What is the value of 2-2cosh(0.1-2) ?

The value of 2-2cosh(0.1-2) is (2e^{19/10}-e^{19/5}-1)/(e^{19/10)}

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2-2cosh(0.1-2)

Solution

2 - 2 cosh (0.1 - 2)

Solution

\frac{2 e ^{\frac{19}{10}} - e ^{\frac{19}{5}} - 1}{e ^{\frac{19}{10}}}

Decimal Notation

- 4.83546 \dots

Solution steps

2 - 2 cosh (0.1 - 2)

= 2 - 2 cosh (\frac{1}{10} - 2)

Simplify:

\frac{1}{10} - 2 = - \frac{19}{10}

\frac{1}{10} - 2

Convert element to fraction:

2 = \frac{2 \cdot 10}{10}

= \frac{1}{10} - \frac{2 \cdot 10}{10}

Since the denominators are equal, combine the fractions:

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

= \frac{1 - 2 \cdot 10}{10}

1 - 2 \cdot 10 = - 19

1 - 2 \cdot 10

Multiply the numbers:

2 \cdot 10 = 20

= 1 - 20

Subtract the numbers:

1 - 20 = - 19

= - 19

= \frac{- 19}{10}

Apply the fraction rule:

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

= - \frac{19}{10}

= 2 - 2 cosh (- \frac{19}{10})

Use the following property:

cosh (- x) = cosh (x)

cosh (- \frac{19}{10}) = cosh (\frac{19}{10})

= 2 - 2 cosh (\frac{19}{10})

Rewrite using trig identities:

cosh (\frac{19}{10}) = \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}}

cosh (\frac{19}{10})

Use the Hyperbolic identity:

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

= \frac{e ^{\frac{19}{10}} + e ^{- \frac{19}{10}}}{2}

\frac{e ^{\frac{19}{10}} + e ^{- \frac{19}{10}}}{2} = \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}}

\frac{e ^{\frac{19}{10}} + e ^{- \frac{19}{10}}}{2}

Apply exponent rule:

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

= \frac{e ^{\frac{19}{10}} + \frac{1}{e ^{\frac{19}{10}}}}{2}

Join

e^{\frac{19}{10}} + \frac{1}{e ^{\frac{19}{10}}} : \frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}}}

e^{\frac{19}{10}} + \frac{1}{e ^{\frac{19}{10}}}

Convert element to fraction:

e^{\frac{19}{10}} = \frac{e ^{\frac{19}{10}} e ^{\frac{19}{10}}}{e ^{\frac{19}{10}}}

= \frac{e ^{\frac{19}{10}} e ^{\frac{19}{10}}}{e ^{\frac{19}{10}}} + \frac{1}{e ^{\frac{19}{10}}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{\frac{19}{10}} e ^{\frac{19}{10}} + 1}{e ^{\frac{19}{10}}}

e^{\frac{19}{10}} e^{\frac{19}{10}} + 1 = e^{2 \cdot \frac{19}{10}} + 1

e^{\frac{19}{10}} e^{\frac{19}{10}} + 1

e^{\frac{19}{10}} e^{\frac{19}{10}} = e^{2 \cdot \frac{19}{10}}

e^{\frac{19}{10}} e^{\frac{19}{10}}

Apply exponent rule:

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

e^{\frac{19}{10}} e^{\frac{19}{10}} = e^{\frac{19}{10} + \frac{19}{10}}

= e^{\frac{19}{10} + \frac{19}{10}}

Add similar elements:

\frac{19}{10} + \frac{19}{10} = 2 \cdot \frac{19}{10}

= e^{2 \cdot \frac{19}{10}}

= e^{2 \cdot \frac{19}{10}} + 1

= \frac{e ^{2 \cdot \frac{19}{10}} + 1}{e ^{\frac{19}{10}}}

Multiply

2 \cdot \frac{19}{10} : \frac{19}{5}

2 \cdot \frac{19}{10}

Multiply fractions:

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

= \frac{19 \cdot 2}{10}

Multiply the numbers:

19 \cdot 2 = 38

= \frac{38}{10}

Cancel the common factor:

2

= \frac{19}{5}

= \frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}}}

= \frac{\frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}}}}{2}

Apply the fraction rule:

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

= \frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}} \cdot 2}

= \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}}

= 2 - 2 \cdot \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}}

2 - 2 \cdot \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}} = \frac{2 e ^{\frac{19}{10}} - e ^{\frac{19}{5}} - 1}{e ^{\frac{19}{10}}}

2 - 2 \cdot \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}}

2 \cdot \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}} = \frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}}}

2 \cdot \frac{e ^{\frac{19}{5}} + 1}{2 e ^{\frac{19}{10}}}

Multiply fractions:

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

= \frac{( e ^{\frac{19}{5}} + 1 ) \cdot 2}{2 e ^{\frac{19}{10}}}

Cancel the common factor:

2

= \frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}}}

= 2 - \frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}}}

Convert element to fraction:

2 = \frac{2 e ^{\frac{19}{10}}}{e ^{\frac{19}{10}}}

= \frac{2 e ^{\frac{19}{10}}}{e ^{\frac{19}{10}}} - \frac{e ^{\frac{19}{5}} + 1}{e ^{\frac{19}{10}}}

Since the denominators are equal, combine the fractions:

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

= \frac{2 e ^{\frac{19}{10}} - ( e ^{\frac{19}{5}} + 1 )}{e ^{\frac{19}{10}}}

- (e^{\frac{19}{5}} + 1) : - e^{\frac{19}{5}} - 1

- (e^{\frac{19}{5}} + 1)

Distribute parentheses

= - (e^{\frac{19}{5}}) - (1)

Apply minus-plus rules

+ (- a) = - a

= - e^{\frac{19}{5}} - 1

= \frac{2 e ^{\frac{19}{10}} - e ^{\frac{19}{5}} - 1}{e ^{\frac{19}{10}}}

= \frac{2 e ^{\frac{19}{10}} - e ^{\frac{19}{5}} - 1}{e ^{\frac{19}{10}}}

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Frequently Asked Questions (FAQ)

What is the value of 2-2cosh(0.1-2) ?
The value of 2-2cosh(0.1-2) is (2e^{19/10}-e^{19/5}-1)/(e^{19/10)}