Art Of Electronics Second Edition - Math Question - Complex Number Current Calculation
First, the author mentions that, "The phase of the voltage is arbitrary," so he's defining the source voltage waveform as a cosine wave.  The current function I(t) will be referenced to this function.

Next, since \$V(t) = Acos(\omega t)\$ and \$Z = -j/\omega C\$, then

$$I(t) = V/Z =  A \omega Ccos(\omega t) / -j = j A \omega C cos(\omega t) $$

Since a multiplication by \$ j \$ represents a 90 degree phase shift, we yield

$$I(t) = A \omega C sin(\omega t) $$

which is what the book describes as well.

Now for the 0.059 coefficient: just check your math.  You're correct in \$ \omega = 2 \pi f\$ where \$f = 60 Hz \$, thus

$$ A \omega C = 156 (2 \pi 60) 1e ^{-6} = 0.059$$