What is the value of arccosh(9) ?

The value of arccosh(9) is ln(9+4sqrt(5))

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arccosh(9)

arccosh (9)

ln (9 + 45)

Decimal

2.88727 \dots

Solution steps

arccosh (9)

Use the Hyperbolic identity:

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

= ln (9 + 9^{2} - 1)

ln (9 + 9^{2} - 1) = ln (9 + 45)

ln (9 + 9^{2} - 1)

9^{2} - 1 = 45

9^{2} - 1

9^{2} = 81

= 81 - 1

Subtract the numbers:

81 - 1 = 80

= 80

Prime factorization of

80 : 2^{4} \cdot 5

80

80

divides by

280 = 40 \cdot 2

= 2 \cdot 40

40

divides by

240 = 20 \cdot 2

= 2 \cdot 2 \cdot 20

20

divides by

220 = 10 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 10

10

divides by

210 = 5 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5

2, 5

are all prime numbers, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 5

= 2^{4} \cdot 5

= 2^{4} \cdot 5

Apply radical rule:

= 5 2^{4}

Apply radical rule:

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

= 2^{2} 5

Refine

= 45

= ln (9 + 45)

= ln (9 + 45)