Answer
$\cos{x}= -\dfrac{2\sqrt{5}}{5}$
$\sin{x}=-\dfrac{\sqrt{5}}{5}$
Work Step by Step
Since $x$ lies in the third quadrant, $\sin{x}$ and $\cos{x}$ are both negative.
Using the identity:
$$\sec{x}=-\sqrt{1+\tan^2{x}}$$
$$\sec{x}=-\sqrt{1+\left(\dfrac{1}{2} \right)^2}$$
$$\sec{x}=-\dfrac{\sqrt{5}}{2}$$
$\because \cos{x}=\dfrac{1}{\sec{x}}$
$\cos{x}=\dfrac{1}{-\dfrac{\sqrt{5}}{2}}$
$\cos{x}= -\dfrac{2\sqrt{5}}{5}$
Using the formula:
$$\sin{x}=-\sqrt{1-\cos^2{x}}$$
$$\sin{x}=-\sqrt{1-\left(-\dfrac{2\sqrt{5}}{5} \right)^2}$$
$$\sin{x}=-\dfrac{\sqrt{5}}{5}$$
