Find the real part, the imaginary part, and the absolute value of

cosh (x)

The given expression is cosh (ix).

The domain of a function refers to the range of values that can be plugged into it. This is the set of x values in a function like f (x). The range of a function is the set of values that the function can take. This is the set of values that the function produces when we enter x value. The y values are what you're looking for.

The basic hyperbolic functions are:

1. Hyperbolic sine(sinh)
2. Hyperbolic cosine(cosh)
3. Hyperbolic tangent(tanh)

$$\begin{gathered} \mathrm{Let}\cosh \left(\mathrm{ix}\right)=\cosh \left(\mathrm{iz}\right). \\ \mathrm{Use}\mathrm{the}\mathrm{formula}\mathrm{mentioned}\mathrm{below}. \\ \sinh \left(\mathrm{iz}\right)=\mathrm{i}\sin \left(\mathrm{z}\right) \\ \tanh \left(\mathrm{iz}\right)=\mathrm{i}\tan \left(\mathrm{z}\right) \\ \mathrm{DivideLeft}\mathrm{Hand}\mathrm{Side}\mathrm{of}\mathrm{first}\mathrm{identity}\mathrm{by}\mathrm{second}\mathrm{identity}. \\ \frac{\sinh \left(\mathrm{iz}\right)}{\tanh \left(\mathrm{iz}\right)}=\cosh \left(\mathrm{iz}\right)\dots .\left(1\right) \\ \mathrm{Divide}\mathrm{Left}\mathrm{Hand}\mathrm{Side}\mathrm{of}\mathrm{first}\mathrm{identity}\mathrm{by}\mathrm{second}\mathrm{identity}.\dots .\left(2\right) \\ \frac{\mathrm{i}\sinh \left(\mathrm{z}\right)}{\mathrm{i}\tanh \left(\mathrm{z}\right)}=\cosh \left(\mathrm{z}\right) \\ \mathrm{Combine}\mathrm{equation}\left(1\right)\mathrm{and}\mathrm{equation}\left(2\right). \\ \cosh \left(\mathrm{iz}\right)=\cosh \left(\mathrm{z}\right) \\ \cosh \left(\mathrm{ix}\right)=\cosh \left(\mathrm{x}\right) \\ \mathrm{Hence}, \\ \mathrm{The}\mathrm{Real}\mathrm{part}\mathrm{of}\cosh \left(\mathrm{ix}\right)\mathrm{is}\cosh \left(\mathrm{x}\right). \\ \mathrm{The}\mathrm{Imaginary}\mathrm{part}\mathrm{of}\cosh \left(\mathrm{ix}\right)\mathrm{is}0. \\ \mathrm{The}\mathrm{absolute}\mathrm{value}\mathrm{of}\cosh \left(\mathrm{ix}\right)\mathrm{is}\cosh \left(\mathrm{x}\right). \end{gathered}$$