos ( - \frac{π}{6} ) + cos ( \frac{π}{6} )}{2} + i \frac{- sin ( - \frac{π}{6} ) + sin ( \frac{π}{6} )}{2}

\frac{e ^{\frac{πi}{6}} - e ^{- \frac{πi}{6}}}{2}

e^{\frac{πi}{6}} - e^{- \frac{πi}{6}} = cos (\frac{π}{6}) + i sin (\frac{π}{6}) - (cos (- \frac{π}{6}) + i sin (- \frac{π}{6}))

e^{\frac{πi}{6}} - e^{- \frac{πi}{6}}

Apply imaginary number rule:

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

= cos (\frac{π}{6}) + i sin (\frac{π}{6}) - e^{- \frac{πi}{6}}

Apply imaginary number rule:

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

= cos (\frac{π}{6}) + i sin (\frac{π}{6}) - (cos (- \frac{π}{6}) + i sin (- \frac{π}{6}))

= \frac{cos ( \frac{π}{6} ) + i sin ( \frac{π}{6} ) - ( cos ( - \frac{π}{6} ) + i sin ( - \frac{π}{6} ) )}{2}

Expand

cos (\frac{π}{6}) + sin (\frac{π}{6}) i - (cos (- \frac{π}{6}) + sin (- \frac{π}{6}) i) : cos (\frac{π}{6}) + sin (\frac{π}{6}) i - cos (- \frac{π}{6}) - sin (- \frac{π}{6}) i

cos (\frac{π}{6}) + sin (\frac{π}{6}) i - (cos (- \frac{π}{6}) + sin (- \frac{π}{6}) i)

= cos (\frac{π}{6}) + i sin (\frac{π}{6}) - (cos (- \frac{π}{6}) + i sin (- \frac{π}{6}))

- (cos (- \frac{π}{6}) + sin (- \frac{π}{6}) i) : - cos (- \frac{π}{6}) - sin (- \frac{π}{6}) i

- (cos (- \frac{π}{6}) + sin (- \frac{π}{6}) i)

Distribute parentheses

= - (cos (- \frac{π}{6})) - (sin (- \frac{π}{6}) i)

Apply minus-plus rules

+ (- a) = - a

= - cos (- \frac{π}{6}) - sin (- \frac{π}{6}) i

= cos (\frac{π}{6}) + sin (\frac{π}{6}) i - cos (- \frac{π}{6}) - sin (- \frac{π}{6}) i

= \frac{cos ( \frac{π}{6} ) + i sin ( \frac{π}{6} ) - cos ( - \frac{π}{6} ) - i sin ( - \frac{π}{6} )}{2}

Rewrite

\frac{cos ( \frac{π}{6} ) + sin ( \frac{π}{6} ) i - cos ( - \frac{π}{6} ) - sin ( - \frac{π}{6} ) i}{2}

in standard complex form:

\frac{cos ( \frac{π}{6} ) - cos ( - \frac{π}{6} )}{2} + \frac{sin ( \frac{π}{6} ) - sin ( - \frac{π}{6} )}{2} i

\frac{cos ( \frac{π}{6} ) + sin ( \frac{π}{6} ) i - cos ( - \frac{π}{6} ) - sin ( - \frac{π}{6} ) i}{2}

Apply the fraction rule:

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

\frac{cos ( \frac{π}{6} ) + sin ( \frac{π}{6} ) i - cos ( - \frac{π}{6} ) - sin ( - \frac{π}{6} ) i}{2} = \frac{cos ( \frac{π}{6} )}{2} + \frac{sin ( \frac{π}{6} ) i}{2} - \frac{cos ( - \frac{π}{6} )}{2} - \frac{sin ( - \frac{π}{6} ) i}{2}

= \frac{cos ( \frac{π}{6} )}{2} + \frac{i sin ( \frac{π}{6} )}{2} - \frac{cos ( - \frac{π}{6} )}{2} - \frac{i sin ( - \frac{π}{6} )}{2}

Group like terms

= \frac{cos ( \frac{π}{6} )}{2} + \frac{i sin ( \frac{π}{6} )}{2} - \frac{cos ( - \frac{π}{6} )}{2} - \frac{i sin ( - \frac{π}{6} )}{2}

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

= (\frac{cos ( \frac{π}{6} )}{2} - \frac{cos ( - \frac{π}{6} )}{2}) + (\frac{sin ( \frac{π}{6} )}{2} - \frac{sin ( - \frac{π}{6} )}{2}) i

\frac{sin ( \frac{π}{6} )}{2} - \frac{sin ( - \frac{π}{6} )}{2} = \frac{sin ( \frac{π}{6} ) - sin ( - \frac{π}{6} )}{2}

\frac{sin ( \frac{π}{6} )}{2} - \frac{sin ( - \frac{π}{6} )}{2}

Apply rule

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

= \frac{sin ( \frac{π}{6} ) - sin ( - \frac{π}{6} )}{2}

= (\frac{cos ( \frac{π}{6} )}{2} - \frac{cos ( - \frac{π}{6} )}{2}) + \frac{sin ( \frac{π}{6} ) - sin ( - \frac{π}{6} )}{2} i

\frac{cos ( \frac{π}{6} )}{2} - \frac{cos ( - \frac{π}{6} )}{2} = \frac{cos ( \frac{π}{6} ) - cos ( - \frac{π}{6} )}{2}

\frac{cos ( \frac{π}{6} )}{2} - \frac{cos ( - \frac{π}{6} )}{2}

Apply rule

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

= \frac{cos ( \frac{π}{6} ) - cos ( - \frac{π}{6} )}{2}

= \frac{cos ( \frac{π}{6} ) - cos ( - \frac{π}{6} )}{2} + \frac{sin ( \frac{π}{6} ) - sin ( - \frac{π}{6} )}{2} i

= \frac{cos ( \frac{π}{6} ) - cos ( - \frac{π}{6} )}{2} + \frac{sin ( \frac{π}{6} ) - sin ( - \frac{π}{6} )}{2} i

= \frac{- cos ( - \frac{π}{6} ) + cos ( \frac{π}{6} )}{2} + i \frac{- sin ( - \frac{π}{6} ) + sin ( \frac{π}{6} )}{2}

Use the following property:

sin (- x) = - sin (x)

sin (- \frac{π}{6}) = - sin (\frac{π}{6})

= \frac{- cos ( - \frac{π}{6} ) + cos ( \frac{π}{6} )}{2} + i \frac{- ( - sin ( \frac{π}{6} ) ) + sin ( \frac{π}{6} )}{2}

Use the following property:

cos (- x) = cos (x)

cos (- \frac{π}{6}) = cos (\frac{π}{6})

= \frac{- cos ( \frac{π}{6} ) + cos ( \frac{π}{6} )}{2} + i \frac{- ( - sin ( \frac{π}{6} ) ) + sin ( \frac{π}{6} )}{2}

Simplify

= i sin (\frac{π}{6})

= i sin (\frac{π}{6})

arcsinh (5 + 12 i)

sin (10 2^{\circ})

- 2 sec (\frac{π}{2})

5 + 4 cos (0)

arctan (0.245)