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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

csch(ln(4))

csch (ln (4))

\frac{8}{15}

Decimal Notation

0.53333 \dots

Solution steps

csch (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)

= csch (2 ln (2))

Rewrite using trig identities:

\frac{8}{15}

csch (2 ln (2))

Use the Hyperbolic identity:

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

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

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

\frac{2}{e ^{2 l n (2)} - e ^{- 2 l n (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{2}{4 - 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{2}{4 - \frac{1}{4}}

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

Apply the fraction rule:

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

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

Multiply the numbers:

2 \cdot 4 = 8

= \frac{8}{15}

= \frac{8}{15}

= \frac{8}{15}

= \frac{8}{15}

- 4 π sin (\frac{5 π}{4})

sec^{2} (6 0^{\circ}) - tan^{2} (6 0^{\circ})

45 cos (6 5^{\circ})

sin (4 5^{\circ}) + cos (4 5^{\circ}) + tan (9 0^{\circ})

arctan (0.95)