What is the value of sinh(ln(4)) ?

The value of sinh(ln(4)) is 15/8

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sinh(ln(4))

sinh (ln (4))

\frac{15}{8}

Decimal Notation

1.875

Solution steps

sinh (ln (4))

Simplify:

ln (4) = 2 ln (2)

ln (4)

Rewrite

4

in power-base form:

4 = 2^{2}

= ln (2^{2})

Apply log rule:

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

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

= 2 ln (2)

= sinh (2 ln (2))

Rewrite using trig identities:

\frac{15}{8}

sinh (2 ln (2))

Use the Hyperbolic identity:

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

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

\frac{e ^{2 l n (2)} - e ^{- 2 l n (2)}}{2} = \frac{15}{8}

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

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

e^{2 l n (2)}

Apply exponent rule:

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

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

Apply log rule:

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

e^{l n (2)} = 2

= 2^{2}

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

e^{- 2 l n (2)}

Apply exponent rule:

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

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

Apply log rule:

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

e^{l n (2)} = 2

= 2^{- 2}

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

Simplify

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

2^{2} = 4

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

2^{- 2} = \frac{1}{4}

2^{- 2}

Apply exponent rule:

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

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

2^{2} = 4

= \frac{1}{4}

= \frac{4 - \frac{1}{4}}{2}

Join

4 - \frac{1}{4} : \frac{15}{4}

4 - \frac{1}{4}

Convert element to fraction:

4 = \frac{4 \cdot 4}{4}

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

Since the denominators are equal, combine the fractions:

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

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

4 \cdot 4 - 1 = 15

4 \cdot 4 - 1

Multiply the numbers:

4 \cdot 4 = 16

= 16 - 1

Subtract the numbers:

16 - 1 = 15

= 15

= \frac{15}{4}

= \frac{\frac{15}{4}}{2}

Apply the fraction rule:

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

= \frac{15}{4 \cdot 2}

Multiply the numbers:

4 \cdot 2 = 8

= \frac{15}{8}

= \frac{15}{8}

= \frac{15}{8}

= \frac{15}{8}

\frac{cos ( 4 5 ^{\circ} )}{cos ( 3 0 ^{\circ} )}

cot (5 π)

300 sin (3 0^{\circ})

2 + sin (π)

\frac{3}{cos ( 3 0 ^{\circ} )}