Question

law of cosines: $a^2 = b^2 + c^2 - 2bccos(a)$
which equation correctly uses the law of cosines to solve for $y$?
$\bigcirc\\ 9^2 = y^2 + 19^2 - 2(y)(19)cos(41^circ)$
$\bigcirc\\ y^2 = 9^2 + 19^2 - 2(y)(19)cos(41^circ)$
$\bigcirc\\ 9^2 = y^2 + 19^2 - 2(9)(19)cos(41^circ)$
$\bigcirc\\ y^2 = 9^2 + 19^2 - 2(9)(19)cos(41^circ)$

Explanation:

Step1: Recall Law of Cosines

The given law is $a^2 = b^2 + c^2 - 2bc\cos(A)$, where $a$ is the side opposite angle $A$, and $b,c$ are the other two sides.

Step2: Match to triangle sides/angle

Side $y$ is opposite the $41^\circ$ angle. The sides adjacent to this angle are 9 and 19. Substitute $a=y$, $b=9$, $c=19$, $A=41^\circ$ into the formula.
<Expression>
$y^2 = 9^2 + 19^2 - 2(9)(19)\cos(41^\circ)$
</Expression>

Answer:

$y^2 = 9^2 + 19^2 - 2(9)(19)\cos(41^\circ)$ (the fourth option)