# Complex Numbers, Phasor Notation, and Eulers Identity

On Page 30 in the Art Of Electronics (2nd Edition).

The two images below describe the conservation of Sinusoidal signals to a complex representation and back. I am perfectly ok with the first image. The next image has me stumped.

In particular, I am stumped at the description of V(t) and I(t)

For V(t), I understand why they say V(t) = Re(V * e^jwt)

I do not understand the next line of the simplification V(t) = Re(V)cos(wt) - Im(V) sin(wt)

e^jwt should simplify to

cos(wt) + jsim(wt), I don't understand how V is distributed to the second equation.

I guess I(t) would follow similarly. Can someone help with the factoring?

So, assume $\mathbf{V}=5 j$. Then:
\begin{align*} V\left(t\right)&=\operatorname{Re}\left(\mathbf{V}\cdot e^{j \omega t}\right) \\ &= \operatorname{Re}\left(\mathbf{V}\cdot \left[\operatorname{cos}\left(\omega t\right)+i\operatorname{sin}\left(\omega t\right)\right]\right) \\ &= \operatorname{Re}\left(5 i\cdot \left[\operatorname{cos}\left(\omega t\right)+i\operatorname{sin}\left(\omega t\right)\right]\right) \\ &= \operatorname{Re}\left( 5 i\cdot\operatorname{cos}\left(\omega t\right)+5 i\cdot i\operatorname{sin}\left(\omega t\right)\right) \\ &=\operatorname{Re}\left( 5 i\cdot\operatorname{cos}\left(\omega t\right)-5 \operatorname{sin}\left(\omega t\right)\right) \\ &= -5 \operatorname{sin}\left(\omega t\right) \end{align*}
But perhaps $\mathbf{V}=3-4 j$? This just provides phase information where the polar magnitude is $5\:\textrm{V}$. But the result is arrived at in the same fashion as above. It's just that you now have sine and cosine terms mixed into the result.