Adding sines and cosines tip to tail

[mbigras]

Homework Statement

Express the following in the form $z = Re[Ae^{j(\omega t+ \alpha)}]$
$$z = 2 \sin(\omega t) + 3 \cos(\omega t)$$

Homework Equations

$$e^{j \theta} = \cos(\theta) + j\sin(\theta)$$

The Attempt at a Solution

$$z = 2 \sin(\omega t) + 3 \cos(\omega t) \\ z = 2 \cos(\omega t - \pi/2) + 3 \cos(\omega t) \\$$
I'm looking at my professors notes for this question. He is then able to add these terms together like vectors, tip to tail (see attached image). I'm pretty sure this works because he's using the real parts, although I don't see the general mechanics of how to treat these cosine terms like vectors. Especially how he seems to be ignoring the $\omega t$ and only using the $-\pi/2$ and $0$ when choosing the angles that the vectors shoot out. I'd like some explanation for why his process works.

 

[Simon Bridge]

He's adding phasors - yes.
sine and cosine are the same vector rotated by 90deg.
But it only works like that if they have the same frequency.