Hyperbolic Functions cosh(2-3i)

[phrygian]

Homework Statement

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

cosh(2-3i)

Homework Equations

The Attempt at a Solution

I know how to write this using exponentials, but when I looked up the answer the book expanded cosh(2-3i) to cosh 2 cos 3 − i sinh 2 sin 3 , how in the world do you get there??

Thanks for the help

 

[Tomtom]

You'll be happy with this answer, I believe.
If you remember from regular sin and cosine rules, you have the double angle formula:

sin(a+b) = sina*cosb + sinb*cosa.

Well, you have equivalent functions for hyperbolic functions:

sinh(a±b) = sinha*coshb ± sinhb*cosha

and

cosh(a±b) = cosha*coshb ± sinha*sinhb

You are going to use the second one. The ± means that if you use minus in the first ±, you use minus in the second part of the expression as well.