Answer:
a) [tex]\cosh 9 \approx 4051.54203[/tex], b) [tex]\cosh (\ln 9) \approx 4.55556[/tex].
Step-by-step explanation:
a) As we remember, hyperbolic cosine is represented by the following expression:
[tex]\cosh x = \frac{e^{x}+e^{-x}}{2}[/tex]
[tex]\cosh x = \left(\frac{1}{2\cdot e^{x}}\right) \cdot (e^{2\cdot x}+1)[/tex]
The numerical value of [tex]\cosh 9[/tex] is:
[tex]\cosh 9 = \left(\frac{1}{2\cdot e^{9}} \right)\cdot (e^{18}+1)[/tex]
[tex]\cosh 9 \approx 4051.54203[/tex]
b) By using the definition presented and developed in item a), we get the following expression:
[tex]\cosh x = \left(\frac{1}{2\cdot e^{\ln 9}}\right) \cdot (e^{2\cdot \ln 9}+1)[/tex]
Now, we simplify this by using power and logarithm properties:
[tex]\cosh (\ln 9) = \left[\frac{1}{(2)\cdot (9)} \right]\cdot (9^{2}+1)[/tex]
[tex]\cosh (\ln 9) = \frac{82}{18}[/tex]
[tex]\cosh (\ln 9) = \frac{41}{9}[/tex]
[tex]\cosh (\ln 9) \approx 4.55556[/tex]
