What is the value of sinh(-arccosh(2)) ?

The value of sinh(-arccosh(2)) is -sqrt(3)

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sinh(-arccosh(2))

Solution

sinh (- arccosh (2))

Solution

- 3

Decimal Notation

- 1.73205 \dots

Solution steps

sinh (- arccosh (2))

Use the following property:

sinh (- x) = - sinh (x)

sinh (- arccosh (2)) = - sinh (arccosh (2))

= - sinh (arccosh (2))

Rewrite using trig identities:

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

sinh (arccosh (2))

Use the Hyperbolic identity:

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

= \frac{e ^{arccosh (2)} - e ^{- arccosh (2)}}{2}

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

\frac{e ^{arccosh (2)} - e ^{- arccosh (2)}}{2}

Apply exponent rule:

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

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

Join

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

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

Convert element to fraction:

e^{arccosh (2)} = \frac{e ^{arccosh (2)} e ^{arccosh (2)}}{e ^{arccosh (2)}}

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

Since the denominators are equal, combine the fractions:

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

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

e^{arccosh (2)} e^{arccosh (2)} - 1 = e^{2 arccosh (2)} - 1

e^{arccosh (2)} e^{arccosh (2)} - 1

e^{arccosh (2)} e^{arccosh (2)} = e^{2 arccosh (2)}

e^{arccosh (2)} e^{arccosh (2)}

Apply exponent rule:

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

e^{arccosh (2)} e^{arccosh (2)} = e^{arccosh (2) + arccosh (2)}

= e^{arccosh (2) + arccosh (2)}

Add similar elements:

arccosh (2) + arccosh (2) = 2 arccosh (2)

= e^{2 arccosh (2)}

= e^{2 arccosh (2)} - 1

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

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

Apply the fraction rule:

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

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

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

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

Rewrite using trig identities:

arccosh (2) = ln (2 + 3)

arccosh (2)

Use the Hyperbolic identity:

arccosh (x) = ln (x + x^{2} - 1)

= ln (2 + 2^{2} - 1)

Simplify

= ln (2 + 3)

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

Simplify

- \frac{e ^{2 l n (2 + 3)} - 1}{2 e ^{l n (2 + 3)}} : - 3

- \frac{e ^{2 l n (2 + 3)} - 1}{2 e ^{l n (2 + 3)}}

e^{l n (2 + 3)} = 2 + 3

e^{l n (2 + 3)}

Apply log rule:

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

= 2 + 3

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

e^{2 l n (2 + 3)} = (2 + 3)^{2}

e^{2 l n (2 + 3)}

Apply exponent rule:

a^{b c} = (a^{b})^{c}

= (e^{l n (2 + 3)})^{2}

Apply log rule:

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

e^{l n (2 + 3)} = 2 + 3

= (2 + 3)^{2}

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

Simplify

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

Expand

(2 + 3)^{2} - 1 : 6 + 43

(2 + 3)^{2} - 1

(2 + 3)^{2} : 7 + 43

Apply Perfect Square Formula:

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

a = 2, b = 3

= 2^{2} + 2 \cdot 23 + (3)^{2}

Simplify

2^{2} + 2 \cdot 23 + (3)^{2} : 7 + 43

2^{2} + 2 \cdot 23 + (3)^{2}

2^{2} = 4

2^{2}

2^{2} = 4

= 4

2 \cdot 23 = 43

2 \cdot 23

Multiply the numbers:

2 \cdot 2 = 4

= 43

(3)^{2} = 3

(3)^{2}

Apply radical rule:

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

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

Apply exponent rule:

(a^{b})^{c} = a^{b c}

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

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

\frac{1}{2} \cdot 2

Multiply fractions:

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

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

Cancel the common factor:

2

= 1

= 3

= 4 + 43 + 3

Add the numbers:

4 + 3 = 7

= 7 + 43

= 7 + 43

= 7 + 43 - 1

Subtract the numbers:

7 - 1 = 6

= 6 + 43

= - \frac{6 + 4 3}{2 ( 2 + 3 )}

Cancel

\frac{6 + 4 3}{2 ( 2 + 3 )} : 3

\frac{6 + 4 3}{2 ( 2 + 3 )}

Factor

6 + 43 : 2 (3 + 23)

6 + 43

Rewrite as

= 2 \cdot 3 + 2 \cdot 23

Factor out common term

2

= 2 (3 + 23)

= \frac{2 ( 3 + 2 3 )}{2 ( 2 + 3 )}

Divide the numbers:

\frac{2}{2} = 1

= \frac{3 + 2 3}{( 2 + 3 )}

Factor

3 + 23 : 3 (3 + 2)

3 + 23

3 = 33

= 33 + 23

Factor out common term

3

= 3 (3 + 2)

= \frac{3 ( 3 + 2 )}{( 2 + 3 )}

Cancel the common factor:

3 + 2

= 3

= - 3

= - 3

= - 3

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Frequently Asked Questions (FAQ)

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