Phases of a set of cosine terms

[S_100]

b) The expression F (t) = (4/ π ) [cos (2 π ft) -( 1/3)cos (6 π ft) + (1/5) cos (10 π ft)]
is an approximation to a unit square-wave of frequency f at time t, Figure 6.2.

For each cosine term give the
(i) amplitude (iii) phase
(ii) frequency (iv) period [4]

I am confused on how to deduce the phase of each cosine term , part (iii)

The answer states for cos (2 π ft) the phase is zero,
(1/3)cos (6 π ft) the phase is +/- π
(1/5) cos (10 π ft)] phase is zero


I know that (a π ft) represents the angle where a is a number, but that's as far as my attempt at a solution goes. I'm unsure how to approach finding the phase of these cosine terms

 

[Dr.Peterson]

They are taking "phase" to be the angle [MATH]\theta[/MATH] in [MATH]A\cos(\omega x - \theta)[/MATH]. For the first and last terms, the phase is clearly zero. The second term is [MATH]-\frac{1}{3}\cos (6 \pi f t) = \frac{1}{3}\cos (6 \pi f t - \pi)[/MATH], because [MATH]\cos(x - \pi) = - cos(x)[/MATH]. So the phase in that case is [MATH]\pi[/MATH].

Did you look up "phase" in your textbook to see their definition?

 

[S_100]

Did you look up "phase" in your textbook to see their definition?

I looked in my physics textbook, but there was no clear mention of phase, only that from the derivation of SHM of a particle moving around in a circle, and stated if x the (vertical displacement of particle from the centre of the circle was zero when t=0 i.e at the starting point, then the equation for the displacement would be x = Asin(wt) - Or Acos(wt) for the horiztonal displacement I think.

The second term is [MATH]-\frac{1}{3}\cos (6 \pi f t) = \frac{1}{3}\cos (6 \pi f t - \pi)[/MATH], because [MATH]\cos(x - \pi) = - cos(x)[/MATH]. So the phase in that case is [MATH]\pi[/MATH].

So the phase is π
But why does the answer state plus or minus π, surely in the case above it is -π ?

 

[Dr.Peterson]

Think about it! If we replace my θ with -π rather than π , how will the result be different? Also, don't miss the fact that my θ is subtracted, not added; do you see how that relates to horizontal shifts in graphs? There's a lot to talk about here.

So this is from a physics book and "phase" doesn't appear in the index at all? The equations you quote have no phase shift, so they have said nothing about phase there ...

But also, simple harmonic motion is not really about moving in a circle at all, so I wonder if you are misinterpreting something they said.

If I were helping you face to face, I would be paging through your book about now, trying to get a feel for what they expect you to know.