Complex exponentials & phasors

[Quincy]

Homework Statement

x(t) = 2sin($$\omega$$₀t + 45^o) + cos($$\omega$$₀t)

Express x(t) in the form x(t) = Acos($$\omega$$₀t + $$\phi$$)

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:
$$2.798cos(\omega_o t - 30.3612^o)$$