Answer
$$\sin x=\frac{12}{13}$$
$$\tan x=-\frac{12}{5}$$
Work Step by Step
$$\cos x=-\frac{5}{13}, x\in\Big[\frac{\pi}{2},\pi\Big]$$
1) As $x\in\Big[\frac{\pi}{2},\pi\Big]$, it means angle $x$ is in the second quadrant, so $\sin x\gt0$ and $\tan x\lt0$.
2) To find $\sin x$, we employ the formula: $$\sin^2 x+\cos^2x=1$$
$$\sin^2x=1-\cos^2x=1-\Big(-\frac{5}{13}\Big)^2=1-\frac{25}{169}=\frac{144}{169}$$
$$|\sin x|=\frac{12}{13}$$
Since $\sin x\gt0$, $$\sin x=\frac{12}{13}$$
3) To find $\tan x$, we employ the formula:
$$\tan x=\frac{\sin x}{\cos x}=\frac{\frac{12}{13}}{-\frac{5}{13}}=-\frac{12}{5}$$
