The Sinusoidal Identity The sum of two sinusoidal functions of the same frequency is another sinusoidal function with that frequency! For any real constants a and b, a cos(ωt) + b sin(ωt) = A cos(ωt − φ) (1) where A and φ can be described in at least two ways: A= � b φ = tan−1 ; a a2 + b2 , (2) a + bi = Aeiφ . (3) a = A cos(φ) and b = A sin(φ). (4) Conversely, we have Geometrically this is summarized by the triangle in the ﬁgure below. A b φ a Fig. 1. a + bi = Aeiφ . One proof of (1) is a simple application of the cosine addition formula cos(α − β) = cos(α) cos( β) + sin(α) sin( β). We will now give an equivalent proof using Euler’s formula and complex arithmetic: The triangle in Figure 1 is the standard polar coordinates trian gle. It shows a + ib = Aeiφ or a − ib = Ae−iφ . Thus A cos(ωt − φ) = Re( Aei(ωt−φ) ) = Re(eiωt · Ae−iφ ) = Re((cos(ωt) + i sin(ωt)) · ( a − ib)) = Re( a cos(ωt) + b sin(ωt) + i ( a sin(ωt) − b cos(ωt))) = a cos(ωt) + b sin(ωt). We should stress the importance of the trigonometric identity (1). It shows that any linear combination of cos(ωt) and sin(ωt) is not only periodic of period 2ωπ , but is also sinusoidal. If you try to add cos(ωt) to sin(ωt) “by hand”, you will probably agree that this is not at all obvious. We will call A cos(ωt − φ) amplitude-phase form and a cos(ωt) + b sin(ωt) rectangular or Cartesian form. You should be familiar with amplitudephase form; we usually prefer it because both amplitude and phase have geometric and physical meaning for us.