What is the value of sinh(ln(2)+pi/2 i) ?

The value of sinh(ln(2)+pi/2 i) is (3cos(pi/2))/4+i(5sin(pi/2))/4

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sinh(ln(2)+pi/2 i)

Solution

sinh (ln (2) + \frac{π}{2} i)

Solution

\frac{3 cos ( \frac{π}{2} )}{4} + i \frac{5 sin ( \frac{π}{2} )}{4}

Solution steps

sinh (ln (2) + \frac{π}{2} i)

Simplify

sinh (ln (2) + \frac{π}{2} i) : sinh (\frac{2 ln ( 2 ) + πi}{2})

sinh (ln (2) + \frac{π}{2} i)

Multiply

\frac{π}{2} i : \frac{πi}{2}

\frac{π}{2} i

Multiply fractions:

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

= \frac{πi}{2}

= sinh (ln (2) + \frac{πi}{2})

Join

ln (2) + \frac{πi}{2} : \frac{2 ln ( 2 ) + πi}{2}

ln (2) + \frac{πi}{2}

Convert element to fraction:

ln (2) = \frac{ln ( 2 ) 2}{2}

= \frac{ln ( 2 ) \cdot 2}{2} + \frac{πi}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{ln ( 2 ) \cdot 2 + πi}{2}

= sinh (\frac{ln ( 2 ) \cdot 2 + πi}{2})

= sinh (\frac{2 ln ( 2 ) + πi}{2})

= sinh (\frac{2 ln ( 2 ) + πi}{2})

Rewrite using trig identities:

\frac{3 cos ( \frac{π}{2} )}{4} + i \frac{5 sin ( \frac{π}{2} )}{4}

sinh (\frac{2 ln ( 2 ) + πi}{2})

Use the Hyperbolic identity:

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

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

Simplify

\frac{e ^{\frac{2 l n ( 2 ) + πi}{2}} - e ^{- \frac{2 l n ( 2 ) + πi}{2}}}{2} : \frac{- cos ( - \frac{π}{2} ) + 4 cos ( \frac{π}{2} )}{4} + i \frac{- sin ( - \frac{π}{2} ) + 4 sin ( \frac{π}{2} )}{4}

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

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

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

Apply imaginary number rule:

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

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

Apply imaginary number rule:

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

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

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

e^{l n (2)} (cos (\frac{π}{2}) + sin (\frac{π}{2}) i) = 2 (cos (\frac{π}{2}) + i sin (\frac{π}{2}))

e^{l n (2)} (cos (\frac{π}{2}) + sin (\frac{π}{2}) i)

e^{l n (2)} = 2

e^{l n (2)}

Apply log rule:

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

= 2

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

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

e^{- l n (2)} (cos (- \frac{π}{2}) + sin (- \frac{π}{2}) i)

Apply exponent rule:

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

e^{- l n (2)} = \frac{1}{e ^{l n (2)}}

= \frac{1}{e ^{l n (2)}} (cos (- \frac{π}{2}) + i sin (- \frac{π}{2}))

Multiply fractions:

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

= \frac{1 \cdot ( cos ( - \frac{π}{2} ) + sin ( - \frac{π}{2} ) i )}{e ^{l n (2)}}

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

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

Multiply:

1 \cdot (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}))

Remove parentheses:

(a) = a

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

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

e^{l n (2)} = 2

e^{l n (2)}

Apply log rule:

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

= 2

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

= \frac{2 ( cos ( \frac{π}{2} ) + i sin ( \frac{π}{2} ) ) - \frac{c o s ( - \frac{π}{2} ) + i s i n ( - \frac{π}{2} )}{2}}{2}

