What is the value of tanh(-6) ?

The value of tanh(-6) is -(e^{12}-1)/(e^{12)+1}

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tanh(-6)

tanh (- 6)

- \frac{e ^{12} - 1}{e ^{12} + 1}

Decimal Notation

- 0.99998 \dots

Solution steps

tanh (- 6)

Use the following property:

tanh (- x) = - tanh (x)

tanh (- 6) = - tanh (6)

= - tanh (6)

Rewrite using trig identities:

tanh (6) = \frac{e ^{12} - 1}{e ^{12} + 1}

tanh (6)

Use the Hyperbolic identity:

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

= \frac{e ^{6} - e ^{- 6}}{e ^{6} + e ^{- 6}}

\frac{e ^{6} - e ^{- 6}}{e ^{6} + e ^{- 6}} = \frac{e ^{12} - 1}{e ^{12} + 1}

\frac{e ^{6} - e ^{- 6}}{e ^{6} + e ^{- 6}}

Apply exponent rule:

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

e^{- 6} = \frac{1}{e ^{6}}

= \frac{e ^{6} - \frac{1}{e ^{6}}}{e ^{6} + \frac{1}{e ^{6}}}

Join

e^{6} + \frac{1}{e ^{6}} : \frac{e ^{12} + 1}{e ^{6}}

e^{6} + \frac{1}{e ^{6}}

Convert element to fraction:

e^{6} = \frac{e ^{6} e ^{6}}{e ^{6}}

= \frac{e ^{6} e ^{6}}{e ^{6}} + \frac{1}{e ^{6}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{6} e ^{6} + 1}{e ^{6}}

e^{6} e^{6} + 1 = e^{12} + 1

e^{6} e^{6} + 1

e^{6} e^{6} = e^{12}

e^{6} e^{6}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{6} e^{6} = e^{6 + 6}

= e^{6 + 6}

Add the numbers:

6 + 6 = 12

= e^{12}

= e^{12} + 1

= \frac{e ^{12} + 1}{e ^{6}}

= \frac{e ^{6} - \frac{1}{e ^{6}}}{\frac{e ^{12} + 1}{e ^{6}}}

Join

e^{6} - \frac{1}{e ^{6}} : \frac{e ^{12} - 1}{e ^{6}}

e^{6} - \frac{1}{e ^{6}}

Convert element to fraction:

e^{6} = \frac{e ^{6} e ^{6}}{e ^{6}}

= \frac{e ^{6} e ^{6}}{e ^{6}} - \frac{1}{e ^{6}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{6} e ^{6} - 1}{e ^{6}}

e^{6} e^{6} - 1 = e^{12} - 1

e^{6} e^{6} - 1

e^{6} e^{6} = e^{12}

e^{6} e^{6}

Apply exponent rule:

a^{b} \cdot a^{c} = a^{b + c}

e^{6} e^{6} = e^{6 + 6}

= e^{6 + 6}

Add the numbers:

6 + 6 = 12

= e^{12}

= e^{12} - 1

= \frac{e ^{12} - 1}{e ^{6}}

= \frac{\frac{e ^{12} - 1}{e ^{6}}}{\frac{e ^{12} + 1}{e ^{6}}}

Divide fractions:

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

= \frac{( e ^{12} - 1 ) e ^{6}}{e ^{6} ( e ^{12} + 1 )}

Cancel the common factor:

e^{6}

= \frac{e ^{12} - 1}{e ^{12} + 1}

= \frac{e ^{12} - 1}{e ^{12} + 1}

= - \frac{e ^{12} - 1}{e ^{12} + 1}

cos (13 7^{\circ})

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