Answer
$$\sin\theta=-\frac{\sqrt{77}}{11}$$
Work Step by Step
$$\tan\theta=-\frac{\sqrt7}{2}\hspace{1cm}\sec\theta\gt0$$
From a Pythagorean Identity
$$\tan^2\theta+1=\sec^2\theta$$
combined with a Reciprocal Identity
$$\sec\theta=\frac{1}{\cos\theta}$$
we have
$$\tan^2\theta+1=\frac{1}{\cos^2\theta}$$
$$\frac{1}{\cos^2\theta}=(-\frac{\sqrt 7}{2})^2+1=\frac{7}{4}+1=\frac{11}{4}$$
$$\cos^2\theta=\frac{4}{11}$$
$$\cos\theta=\pm\frac{2}{\sqrt{11}}=\pm\frac{2\sqrt{11}}{11}$$
As shown above, $\sec\theta=\frac{1}{\cos\theta}$. This means the signs of $\sec\theta$ and $\cos\theta$ are the same.
Therefore, since $\sec\theta\gt0$, $\cos\theta\gt0$.
$$\cos\theta=\frac{2\sqrt{11}}{11}$$
Also, from Quotient Identities,
$$\tan\theta=\frac{\sin\theta}{\cos\theta}$$
which means
$$\sin\theta=\tan\theta\times\cos\theta$$
$$\sin\theta=(-\frac{\sqrt7}{2})\times(\frac{2\sqrt{11}}{11})=-\frac{\sqrt{77}}{11}$$