Join

2 (cos (\frac{π}{2}) + sin (\frac{π}{2}) i) - \frac{cos ( - \frac{π}{2} ) + sin ( - \frac{π}{2} ) i}{2} : \frac{4 cos ( \frac{π}{2} ) + 4 i sin ( \frac{π}{2} ) - cos ( - \frac{π}{2} ) - i sin ( - \frac{π}{2} )}{2}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 ( cos ( \frac{π}{2} ) + i sin ( \frac{π}{2} ) ) - ( cos ( - \frac{π}{2} ) + i sin ( - \frac{π}{2} ) )}{2}

Expand

4 (cos (\frac{π}{2}) + sin (\frac{π}{2}) i) - (cos (- \frac{π}{2}) + sin (- \frac{π}{2}) i) : 4 cos (\frac{π}{2}) + 4 i sin (\frac{π}{2}) - cos (- \frac{π}{2}) - sin (- \frac{π}{2}) i

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

= 4 (cos (\frac{π}{2}) + i sin (\frac{π}{2})) - (cos (- \frac{π}{2}) + i sin (- \frac{π}{2}))

Expand

4 (cos (\frac{π}{2}) + sin (\frac{π}{2}) i) : 4 cos (\frac{π}{2}) + 4 i sin (\frac{π}{2})

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

Apply the distributive law:

a (b + c) = ab + a c

a = 4, b = cos (\frac{π}{2}), c = sin (\frac{π}{2}) i

= 4 cos (\frac{π}{2}) + 4 sin (\frac{π}{2}) i

= 4 cos (\frac{π}{2}) + 4 i sin (\frac{π}{2})

= 4 cos (\frac{π}{2}) + 4 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

- (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

= 4 cos (\frac{π}{2}) + 4 i sin (\frac{π}{2}) - cos (- \frac{π}{2}) - sin (- \frac{π}{2}) i

= \frac{4 cos ( \frac{π}{2} ) + 4 i sin ( \frac{π}{2} ) - cos ( - \frac{π}{2} ) - i sin ( - \frac{π}{2} )}{2}

= \frac{\frac{4 c o s ( \frac{π}{2} ) + 4 i s i n ( \frac{π}{2} ) - c o s ( - \frac{π}{2} ) - i s i n ( - \frac{π}{2} )}{2}}{2}

Apply the fraction rule:

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

= \frac{4 cos ( \frac{π}{2} ) + 4 i sin ( \frac{π}{2} ) - cos ( - \frac{π}{2} ) - sin ( - \frac{π}{2} ) i}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{4 cos ( \frac{π}{2} ) + 4 i sin ( \frac{π}{2} ) - cos ( - \frac{π}{2} ) - i sin ( - \frac{π}{2} )}{4}

Rewrite

\frac{4 cos ( \frac{π}{2} ) + 4 i sin ( \frac{π}{2} ) - cos ( - \frac{π}{2} ) - sin ( - \frac{π}{2} ) i}{4}

in standard complex form:

\frac{4 cos ( \frac{π}{2} ) - cos ( - \frac{π}{2} )}{4} + \frac{4 sin ( \frac{π}{2} ) - sin ( - \frac{π}{2} )}{4} i

\frac{4 cos ( \frac{π}{2} ) + 4 i sin ( \frac{π}{2} ) - cos ( - \frac{π}{2} ) - sin ( - \frac{π}{2} ) i}{4}

Apply the fraction rule:

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

\frac{4 cos ( \frac{π}{2} ) + 4 i sin ( \frac{π}{2} ) - cos ( - \frac{π}{2} ) - sin ( - \frac{π}{2} ) i}{4} = \frac{4 cos ( \frac{π}{2} )}{4} + \frac{4 i sin ( \frac{π}{2} )}{4} - \frac{cos ( - \frac{π}{2} )}{4} - \frac{sin ( - \frac{π}{2} ) i}{4}

= \frac{4 cos ( \frac{π}{2} )}{4} + \frac{4 i sin ( \frac{π}{2} )}{4} - \frac{cos ( - \frac{π}{2} )}{4} - \frac{i sin ( - \frac{π}{2} )}{4}

