Since [tex]\sin(\pi-x)=\sin x[/tex], you have
[tex]\sin\dfrac{7\pi}6=\sin\left(\pi-\dfrac{7\pi}6\right)=\sin\left(-\dfrac\pi6\right)[/tex]
and since [tex]\sin x[/tex] is odd, i.e. [tex]\sin(-x)=-\sin x[/tex], you have
[tex]\sin\dfrac{7\pi}6=-\sin\dfrac\pi6=-\dfrac12[/tex]
