What is the value of csch(ln(3)) ?

The value of csch(ln(3)) is 3/4

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csch(ln(3))

csch (ln (3))

\frac{3}{4}

Decimal Notation

0.75

Solution steps

csch (ln (3))

Use the Hyperbolic identity:

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

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

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

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

e^{l n (3)} = 3

e^{l n (3)}

Apply log rule:

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

= 3

e^{- l n (3)} = 3^{- 1}

e^{- l n (3)}

Apply exponent rule:

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

= (e^{l n (3)})^{- 1}

Apply log rule:

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

e^{l n (3)} = 3

= 3^{- 1}

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

Simplify

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

Apply exponent rule:

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

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

Join

3 - \frac{1}{3} : \frac{8}{3}

3 - \frac{1}{3}

Convert element to fraction:

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

= \frac{3 \cdot 3}{3} - \frac{1}{3}

Since the denominators are equal, combine the fractions:

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

= \frac{3 \cdot 3 - 1}{3}

3 \cdot 3 - 1 = 8

3 \cdot 3 - 1

Multiply the numbers:

3 \cdot 3 = 9

= 9 - 1

Subtract the numbers:

9 - 1 = 8

= 8

= \frac{8}{3}

= \frac{2}{\frac{8}{3}}

Apply the fraction rule:

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

= \frac{2 \cdot 3}{8}

Multiply the numbers:

2 \cdot 3 = 6

= \frac{6}{8}

Cancel the common factor:

2

= \frac{3}{4}

= \frac{3}{4}

= \frac{3}{4}

- cosh (0)

csch (ln (4))

- 4 π sin (\frac{5 π}{4})

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

45 cos (6 5^{\circ})