Answer
$m \angle x \approx 36.9^{\circ}$
$m \angle y \approx 53.1^{\circ}$
Work Step by Step
First, we want to know what side is opposite the angle in question. The side that is opposite to the angle we are looking for, $\angle x$, has a measure of $3$, so let's plug in what we know into the formula for the law of cosines:
$3^2 = 4^2 + 5^2 - 2(4)(5)$ cos $\angle x$
Evaluate exponents first, according to order of operations:
$9 = 16 + 25 - 2(4)(5)$ cos $\angle x$
Add to simplify on the right side of the equation:
$9 = 41 - 2(4)(5)$ cos $\angle x$
Multiply to simplify:
$9 = 41 - 40$ cos $\angle x$
Subtract $41$ from each side of the equation to move constants to the left side of the equation:
$-32= -40$ cos $m \angle x$
Divide each side by $-40$:
cos $m \angle x = \frac{-32}{-40}$
Take $cos^{-1}$ to solve for $\angle x$:
$m \angle x \approx 36.9^{\circ}$
Now, we use the law of cosines to find $m \angle y$. The line opposite to $\angle y$ measures $4$, so now, we can set up the equation:
$4^2 = 3^2 + 5^2 - 2(3)(5)$ cos $\angle y$
Evaluate exponents first, according to order of operations:
$16 = 9 + 25 - 2(3)(5)$ cos $\angle y$
Add to simplify on the right side of the equation:
$16 = 34 - 2(3)(5)$ cos $\angle y$
Multiply to simplify:
$16 = 34 - 30$ cos $\angle y$
Subtract $34$ from each side of the equation to move constants to the left side of the equation:
$-18 = -30$ cos $m \angle y$
Divide each side by $-30$:
cos $m \angle y = \frac{-18}{-30}$
Take $cos^{-1}$ to solve for $\angle y$:
$m \angle y \approx 53.1^{\circ}$
