Complex exponentials & phasors

[Quincy]

Homework Statement

x(t) = 2sin(\omega₀t + 45^o) + cos(\omega₀t)

Express x(t) in the form x(t) = Acos(\omega₀t + \phi)

The Attempt at a Solution

I don't really know when to begin; I can't find anything about it in the textbook.

 

[rock.freak667]

well expand out Acos(ω₀t+φ) and then equate coefficients.

 

[xcvxcvvc]

I'm not sure what that = 45 is doing. I'm going to assume you meant + 45

2sin(\omega_0 t + 45^o) + cos(\omega_0t)
our first step in using phasor notation is to define each sinusoid as either a sine or cosine:
2cos(\omega_0 t +45^o - 90^o) + cos(\omega_0 t)
2cos(\omega_0 t -45^o) + cos(\omega_0 t)
we then define what our phasor is at 0 degrees:
cos(\omega_0t) = (1 \angle 0^o)
apply it:
(2 \angle -45^o) + (1 \angle 0^o)
break up into rectangular coordinates:
2cos(-45^o) + 2sin(-45^o)j + 1
use common real and imaginary part arithmetic to bring back into polar form:
(2.798 \angle -30.3612)
bring out of phasor form:
2.798cos(\omega_o t - 30.3612^o)