Answer
$18$ miles south and $36$ miles east
Work Step by Step
The trigonometric ratios are as follows: $\sin \theta= \dfrac{opposite}{hypotenuse}$ ; $\cos \theta= \dfrac{Adjacent}{hypotenuse}$ and $\tan \theta= \dfrac{Opposite}{Adjacent}$
We can use the sine and cosine of the triangle to solve for the distance.
Since, $\cos \theta= \dfrac{Adjacent}{hypotenuse}$
Now, miles to the south $ = 40 \times \cos 64^{\circ}=18$ miles
and $\sin \theta= \dfrac{opposite}{hypotenuse}$
Now, miles to the east $ = 40 \times \sin 64^{\circ}=36$ miles
