What is the value of arccosh(3) ?

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

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arccosh(3)

arccosh (3)

ln (3 + 22)

Decimal Notation

1.76274 \dots

Solution steps

arccosh (3)

Use the Hyperbolic identity:

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

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

ln (3 + 3^{2} - 1) = ln (3 + 22)

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

3^{2} - 1 = 22

3^{2} - 1

3^{2} = 9

= 9 - 1

Subtract the numbers:

9 - 1 = 8

= 8

Prime factorization of

8 : 2^{3}

8

8

divides by

28 = 4 \cdot 2

= 2 \cdot 4

4

divides by

24 = 2 \cdot 2

= 2 \cdot 2 \cdot 2

2

is a prime number, therefore no further factorization is possible

= 2 \cdot 2 \cdot 2

= 2^{3}

= 2^{3}

Apply exponent rule:

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

= 2^{2} \cdot 2

Apply radical rule:

= 2 2^{2}

Apply radical rule:

2^{2} = 2

= 22

= ln (3 + 22)

= ln (3 + 22)

sin (4 5^{\circ}) sin (1 5^{\circ})

1 - 2 sin^{2} (\frac{5 π}{8})

cos (4.1)

\frac{4}{tan ( 3 4 ^{\circ} )}

tan (- 45 0^{\circ})