What is the value of cosh(ln(9)) ?

The value of cosh(ln(9)) is 41/9

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cosh(ln(9))

cosh (ln (9))

\frac{41}{9}

Decimal Notation

4.55555 \dots

Solution steps

cosh (ln (9))

Simplify:

ln (9) = 2 ln (3)

ln (9)

Rewrite

9

in power-base form:

9 = 3^{2}

= ln (3^{2})

Apply log rule:

lo g_{a} (x^{b}) = b \cdot lo g_{a} (x)

ln (3^{2}) = 2 ln (3)

= 2 ln (3)

= cosh (2 ln (3))

Rewrite using trig identities:

\frac{41}{9}

cosh (2 ln (3))

Use the Hyperbolic identity:

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

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

\frac{e ^{2 l n (3)} + e ^{- 2 l n (3)}}{2} = \frac{41}{9}

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

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

e^{2 l n (3)}

Apply exponent rule:

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

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

Apply log rule:

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

e^{l n (3)} = 3

= 3^{2}

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

e^{- 2 l n (3)}

Apply exponent rule:

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

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

Apply log rule:

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

e^{l n (3)} = 3

= 3^{- 2}

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

Simplify

\frac{3 ^{2} + 3 ^{- 2}}{2}

3^{2} = 9

= \frac{9 + 3 ^{- 2}}{2}

3^{- 2} = \frac{1}{9}

3^{- 2}

Apply exponent rule:

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

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

3^{2} = 9

= \frac{1}{9}

= \frac{9 + \frac{1}{9}}{2}

Join

9 + \frac{1}{9} : \frac{82}{9}

9 + \frac{1}{9}

Convert element to fraction:

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

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

Since the denominators are equal, combine the fractions:

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

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

9 \cdot 9 + 1 = 82

9 \cdot 9 + 1

Multiply the numbers:

9 \cdot 9 = 81

= 81 + 1

Add the numbers:

81 + 1 = 82

= 82

= \frac{82}{9}

= \frac{\frac{82}{9}}{2}

Apply the fraction rule:

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

= \frac{82}{9 \cdot 2}

Multiply the numbers:

9 \cdot 2 = 18

= \frac{82}{18}

Cancel the common factor:

2

= \frac{41}{9}

= \frac{41}{9}

= \frac{41}{9}

= \frac{41}{9}

csc (42 0^{\circ})

sin (arcsin (0.9))

cos (- 5)

csc (\frac{1}{2})

12 cos (6 0^{\circ})