How Is Total Power Calculated in AC Circuits?

[Simon.T]

Ok guys this one is straight out of a textbook, (Tooley, M. and Dingle, L. (2008) Higher National Engineering, 2nd ed. Oxford: Elsevier, p.353.)

Homework Statement

$$v=50sin(\omega t)+10sin(3\omega t + \pi /2)$$
$$i=3.54sin(\omega t + \pi /4) + 0.316sin(3\omega t +0.321)$$

Homework Equations

$$\sum_{n=1}^{n=m} \frac {V_nI_ncos\phi_n} 2$$

The Attempt at a Solution

The solution given is
$$P_{TOT} =\frac {50\times 3.54} 2 cos(-\pi /4) + \frac {10\times 0.316} 2 cos(\pi /2 - 0.321)$$

I don't understand how the solution in part 3 relates to the equation in part 2. The solution implies $$cos(a-b) = sin(wt + \phi_1)\times sin(wt+\phi_2)$$ but I cannot find this written anywhere apart from here

$$cos(a-b)=cos(a)cos(b)+sin(a)sin(b)$$ So we are just ignoring the cos terms?

I'm really quite tired so I am sorry if this seems like a stupid question, I swear this textbook is useless.

 

[SteamKing]

Try this reference: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/powerac.html

Scroll down and look for the section on Average Power calculation.