What is the value of cosh(2)-sinh(sqrt(3)) ?

The value of cosh(2)-sinh(sqrt(3)) is (e^4+1-e^{sqrt(3)+2}+e^{-sqrt(3)+2})/(2e^2)

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cosh(2)-sinh(sqrt(3))

Solution

cosh (2) - sinh (3)

Solution

\frac{e ^{4} + 1 - e ^{3 + 2} + e ^{- 3 + 2}}{2 e ^{2}}

Decimal Notation

1.02453 \dots

Solution steps

cosh (2) - sinh (3)

Rewrite using trig identities:

sinh (3) = \frac{e ^{23} - 1}{2 e ^{3}}

sinh (3)

Use the Hyperbolic identity:

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

= \frac{e ^{3} - e ^{- 3}}{2}

\frac{e ^{3} - e ^{- 3}}{2} = \frac{e ^{23} - 1}{2 e ^{3}}

\frac{e ^{3} - e ^{- 3}}{2}

Apply exponent rule:

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

= \frac{e ^{3} - \frac{1}{e ^{3}}}{2}

Join

e^{3} - \frac{1}{e ^{3}} : \frac{e ^{23} - 1}{e ^{3}}

e^{3} - \frac{1}{e ^{3}}

Convert element to fraction:

e^{3} = \frac{e ^{3} e ^{3}}{e ^{3}}

= \frac{e ^{3} e ^{3}}{e ^{3}} - \frac{1}{e ^{3}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{3} e ^{3} - 1}{e ^{3}}

e^{3} e^{3} - 1 = e^{23} - 1

e^{3} e^{3} - 1

e^{3} e^{3} = e^{23}

e^{3} e^{3}

Apply exponent rule:

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

e^{3} e^{3} = e^{3 + 3}

= e^{3 + 3}

Add similar elements:

3 + 3 = 23

= e^{23}

= e^{23} - 1

= \frac{e ^{23} - 1}{e ^{3}}

= \frac{\frac{e ^{23} - 1}{e ^{3}}}{2}

Apply the fraction rule:

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

= \frac{e ^{23} - 1}{e ^{3} \cdot 2}

= \frac{e ^{23} - 1}{2 e ^{3}}

= cosh (2) - \frac{e ^{23} - 1}{2 e ^{3}}

cosh (2) - \frac{e ^{23} - 1}{2 e ^{3}} = \frac{2 e ^{3} cosh ( 2 ) - e ^{23} + 1}{2 e ^{3}}

cosh (2) - \frac{e ^{23} - 1}{2 e ^{3}}

Convert element to fraction:

cosh (2) = \frac{cosh ( 2 ) 2 e ^{3}}{2 e ^{3}}

= \frac{cosh ( 2 ) \cdot 2 e ^{3}}{2 e ^{3}} - \frac{e ^{23} - 1}{2 e ^{3}}

Since the denominators are equal, combine the fractions:

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

= \frac{cosh ( 2 ) \cdot 2 e ^{3} - ( e ^{23} - 1 )}{2 e ^{3}}

Expand

cosh (2) \cdot 2 e^{3} - (e^{23} - 1) : cosh (2) \cdot 2 e^{3} - e^{23} + 1

cosh (2) \cdot 2 e^{3} - (e^{23} - 1)

= 2 e^{3} cosh (2) - (e^{23} - 1)

- (e^{23} - 1) : - e^{23} + 1

- (e^{23} - 1)

Distribute parentheses

= - (e^{23}) - (- 1)

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - e^{23} + 1

= cosh (2) \cdot 2 e^{3} - e^{23} + 1

= \frac{2 e ^{3} cosh ( 2 ) - e ^{23} + 1}{2 e ^{3}}

= \frac{2 e ^{3} cosh ( 2 ) - e ^{23} + 1}{2 e ^{3}}

Rewrite using trig identities:

cosh (2) = \frac{e ^{4} + 1}{2 e ^{2}}

cosh (2)

Use the Hyperbolic identity:

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

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

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

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

Apply exponent rule:

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

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

Join

e^{2} + \frac{1}{e ^{2}} : \frac{e ^{4} + 1}{e ^{2}}

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

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

e^{2} e^{2} + 1 = e^{4} + 1

e^{2} e^{2} + 1

e^{2} e^{2} = e^{4}

e^{2} e^{2}

Apply exponent rule:

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

e^{2} e^{2} = e^{2 + 2}

= e^{2 + 2}

Add the numbers:

2 + 2 = 4

= e^{4}

= e^{4} + 1

= \frac{e ^{4} + 1}{e ^{2}}

= \frac{\frac{e ^{4} + 1}{e ^{2}}}{2}

Apply the fraction rule:

