In electromagetics we usually use phasors to simply our calculations. This video shows a short and easy way to derive the phasor relationship to the instantaneous time-domain function. This derivation uses the Euler identity and is valid for all waves including traveling waves.
I am teaching using the Fawwaz T. Ulaby textbook as a reference. This is covered in chapter 1-7 of Fundamentals of Applied Electromagnetics, 8th edition.
For more information about Fundamentals of Applied Electromagnetics by Ulaby please visit this website: https://em8e.eecs.umich.edu/