So, here's a question:
$ \cos( \omega t ) + 2 \cos( \omega t + \frac{\pi}{4} ) + 3 \cos( \omega t + \frac{\pi}{2} ) $
To add these together, I figure there should be at least 2 ways:
1) Cosine addition laws:
$ \cos( \omega t ) + 2 \left( \cos( \omega t ) \cos( \frac{\pi}{4} ) - \sin( \omega t ) \sin( \frac{\pi}{4} ) \right) + 3 \left( \cos( \omega t ) \cos( \frac{\pi}{2} ) - \sin( \omega t ) \sin( \frac{\pi}{2} ) \right) \\ =\cos( \omega t ) \left( 1 + \sqrt{2} \right) - \sin( \omega t ) \left( 3 + \sqrt{2} \right) $
2) Phasors / complex addition
$ 1 \angle 0 + 2 \angle 45 ^\circ + 3 \angle 90^\circ $
$ = 1 + \sqrt{2} + j \sqrt{2} + j 3 $
$ = 1 + \sqrt{2} + j ( 3 + \sqrt{2} ) $
Which has
$ A = \sqrt{ 14 + 8 \sqrt{2} } \approx 5.03 $
$ \phi = \arctan{ \left( \frac{ 3 + \sqrt{2} }{ 1 + \sqrt{2} } \right) } \approx 1.07 rad \approx 61 ^\circ $
Thus answer is $ 5 \angle 61^\circ $, or $5 \cos( \omega t + 1.07 )$
If you graph them, $5 \cos( \omega t + 1.07 )$ produces the same graph as $ \cos( \omega t ) \left( 1 + \sqrt{2} \right) - \sin( \omega t ) \left( 3 + \sqrt{2} \right) $
So how can you convert between them?