What is the value of arccosh(2/3) ?

The value of arccosh(2/3) is ln((2+sqrt(5)i)/3)

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arccosh(2/3)

arccosh (\frac{2}{3})

ln (\frac{2 + 5 i}{3})

Solution steps

arccosh (\frac{2}{3})

Use the Hyperbolic identity:

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

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

Simplify

ln \frac{2}{3} + (\frac{2}{3})^{2} - 1 : ln (\frac{2 + 5 i}{3})

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

(\frac{2}{3})^{2} - 1 = i \frac{5}{3}

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

(\frac{2}{3})^{2} = \frac{4}{9}

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

Apply exponent rule:

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

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

2^{2} = 4

= \frac{4}{3 ^{2}}

3^{2} = 9

= \frac{4}{9}

= \frac{4}{9} - 1

Join

\frac{4}{9} - 1 : - \frac{5}{9}

\frac{4}{9} - 1

Convert element to fraction:

1 = \frac{1 \cdot 9}{9}

= \frac{4}{9} - \frac{1 \cdot 9}{9}

Since the denominators are equal, combine the fractions:

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

= \frac{4 - 1 \cdot 9}{9}

4 - 1 \cdot 9 = - 5

4 - 1 \cdot 9

Multiply the numbers:

1 \cdot 9 = 9

= 4 - 9

Subtract the numbers:

4 - 9 = - 5

= - 5

= \frac{- 5}{9}

Apply the fraction rule:

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

= - \frac{5}{9}

= - \frac{5}{9}

Apply radical rule:

- a = - 1 a

- \frac{5}{9} = - 1 \frac{5}{9}

= - 1 \frac{5}{9}

Apply imaginary number rule:

- 1 = i

= i \frac{5}{9}

Apply radical rule:

assuming

a \geq 0, b \geq 0

\frac{5}{9} = \frac{5}{9}

= i \frac{5}{9}

9 = 3

9

Factor the number:

9 = 3^{2}

= 3^{2}

Apply radical rule:

3^{2} = 3

= 3

= i \frac{5}{3}

= ln (\frac{2}{3} + i \frac{5}{3})

Join

\frac{2}{3} + i \frac{5}{3} : \frac{2 + 5 i}{3}

\frac{2}{3} + i \frac{5}{3}

Multiply

i \frac{5}{3} : \frac{5 i}{3}

i \frac{5}{3}

Multiply fractions:

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

= \frac{5 i}{3}

= \frac{2}{3} + \frac{5 i}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{2 + 5 i}{3}

= ln (\frac{2 + 5 i}{3})

= ln (\frac{2 + 5 i}{3})

arctan (\frac{4}{11})

sin (1 5^{\circ}) sin (6)

tan (6 0^{\circ}) - sin (6 0^{\circ}) - tan (6 0^{\circ})

tan (5 0^{\circ}) \cdot 1.3

15.24 tan (4 1^{\circ})

What is the value of arccosh(2/3) ?
The value of arccosh(2/3) is ln((2+sqrt(5)i)/3)