Group like terms

= \frac{4 cos ( \frac{π}{2} )}{4} - \frac{cos ( - \frac{π}{2} )}{4} + \frac{4 i sin ( \frac{π}{2} )}{4} - \frac{i sin ( - \frac{π}{2} )}{4}

Divide the numbers:

\frac{4}{4} = 1

= cos (\frac{π}{2}) - \frac{cos ( - \frac{π}{2} )}{4} + i sin (\frac{π}{2}) - \frac{i sin ( - \frac{π}{2} )}{4}

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

= (cos (\frac{π}{2}) - \frac{cos ( - \frac{π}{2} )}{4}) + (sin (\frac{π}{2}) - \frac{sin ( - \frac{π}{2} )}{4}) i

sin (\frac{π}{2}) - \frac{sin ( - \frac{π}{2} )}{4} = \frac{4 sin ( \frac{π}{2} ) - sin ( - \frac{π}{2} )}{4}

sin (\frac{π}{2}) - \frac{sin ( - \frac{π}{2} )}{4}

Convert element to fraction:

sin (\frac{π}{2}) = \frac{sin ( \frac{π}{2} ) 4}{4}

= \frac{sin ( \frac{π}{2} ) \cdot 4}{4} - \frac{sin ( - \frac{π}{2} )}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{sin ( \frac{π}{2} ) \cdot 4 - sin ( - \frac{π}{2} )}{4}

= (cos (\frac{π}{2}) - \frac{cos ( - \frac{π}{2} )}{4}) + \frac{4 sin ( \frac{π}{2} ) - sin ( - \frac{π}{2} )}{4} i

cos (\frac{π}{2}) - \frac{cos ( - \frac{π}{2} )}{4} = \frac{4 cos ( \frac{π}{2} ) - cos ( - \frac{π}{2} )}{4}

cos (\frac{π}{2}) - \frac{cos ( - \frac{π}{2} )}{4}

Convert element to fraction:

cos (\frac{π}{2}) = \frac{cos ( \frac{π}{2} ) 4}{4}

= \frac{cos ( \frac{π}{2} ) \cdot 4}{4} - \frac{cos ( - \frac{π}{2} )}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{cos ( \frac{π}{2} ) \cdot 4 - cos ( - \frac{π}{2} )}{4}

= \frac{4 cos ( \frac{π}{2} ) - cos ( - \frac{π}{2} )}{4} + \frac{4 sin ( \frac{π}{2} ) - sin ( - \frac{π}{2} )}{4} i

= \frac{4 cos ( \frac{π}{2} ) - cos ( - \frac{π}{2} )}{4} + \frac{4 sin ( \frac{π}{2} ) - sin ( - \frac{π}{2} )}{4} i

= \frac{- cos ( - \frac{π}{2} ) + 4 cos ( \frac{π}{2} )}{4} + i \frac{- sin ( - \frac{π}{2} ) + 4 sin ( \frac{π}{2} )}{4}

Use the following property:

sin (- x) = - sin (x)

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

= \frac{- cos ( - \frac{π}{2} ) + 4 cos ( \frac{π}{2} )}{4} + i \frac{- ( - sin ( \frac{π}{2} ) ) + 4 sin ( \frac{π}{2} )}{4}

Use the following property:

cos (- x) = cos (x)

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

= \frac{- cos ( \frac{π}{2} ) + 4 cos ( \frac{π}{2} )}{4} + i \frac{- ( - sin ( \frac{π}{2} ) ) + 4 sin ( \frac{π}{2} )}{4}

Simplify

= \frac{3 cos ( \frac{π}{2} )}{4} + i \frac{5 sin ( \frac{π}{2} )}{4}

= \frac{3 cos ( \frac{π}{2} )}{4} + i \frac{5 sin ( \frac{π}{2} )}{4}

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Frequently Asked Questions (FAQ)

What is the value of sinh(ln(2)+pi/2 i) ?
The value of sinh(ln(2)+pi/2 i) is (3cos(pi/2))/4+i(5sin(pi/2))/4