cos 96o + sin 96o
tan 19No = -----------------
           cos 96o - sin 96o

I did the same , but copied the wrong coefficient  (17  instead of 159):

 LS= -0.809784 approx.   

arctan(19N)= arctan( -0.809784 )=-31^o +N*180^o

^{This angle should be divisible by 19}

for k=17 it is  3021

LS=RS= -0.809784 approx.

so the answer is N  =   3021/19= 159

I was exhausted  when I have typed in my answer,- since I HAVE SPENT A LOT OF TIME trying to solve it analytically.