Answer
$$\sin x=-\frac{2\sqrt2}{3}$$
$$\tan x=-2\sqrt2$$
Work Step by Step
$$\cos x=\frac{1}{3}, x\in\Big[-\frac{\pi}{2},0\Big]$$
1) As $x\in\Big[-\frac{\pi}{2},0\Big]$, it means angle $x$ is in the fourth quadrant, so $\sin x\lt0$ and $\tan x\lt0$.
2) To find $\sin x$, we employ the formula: $$\sin^2 x+\cos^2x=1$$
$$\sin^2x=1-\cos^2x=1-\Big(\frac{1}{3}\Big)^2=1-\frac{1}{9}=\frac{8}{9}$$
$$|\sin x|=\frac{2\sqrt2}{3}$$
Since $\sin x\lt0$, $$\sin x=-\frac{2\sqrt2}{3}$$
3) To find $\tan x$, we employ the formula:
$$\tan x=\frac{\sin x}{\cos x}=\frac{-\frac{2\sqrt2}{3}}{\frac{1}{3}}=-2\sqrt2$$
