What is the value of arcsinh(1/2) ?

The value of arcsinh(1/2) is ln((1+sqrt(5))/2)

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arcsinh(1/2)

arcsinh (\frac{1}{2})

ln (\frac{1 + 5}{2})

Decimal

0.48121 \dots

Solution steps

arcsinh (\frac{1}{2})

Use the Hyperbolic identity:

arcsinh (x) = ln (x + x^{2} + 1)

= ln \frac{1}{2} + (\frac{1}{2})^{2} + 1

ln \frac{1}{2} + (\frac{1}{2})^{2} + 1 = ln (\frac{1 + 5}{2})

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

(\frac{1}{2})^{2} + 1 = \frac{5}{2}

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

(\frac{1}{2})^{2} = \frac{1}{4}

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

Apply exponent rule:

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

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

Apply rule

1^{a} = 1

1^{2} = 1

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

2^{2} = 4

= \frac{1}{4}

= \frac{1}{4} + 1

Join

\frac{1}{4} + 1 : \frac{5}{4}

\frac{1}{4} + 1

Convert element to fraction:

1 = \frac{1 \cdot 4}{4}

= \frac{1}{4} + \frac{1 \cdot 4}{4}

Since the denominators are equal, combine the fractions:

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

= \frac{1 + 1 \cdot 4}{4}

1 + 1 \cdot 4 = 5

1 + 1 \cdot 4

Multiply the numbers:

1 \cdot 4 = 4

= 1 + 4

Add the numbers:

1 + 4 = 5

= 5

= \frac{5}{4}

= \frac{5}{4}

Apply radical rule:

assuming

a \geq 0, b \geq 0

= \frac{5}{4}

4 = 2

4

Factor the number:

4 = 2^{2}

= 2^{2}

Apply radical rule:

2^{2} = 2

= 2

= \frac{5}{2}

= ln (\frac{1}{2} + \frac{5}{2})

Join

\frac{1}{2} + \frac{5}{2} : \frac{1 + 5}{2}

\frac{1}{2} + \frac{5}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{1 + 5}{2}

= ln (\frac{1 + 5}{2})

= ln (\frac{1 + 5}{2})

What is the value of arcsinh(1/2) ?
The value of arcsinh(1/2) is ln((1+sqrt(5))/2)