Chapter 8 - Right Triangles and Trigonometry - 8-6 Law of Cosines - Practice and Problem-Solving Exercises - Page 530: 10

$m \angle E \approx 88.5^{\circ}$

Work Step by Step

First, we want to know what angle is opposite the side in question. The side that is opposite to the angle we are looking for is $\overline{FD}$, so let's plug in what we know into the formula for the law of cosines: $27^2 = 13^2 + 24^2 - 2(13)(24)$ cos $\angle E$ Evaluate exponents first, according to order of operations: $729 = 169 + 576 - 2(13)(24)$ cos $m \angle E$ Add to simplify on the right side of the equation: $729 = 745 - 2(13)(24)$ cos $m \angle E$ Multiply to simplify: $729 = 745 - 624$ cos $m \angle E$ Subtract $745$ from each side of the equation to move constants to the left side of the equation: $-16 = -624$ cos $m \angle E$ Divide each side by $-624$: cos $m \angle E \approx 0.02564$ Take $cos^{-1}$ to solve for $\angle E$: $m \angle E \approx 88.5^{\circ}$

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