Answer
$$\sin\theta=-\frac{3\sqrt 5}{7}$$
Work Step by Step
$$\sec\theta=\frac{7}{2}\hspace{1cm}\tan\theta\lt0$$
From a Reciprocal Identity,
$$\sec\theta=\frac{1}{\cos\theta}$$
So, $$\cos\theta=\frac{1}{\sec\theta}$$
$$\cos\theta=\frac{1}{\frac{7}{2}}=\frac{2}{7}$$
From a Pythagorean Identity, we also have
$$\cos^2\theta+\sin^2\theta=1$$
$$\sin^2\theta=1-\cos^2\theta$$
$$\sin^2\theta=1-(\frac{2}{7})^2=1-\frac{4}{49}=\frac{45}{49}$$
$$\sin\theta=\pm\frac{\sqrt{45}}{7}=\pm\frac{3\sqrt5}{7}$$
From a Quotient Identity:
$$\tan\theta=\frac{\sin\theta}{\cos\theta}$$
As $\tan\theta\lt0$, this identity means that the signs of $\sin\theta$ and $\cos\theta$ must be opposite from each other.
Therefore, since $\cos\theta=\frac{2}{7}\gt0$, $\sin\theta\lt0$.
$$\sin\theta=-\frac{3\sqrt 5}{7}$$
