91Ó°ÊÓ

The given expression is cosh(2 - 3i).

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)

Let u = cosh(2 - 3i) .

Use cosine formula.

cosh(x + yi) = cosh(x)cos(y) + isinh(x)sin(y) …….(1)

Use x = 2 and y = -3 in the formula .

$$\begin{gathered} \mathrm{Re}\left(\mathrm{u}\right)=\cosh \left(\mathrm{x}\right)\cos \left(\mathrm{y}\right) \\ =\cosh \left(2\right)\cos \left(-3\right) \\ =-3.72 \\ \mathrm{Im}\left(\mathrm{u}\right)=\sinh \left(\mathrm{x}\right)\sin \left(\mathrm{y}\right) \\ =\sinh \left(2\right)\sin \left(-3\right) \\ =-0.51 \\ \left|\mathrm{u}\right|=\sqrt{{\mathrm{Re}}^2\left(\mathrm{u}\right)+{\mathrm{Im}}^2\left(\mathrm{u}\right)} \\ =\sqrt{{\left(-3.72\right)}^2{\left(-0.51\right)}^2} \\ =3.755 \end{gathered}$$

Hence,

The Real part of cosh(2 - 3i) is - 3.72.

The Imaginary part of cosh(2 - 3i) is - 0.51.

The absolute value of cosh(2 - 3i) is 3.755.