What is the value of cosh(pi) ?

The value of cosh(pi) is (e^{2pi}+1)/(2e^pi)

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cosh(pi)

cosh (π)

\frac{e ^{2 π} + 1}{2 e ^{π}}

Decimal

11.59195 \dots

Solution steps

cosh (π)

Use the Hyperbolic identity:

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

= \frac{e ^{π} + e ^{- π}}{2}

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

\frac{e ^{π} + e ^{- π}}{2}

Apply exponent rule:

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

= \frac{e ^{π} + \frac{1}{e ^{π}}}{2}

Join

e^{π} + \frac{1}{e ^{π}} : \frac{e ^{2 π} + 1}{e ^{π}}

e^{π} + \frac{1}{e ^{π}}

Convert element to fraction:

e^{π} = \frac{e ^{π} e ^{π}}{e ^{π}}

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

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{π} e ^{π} + 1}{e ^{π}}

e^{π} e^{π} + 1 = e^{2 π} + 1

e^{π} e^{π} + 1

e^{π} e^{π} = e^{2 π}

e^{π} e^{π}

Apply exponent rule:

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

e^{π} e^{π} = e^{π + π}

= e^{π + π}

Add similar elements:

π + π = 2 π

= e^{2 π}

= e^{2 π} + 1

= \frac{e ^{2 π} + 1}{e ^{π}}

= \frac{\frac{e ^{2 π} + 1}{e ^{π}}}{2}

Apply the fraction rule:

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

= \frac{e ^{2 π} + 1}{e ^{π} \cdot 2}

= \frac{e ^{2 π} + 1}{2 e ^{π}}