What is the value of cosh(ln(2))+sinh(ln(2)) ?

The value of cosh(ln(2))+sinh(ln(2)) is 2

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cosh(ln(2))+sinh(ln(2))

Solution

cosh (ln (2)) + sinh (ln (2))

Solution

2

Solution steps

cosh (ln (2)) + sinh (ln (2))

Rewrite using trig identities:

cosh (ln (2)) = \frac{5}{4}

cosh (ln (2))

Use the Hyperbolic identity:

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

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

\frac{e ^{l n (2)} + e ^{- l n (2)}}{2} = \frac{5}{4}

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

e^{l n (2)} = 2

e^{l n (2)}

Apply log rule:

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

= 2

e^{- l n (2)} = 2^{- 1}

e^{- l n (2)}

Apply exponent rule:

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

= (e^{l n (2)})^{- 1}

Apply log rule:

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

e^{l n (2)} = 2

= 2^{- 1}

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

Simplify

\frac{2 + 2 ^{- 1}}{2}

Apply exponent rule:

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

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

Join

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

2 + \frac{1}{2}

Convert element to fraction:

2 = \frac{2 \cdot 2}{2}

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

Since the denominators are equal, combine the fractions:

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

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

2 \cdot 2 + 1 = 5

2 \cdot 2 + 1

Multiply the numbers:

2 \cdot 2 = 4

= 4 + 1

Add the numbers:

4 + 1 = 5

= 5

= \frac{5}{2}

= \frac{\frac{5}{2}}{2}

Apply the fraction rule:

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

= \frac{5}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{5}{4}

= \frac{5}{4}

= \frac{5}{4}

Rewrite using trig identities:

sinh (ln (2)) = \frac{3}{4}

sinh (ln (2))

Use the Hyperbolic identity:

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

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

\frac{e ^{l n (2)} - e ^{- l n (2)}}{2} = \frac{3}{4}

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

e^{l n (2)} = 2

e^{l n (2)}

Apply log rule:

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

= 2

e^{- l n (2)} = 2^{- 1}

e^{- l n (2)}

Apply exponent rule:

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

= (e^{l n (2)})^{- 1}

Apply log rule:

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

e^{l n (2)} = 2

= 2^{- 1}

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

Simplify

\frac{2 - 2 ^{- 1}}{2}

Apply exponent rule:

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

= \frac{2 - \frac{1}{2}}{2}

Join

2 - \frac{1}{2} : \frac{3}{2}

2 - \frac{1}{2}

Convert element to fraction:

2 = \frac{2 \cdot 2}{2}

= \frac{2 \cdot 2}{2} - \frac{1}{2}

Since the denominators are equal, combine the fractions:

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

= \frac{2 \cdot 2 - 1}{2}

2 \cdot 2 - 1 = 3

2 \cdot 2 - 1

Multiply the numbers:

2 \cdot 2 = 4

= 4 - 1

Subtract the numbers:

4 - 1 = 3

= 3

= \frac{3}{2}

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

Apply the fraction rule:

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

= \frac{3}{2 \cdot 2}

Multiply the numbers:

2 \cdot 2 = 4

= \frac{3}{4}

= \frac{3}{4}

= \frac{3}{4}

= \frac{5}{4} + \frac{3}{4}

Simplify

\frac{5}{4} + \frac{3}{4} : 2

\frac{5}{4} + \frac{3}{4}

Apply rule

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

= \frac{5 + 3}{4}

Add the numbers:

5 + 3 = 8

= \frac{8}{4}

Divide the numbers:

\frac{8}{4} = 2

= 2

= 2

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Frequently Asked Questions (FAQ)

What is the value of cosh(ln(2))+sinh(ln(2)) ?
The value of cosh(ln(2))+sinh(ln(2)) is 2