Answer
$x \approx 14.7$ cm
Work Step by Step
We would use the Law of Cosines here because we have the measure of an included angle and the lengths of two sides, and we want to find the length of the third side.
Let's set up the equation according to the Law of Cosines. First, we want to know what angle is opposite the side in question. The angle that is opposite to the side we are looking for, $x$, is the angle that measures $65^{\circ}$, so let's plug in what we know into the formula for the law of cosines:
$x^2 = 15^2 + 12^2 - 2(15)(12)$ cos $65^{\circ}$
Evaluate exponents first, according to order of operations:
$x^2 = 225 + 144 - 2(15)(12)$ cos $65^{\circ}$
Add to simplify on the right side of the equation:
$x^2 = 369 - 2(15)(12)$ cos $65^{\circ}$
Take the square root of both sides of the equation:
$x \approx 14.7$ cm
