Understanding Phasors and Complex Numbers in Harmonic Functions

[homad2000]

Homework Statement

What are the phasors F(t) and G(t) corresponding to the following functions:
f(t) = Acosω₁t and g(t) = Acosω₂t

Draw the phasors on Argand diagram as well as F(t)+G(t) at t = $\pi$/(2ω₁)
and from the diagram get f(t)+g(t) as a cosine identity in the simplest form.


I tried plotting the F(t) + G(t),, but I couldn't get the angle nor the magnitude of it! any help will be appreciated ;)

 

[vela]

Can you describe in detail the Argand diagram you drew?

 

[homad2000]

well, assuming that we write the functions on a complex form, we get F(t) = Ae^(iω1t) and G(t) = Ae^(iω2t). And by the way, it is given that ω1<w2 .. so, at the given t, the first argument is pi/2, the second one is not exact, but it's bigger than pi/2.. so, F(t) + G(t) is sum of two vectors drawn in the argand diagram.. but its argument is really complicated, and I'm not sure of it.

 

[vela]

The sum is going to bisect the angle between the two phasors, and you can use some geometrical reasoning and the law of cosines to find the magnitude of the resultant.

Can you take it from there?

 

[homad2000]

I got the argument to be pi*(ω2-ω1/4ω1 ... I could use the cosine rule to get the argument, however, I got a complicated result! I don't know if I'm on the right track!