trigonometric conversion of products to sums

[feline]

Hi, I really need help with this question, have used one identify the double angle formula but can get no further, really stuck any help would be grateful received, many thanks.

Radio carrier wave: v_R = cos w_Rt
Audio modulating wave : v_A = 1.5 + sin w_At
Resulting modulated signal: v_M = (1..5 + sin w_At) cos w_Rt

Given that w_R = 500000 rad/sec and w_{A =} 10000 rad/sec, using the trigonometric conversion of Products to Sums, determine the frequencies contained in the modulated signal.

 

[HallsofIvy]

I think you need \(\displaystyle sin(A+ B)= sin(A)cos(B)+ cos(A)sin(B)\) and \(\displaystyle cos(A+ B)= cos(A)cos(B)- Sin(A)Sin(B)\). Those go from "products" to "sums" but you can reverse that by using the fact that cos(-B)= cos(B) and sin(-B)= -sin(B). That is, if you replace B with -B in those, you get \(\displaystyle sin(A- B)= sin(A)cos(B)- cos(A)sin(B)\) and, adding that to the first equation above, \(\displaystyle sin(A)cos(B)=\frac{1}{2}\left(sin(A+B)+ sin(A-B)\right)\). Similarly, \(\displaystyle cos(A- B)= cos(A)cos(B)+ sin(A)sin(B)\) and adding that to the second equation above, \(\displaystyle cos(A)cos(B)= \frac{1}{2}\left(cos(A+ B)cos(A- B)\right)\) and, subtracting, \(\displaystyle sin(A)sin(B)= \frac{1}{2}\left(cos(A- B)- cos(A+B)\right)\).

 

[feline]

getting frequencies out

Many Thanks for your help, I have managed to do the trig conversion of products to sums but I can not get the frequencies out, do I need to use 2Pi divided by omega?