What is the value of csch(-0.5) ?

The value of csch(-0.5) is -(2e^{3/2}+2sqrt(e))/(e^2-1)

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csch(-0.5)

Solution

csch (- 0.5)

Solution

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

Decimal Notation

- 1.91903 \dots

Solution steps

csch (- 0.5)

= csch (- \frac{1}{2})

Use the following property:

csch (- x) = - csch (x)

csch (- \frac{1}{2}) = - csch (\frac{1}{2})

= - csch (\frac{1}{2})

Rewrite using trig identities:

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

csch (\frac{1}{2})

Use the Hyperbolic identity:

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

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

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

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

Apply exponent rule:

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

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

Join

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

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

Convert element to fraction:

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

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

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

Apply the fraction rule:

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

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

Rationalize

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

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

Multiply by the conjugate

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

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

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

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

Apply the distributive law:

a (b + c) = ab + a c

a = 2 e, b = e^{\frac{1}{2}} e, c = 1

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

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

Simplify

2 e e^{\frac{1}{2}} e + 2 \cdot 1 \cdot e : 2 e^{\frac{3}{2}} + 2 e

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

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

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

Apply radical rule:

a a = a

e e = e

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

Apply exponent rule:

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

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

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

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

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

Join

\frac{1}{2} + 1 : \frac{3}{2}

\frac{1}{2} + 1

Convert element to fraction:

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

= \frac{1}{2} + \frac{1 \cdot 2}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{1 + 1 \cdot 2}{2}

1 + 1 \cdot 2 = 3

1 + 1 \cdot 2

Multiply the numbers:

1 \cdot 2 = 2

= 1 + 2

Add the numbers:

1 + 2 = 3

= 3

= \frac{3}{2}

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

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

2 \cdot 1 \cdot e = 2 e

2 \cdot 1 \cdot e

Multiply the numbers:

2 \cdot 1 = 2

= 2 e

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

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

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

(e^{\frac{1}{2}} e - 1) (e^{\frac{1}{2}} e + 1)

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

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

Apply exponent rule:

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

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

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

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

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

Multiply

2 \cdot \frac{1}{2} : 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= e^{1}

Apply rule

a^{1} = a

= e

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

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

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

Apply exponent rule:

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

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

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

Add similar elements:

\frac{1}{2} + \frac{1}{2} = 2 \cdot \frac{1}{2}

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

Multiply

2 \cdot \frac{1}{2} : 1

2 \cdot \frac{1}{2}

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= e^{1}

Apply rule

a^{1} = a

= e

= (e - 1) (e + 1)

Apply Difference of Two Squares Formula:

(a - b) (a + b) = a^{2} - b^{2}

a = e, b = 1

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

Apply rule

1^{a} = 1

1^{2} = 1

= e^{2} - 1

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

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

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

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

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Frequently Asked Questions (FAQ)

What is the value of csch(-0.5) ?
The value of csch(-0.5) is -(2e^{3/2}+2sqrt(e))/(e^2-1)