## Geometry: Common Core (15th Edition)

$m \angle x \approx 46.8^{\circ}$ $m \angle y \approx 35.0^{\circ}$
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 $14$, so let's plug in what we know into the formula for the law of cosines: $14^2 = 11^2 + 19^2 - 2(11)(19)$ cos $\angle x$ Evaluate exponents first, according to order of operations: $196 = 121 + 361 - 2(11)(19)$ cos $\angle x$ Add to simplify on the right side of the equation: $196 = 482 - 2(11)(19)$ cos $\angle x$ Multiply to simplify: $196 = 482 - 418$ cos $\angle x$ Subtract $418$ from each side of the equation to move constants to the left side of the equation: $-286 = -418$ cos $m \angle x$ Divide each side by $-418$: cos $m \angle x = \frac{-286}{-418}$ Take $cos^{-1}$ to solve for $\angle x$: $m \angle x \approx 46.8^{\circ}$ Now, we use the law of cosines to find $m \angle y$. The line opposite to $\angle y$ measures $11$, so now, we can set up the equation: $11^2 = 14^2 + 19^2 - 2(14)(19)$ cos $\angle y$ Evaluate exponents first, according to order of operations: $121 = 196 + 361 - 2(14)(19)$ cos $\angle y$ Add to simplify on the right side of the equation: $121 = 557 - 2(14)(19)$ cos $\angle y$ Multiply to simplify: $121 = 557 - 532$ cos $\angle y$ Subtract $557$ from each side of the equation to move constants to the left side of the equation: $-436 = -532$ cos $m \angle y$ Divide each side by $-532$: cos $m \angle y = \frac{-436}{-532}$ Take $cos^{-1}$ to solve for $\angle y$: $m \angle y \approx 35.0^{\circ}$