Frequency spectrum of the modulated signal g(t)

[raymond23]

Homework Statement

Let the baseband signal be
s(t)=cos(2πfst+π3), where fs=5kHz. Radio carrier is
c(t)=sin(2πfct), where fc=100MHz. Using the amplitude modulation of g(t)=(1+s(t))c(t), what is the frequency spectrum of the modulated signal g(t)?
What are the amplitude and phase shift of each frequency component in g(t)?

Homework Equations

g(t)=(1+s(t))c(t)

The Attempt at a Solution

g(t)= (1+ cos(2πfst+π3)) sin(2πfct)
g(t)= sin(2πfct)+ cos(2πfst+π3) sin(2πfct)
g(t)=

 

[Redbelly98]

g(t)= sin(2πfct)+ cos(2πfst+π3) sin(2πfct)

Looks good so far. The first sin(...) term gives you one of the frequency components.

You'll need to express "cos(2πfst+π3) sin(2πfct)" as the sum of distinct, single-frequency sin and/or cos terms. You can do that using these trig identities:

sin(x + y) = sin(x)·cos(y) + cos(x)·sin(y)
sin(x - y) = sin(x)·cos(y) - cos(x)·sin(y)

p.s. welcome to PF

 

[raymond23]

I try this
g(t)= sin(2πfct)+ cos(2πfst+π3) sin(2πfct)
g(t)= sin(2πfct)+ (cos(2πfst)+cos(π3)) sin(2πfct)
g(t)= sin(2πfct)+ (cos(2πfst)sin(2πfct)+cos(π3)sin(2πfct))
but don't know how to continue

 

[Redbelly98]

I try this
g(t)= sin(2πfct)+ cos(2πfst+π3) sin(2πfct)
g(t)= sin(2πfct)+ (cos(2πfst)+cos(π3)) sin(2πfct)

There's a problem there, because

cos(2πfst+π3) and cos(2πfst)+cos(π3)​

are not equivalent.

If you add the two equations in my earlier post, you'll have an expression for

sin(x) cos(y)​

which will be useful here.