AC circuit with unusually placed transformer
What I don't know is what to do with the transformer. When it's in parallel I know. I just need help with that, else I think I would manage.


It's called an "auto transformer.
It obeys the usual rules for an ideal transformer  

Call the voltage at the right hand end of R2 "Vin"

Short: 


Vin is divided between N1 &amp; N2 in the ratio of their turns
(as in any transformer) so - 
\$ Vout = \dfrac{N2}{N1+N2} \$


Longer:

$$ Vn2 = Vn1 \times N2/N1 \tag1 $$

So 
$$ Vn1 = Vn2 \times N1/N2 \tag2 $$

By inspection 

$$ Vin = Vn1 + Vn2 \tag3 $$

So 

$$ Vn1 = Vin - Vn2 ... \tag4 $$

From 2 &amp; 4  

$$ Vn2 \times N1/N2 = Vin - Vn2 $$
$$ Vn2 \times (N1/N2 +1 ) = Vin $$
$$ Vn2 = \dfrac{Vin}{N1/N2 + 1} $$ 
$$ Vn2 = \dfrac{Vin}{(N1 + N2) / N2} $$
$$ Vn2 = \dfrac{Vin \times N2}{N1 + N2} $$ 

i.e., MUCH more simply,
Vin is divided across N1 and N2 in the ratio of their turns.

Example 

If N1:N2 = 1:9 Vout is 90% of Vin.
If N1:N2 = 9:1 Vout is 10% of Vin.  



Wikipedia - autotransformer

Autotransformer basics

Autotransformers

Many examples