Answer
$ -2\sqrt2$
Work Step by Step
The terminal side of the angle $\dfrac{-\pi}{4}$ is in Quadrant IV.
The reference angle of an angle is equal to the smallest acute angle that its terminal side makes with the x-axis. The terminal side of $-\frac{\pi}{4}$ makes a $\frac{\pi}{4}$ angle with the positive x-axis.
Thus, the reference angle of the given angle is $\dfrac{\pi}{4}$.
Recall that an angle and its reference angle have the same sine values, except possibly for their signs.
$\dfrac{\pi}{4}$ is a special angle whose sine value if $\dfrac{\sqrt2}{2}$. Since $-\dfrac{\pi}{4}$ in Quadrant IV, its sine is negative.
Therefore $\sin{(-\dfrac{\pi}{4})} = -\dfrac{\sqrt2}{2}$.
Substitute this value into the given expression to obtain:
$4 \sin{\left(-\dfrac{\pi}{4}\right)}=4 \cdot \dfrac{-\sqrt2}{2} = -2\sqrt2$
