Question

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

cosh (x)

Short answer

The Real part of cosh (ix) is cos(x).

The Imaginary part of cosh (ix) is 0.

The absolute value of cosh (ix) is |cos(x)|.

Step-by-step solution

Given Information.

The given expression is cosh (ix).

Definition of Complex Numberand Hyperbolic functions.

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)

Find the real, imaginary, absolute value of cosh(ix).

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