Answer
$$\cos x=-\frac{2\sqrt5}{5}$$
$$\sin x=-\frac{\sqrt5}{5}$$
Work Step by Step
$$\tan x=\frac{1}{2}, x\in\Big[\pi,\frac{3\pi}{2}\Big]$$
1) As $x\in\Big[\pi,\frac{3\pi}{2}\Big]$, it means angle $x$ is in the third quadrant, so $\sin x\lt0$ and $\cos x\lt0$.
2) To find $\cos x$, we employ the formula: $$1+\tan^2x=\frac{1}{\cos^2x}$$
$$\frac{1}{\cos^2x}=1+\Big(\frac{1}{2}\Big)^2=1+\frac{1}{4}=\frac{5}{4}$$
$$\cos^2 x=\frac{4}{5}$$
$$|\cos x|=\frac{2}{\sqrt5}=\frac{2\sqrt5}{5}$$
Since $\cos x\lt0$, $$\cos x=-\frac{2\sqrt5}{5}$$
3) To find $\sin x$, we employ the formula:
$$\tan x=\frac{\sin x}{\cos x}$$
$$\sin x=\tan x\times\cos x=-\frac{2\sqrt5}{5}\times\frac{1}{2}=-\frac{\sqrt5}{5}$$