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

= \frac{e ^{4} + 1}{e ^{2} \cdot 2}

= \frac{e ^{4} + 1}{2 e ^{2}}

= \frac{2 e ^{3} \frac{e ^{4} + 1}{2 e ^{2}} - e ^{23} + 1}{2 e ^{3}}

Simplify

\frac{2 e ^{3} \frac{e ^{4} + 1}{2 e ^{2}} - e ^{23} + 1}{2 e ^{3}} : \frac{e ^{4} + 1 - e ^{3 + 2} + e ^{- 3 + 2}}{2 e ^{2}}

\frac{2 e ^{3} \frac{e ^{4} + 1}{2 e ^{2}} - e ^{23} + 1}{2 e ^{3}}

2 e^{3} \frac{e ^{4} + 1}{2 e ^{2}} = \frac{e ^{4} + 1}{e ^{- 3 + 2}}

2 e^{3} \frac{e ^{4} + 1}{2 e ^{2}}

Multiply fractions:

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

= \frac{( e ^{4} + 1 ) \cdot 2 e ^{3}}{2 e ^{2}}

Cancel the common factor:

2

= \frac{( e ^{4} + 1 ) e ^{3}}{e ^{2}}

Apply exponent rule:

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

\frac{e ^{3}}{e ^{2}} = \frac{1}{e ^{2 - 3}}

= \frac{e ^{4} + 1}{e ^{2 - 3}}

Subtract the numbers:

2 - 3 = - 3 + 2

= \frac{e ^{4} + 1}{e ^{- 3 + 2}}

= \frac{\frac{e ^{4} + 1}{e ^{2 - 3}} - e ^{23} + 1}{2 e ^{3}}

Join

\frac{e ^{4} + 1}{e ^{- 3 + 2}} - e^{23} + 1 : \frac{e ^{4} + 1 - e ^{2 + 3} + e ^{2 - 3}}{e ^{- 3 + 2}}

\frac{e ^{4} + 1}{e ^{- 3 + 2}} - e^{23} + 1

Convert element to fraction:

e^{23} = \frac{e ^{2 \cdot 3} e ^{- 3 + 2}}{e ^{- 3 + 2}}, 1 = \frac{1 e ^{- 3 + 2}}{e ^{- 3 + 2}}

= \frac{e ^{4} + 1}{e ^{- 3 + 2}} - \frac{e ^{23} e ^{- 3 + 2}}{e ^{- 3 + 2}} + \frac{1 \cdot e ^{- 3 + 2}}{e ^{- 3 + 2}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{4} + 1 - e ^{23} e ^{- 3 + 2} + 1 \cdot e ^{- 3 + 2}}{e ^{- 3 + 2}}

e^{4} + 1 - e^{23} e^{- 3 + 2} + 1 \cdot e^{- 3 + 2} = e^{4} + 1 - e^{3 + 2} + e^{- 3 + 2}

e^{4} + 1 - e^{23} e^{- 3 + 2} + 1 \cdot e^{- 3 + 2}

e^{23} e^{- 3 + 2} = e^{3 + 2}

e^{23} e^{- 3 + 2}

Apply exponent rule:

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

e^{23} e^{- 3 + 2} = e^{23 - 3 + 2}

= e^{23 - 3 + 2}

Add similar elements:

23 - 3 = 3

= e^{3 + 2}

1 \cdot e^{- 3 + 2} = e^{- 3 + 2}

1 \cdot e^{- 3 + 2}

Multiply:

1 \cdot e^{- 3 + 2} = e^{- 3 + 2}

= e^{- 3 + 2}

= e^{4} + 1 - e^{2 + 3} + e^{2 - 3}

= \frac{e ^{4} + 1 - e ^{2 + 3} + e ^{2 - 3}}{e ^{- 3 + 2}}

= \frac{\frac{e ^{4} + 1 - e ^{2 + 3} + e ^{2 - 3}}{e ^{- 3 + 2}}}{2 e ^{3}}

Apply the fraction rule:

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

= \frac{e ^{4} + 1 - e ^{3 + 2} + e ^{- 3 + 2}}{e ^{- 3 + 2} \cdot 2 e ^{3}}

e^{- 3 + 2} \cdot 2 e^{3} = 2 e^{2}

e^{- 3 + 2} \cdot 2 e^{3}

Apply exponent rule:

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

e^{- 3 + 2} e^{3} = e^{- 3 + 2 + 3}

= 2 e^{- 3 + 2 + 3}

Add similar elements:

- 3 + 3 = 0

= 2 e^{2}

= \frac{e ^{4} + 1 - e ^{2 + 3} + e ^{2 - 3}}{2 e ^{2}}

= \frac{e ^{4} + 1 - e ^{3 + 2} + e ^{- 3 + 2}}{2 e ^{2}}

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Frequently Asked Questions (FAQ)

What is the value of cosh(2)-sinh(sqrt(3)) ?
The value of cosh(2)-sinh(sqrt(3)) is (e^4+1-e^{sqrt(3)+2}+e^{-sqrt(3)+2})/(2e^2)