There's probably supposed to be a factor of [tex]i[/tex] before the sine term. We have
[tex]\cos\dfrac{7\pi}6=\cos\dfrac{5\pi}6=-\cos\dfrac\pi6=-\dfrac{\sqrt3}2[/tex]
[tex]\sin\dfrac{7\pi}6=-\sin\dfrac{5\pi}6=-\sin\dfrac\pi6=-\dfrac12[/tex]
[tex]\cos\dfrac{7\pi}6+i\sin\dfrac{7\pi}6=-\dfrac{\sqrt3}2-i\dfrac12=-\dfrac{\sqrt3+i}2[/tex]
