I know that the tangent function is periodic, meaning it repeats the same values every $180^\circ$. So, to find an equivalent angle to $312^\circ$ in the given range, I can subtract $180^\circ$ from it until I get there. $312^\circ - 180^\circ = 132^\circ,$ which is still too big. $132^\circ - 180^\circ = -48^\circ,$ which is in the range $-90 < n < 90.$ Therefore, $n = -48.$ 
Find the integer $n,$ $-90 < n < 90,$ such that $\tan n^\circ = \tan 312^\circ.$