What is the value of cosh(45) ?

The value of cosh(45) is (e^{90}+1)/(2e^{45)}

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cosh(45)

cosh (45)

\frac{e ^{90} + 1}{2 e ^{45}}

Decimal Notation

1.74671 E 19

Solution steps

cosh (45)

Use the Hyperbolic identity:

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

= \frac{e ^{45} + e ^{- 45}}{2}

\frac{e ^{45} + e ^{- 45}}{2} = \frac{e ^{90} + 1}{2 e ^{45}}

\frac{e ^{45} + e ^{- 45}}{2}

Apply exponent rule:

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

= \frac{e ^{45} + \frac{1}{e ^{45}}}{2}

Join

e^{45} + \frac{1}{e ^{45}} : \frac{e ^{90} + 1}{e ^{45}}

e^{45} + \frac{1}{e ^{45}}

Convert element to fraction:

e^{45} = \frac{e ^{45} e ^{45}}{e ^{45}}

= \frac{e ^{45} e ^{45}}{e ^{45}} + \frac{1}{e ^{45}}

Since the denominators are equal, combine the fractions:

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

= \frac{e ^{45} e ^{45} + 1}{e ^{45}}

e^{45} e^{45} + 1 = e^{90} + 1

e^{45} e^{45} + 1

e^{45} e^{45} = e^{90}

e^{45} e^{45}

Apply exponent rule:

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

e^{45} e^{45} = e^{45 + 45}

= e^{45 + 45}

Add the numbers:

45 + 45 = 90

= e^{90}

= e^{90} + 1

= \frac{e ^{90} + 1}{e ^{45}}

= \frac{\frac{e ^{90} + 1}{e ^{45}}}{2}

Apply the fraction rule:

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

= \frac{e ^{90} + 1}{e ^{45} \cdot 2}

= \frac{e ^{90} + 1}{2 e ^{45}}

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