What is the value of sqrt(2/pi)*sinh(1) ?

The value of sqrt(2/pi)*sinh(1) is (sqrt(2)sqrt(pi)(e^2-1))/(2epi)

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sqrt(2/pi)*sinh(1)

Solution

\frac{2}{π} \cdot sinh (1)

Solution

\frac{2 π ( e ^{2} - 1 )}{2 e π}

Decimal Notation

0.93767 \dots

Solution steps

\frac{2}{π} sinh (1)

Rewrite using trig identities:

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

sinh (1)

Use the Hyperbolic identity:

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

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

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

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

Apply rule

a^{1} = a

e^{1} = e

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

Apply exponent rule:

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

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

Join

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

e - \frac{1}{e}

Convert element to fraction:

e = \frac{ee}{e}

= \frac{ee}{e} - \frac{1}{e}

Since the denominators are equal, combine the fractions:

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

= \frac{ee - 1}{e}

ee - 1 = e^{2} - 1

ee - 1

ee = e^{2}

ee

Apply exponent rule:

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

ee = e^{1 + 1}

= e^{1 + 1}

Add the numbers:

1 + 1 = 2

= e^{2}

= e^{2} - 1

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

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

Apply the fraction rule:

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

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

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

= \frac{2}{π} \frac{e ^{2} - 1}{2 e}

Simplify

\frac{2}{π} \frac{e ^{2} - 1}{2 e} : \frac{2 π ( e ^{2} - 1 )}{2 e π}

\frac{2}{π} \frac{e ^{2} - 1}{2 e}

Multiply fractions:

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

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

\frac{2}{π} = \frac{2}{π}

\frac{2}{π}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{2}{π}

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

Multiply

(e^{2} - 1) \frac{2}{π} : \frac{2 ( e ^{2} - 1 )}{π}

(e^{2} - 1) \frac{2}{π}

Multiply fractions:

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

= \frac{2 ( e ^{2} - 1 )}{π}

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

Apply the fraction rule:

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

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

Apply radical rule:

2 = 2^{\frac{1}{2}}

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

Apply exponent rule:

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

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

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

Subtract the numbers:

1 - \frac{1}{2} = \frac{1}{2}

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

Apply radical rule:

2^{\frac{1}{2}} = 2

= \frac{e ^{2} - 1}{2 e π}

Rationalize

\frac{e ^{2} - 1}{2 e π} : \frac{2 π ( e ^{2} - 1 )}{2 e π}

\frac{e ^{2} - 1}{2 e π}

Multiply by the conjugate

\frac{π}{π}

= \frac{( e ^{2} - 1 ) π}{π 2 e π}

π 2 e π = 2 e π

π 2 e π

Apply radical rule:

a a = a

π π = π

= 2 e π

= \frac{π ( e ^{2} - 1 )}{2 e π}

Multiply by the conjugate

\frac{2}{2}

= \frac{π ( e ^{2} - 1 ) 2}{2 e π 2}

2 e π 2 = 2 e π

2 e π 2

Apply radical rule:

a a = a

22 = 2

= 2 e π

= \frac{2 π ( e ^{2} - 1 )}{2 e π}

= \frac{2 π ( e ^{2} - 1 )}{2 e π}

= \frac{2 π ( e ^{2} - 1 )}{2 e π}

Popular Examples

tan(pi+pi/3)

tan (π + \frac{π}{3})

6sin^2(x)-5sin(x)=0

6 sin^{2} (x) - 5 sin (x) = 0

sinh(4x)= 3/4

sinh (4 x) = \frac{3}{4}

tan(x)+sec(x)=3

tan (x) + sec (x) = 3

cos^2(3x+pi/4)= 3/4

cos^{2} (3 x + \frac{π}{4}) = \frac{3}{4}

Frequently Asked Questions (FAQ)

What is the value of sqrt(2/pi)*sinh(1) ?
The value of sqrt(2/pi)*sinh(1) is (sqrt(2)sqrt(pi)(e^2-1))/(2epi)