What is the output of two cosine functions?

[Jimmy Johnson]

Homework Statement

inputs x1(t) = cos(ω1t), x2(t) = cos(ω2t).
Show that output g(t) (sum of x1 + x2) = 0.5cos[(ω2-ω1)t] + 0.5cos[(ω2+ω1)t]

Homework Equations

included in upload of attempted solution. Trig identities.

The Attempt at a Solution

Uploaded in pdf. A lot more has been done on the solution however including it all would have taken forever, problem is I keep circling round and winding up back at the same place I started (due to the nature of the identities). It's been a while since I've did a question like this and understand half the battle is spotting the connection, and will probably be kicking myself if I get it done. Any help would be greatly appreciated.

 

[gneill]

It doesn't look like a true identity to me. Try a disproof by counterexample. Choose a couple of representative values for the ω's and a handy value of t (say, t = 1). Does the identity hold?

 

[Jimmy Johnson]

You seem to have a point. I did ω1 = 30 ,ω2 = 60 t = 1. The given proof that I am to aim towards gave (√3)/4 while adding the two functions separately gave ((1+√3)/2).
ω1 = 30 ,ω2 = 60 t = 1... proof = ((-√6+√2)/4) as opposed to ((-√3+√2)/2). So it would seem the identity doesn't hold?

In the question it says X1 = cos(ω1t), X1' = cos(ω2t). I assumed from the layout of the diagram (the circled x indicating a sum) and the two inputs of X1 and X2 pointing logically towards a typo and X1' (no dash in question only for indicative purposes writing this) being X2 in connection with ω2 would make sense.

That being said if they were to be both of X1, assuming it wasn't a typo, would that make a difference? I would be confused as to the purpose of X2 if it were the case and how it would be any different under those circumstances anyhow. Unless it may then fall under the multiple angle formula and use recurrence relationships?

 

[gneill]

Can you post the original question? Perhaps a picture? It's difficult to make out what the situation is without context.

Edit: Nevermind. I just realized that you did post the question in your pdf. Sorry about that.

 

[Jimmy Johnson]

the original question is in the pdf attatched? or is it difficult to make out?

 

[gneill]

The circled X represents a product, not a sum.

 

[Jimmy Johnson]

wow. Well that will hopefully clear it up, I've to head to work for a few hours here then i'll get back to the drawing board. In the question it says that it can be shown as a sum? Would that not indicate that it should be proven as a sum i.e. the two x terms added and that my working out with the counter examples may be changed and hold?

I can't do much on it now but hopefully that will sort it out. Thanks a lot :)

 

[DrClaude]

In the question it says that it can be shown as a sum? Would that not indicate that it should be proven as a sum i.e. the two x terms added and that my working out with the counter examples may be changed and hold?

No, the two x terms are multiplied, but the result can be expressed as a sum.

 

[Jimmy Johnson]

Yeah I got it pretty much instantly after it being pointed out to be a product. Fell pretty dumb haha Thanks for the help!