Answer
The measure of the angle with its vertex at the pitcher is about $54.7^{\circ}$.
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, $x$, is the side that measures $90$ ft., so let's plug in what we know into the formula for the law of cosines:
$90^2 = 57^2 + 110^2 - 2(57)(110)$ cos $x$
Evaluate exponents first, according to order of operations:
$8100 = 3249 + 12100 - 2(57)(110)$ cos $x$
Add to simplify on the right side of the equation:
$8100 = 15349 - 2(57)(110)$ cos $x$
Multiply on the right side of the equation:
$8100 = 15349 - 12540$ cos $x$
Subtract $15349$ from each side of the equation to move constants to the left side of the equation:
$-7249 = -12540$ cos $x$
Divide each side by $-12540$:
cos $\angle x = \frac{-7249}{-12540}$
Take $cos^{-1}$ to solve for $\angle x$:
$m \angle x \approx 54.7^{\circ}$
The measure of the angle with its vertex at the pitcher is about $54.7^{\circ}$.
