Question

which equation correctly uses the law of cosines to solve for y?
9² = y² + 19² - 2(y)(19)cos(41°)
y² = 9² + 19² - 2(y)(19)cos(41°)
9² = y² + 19² - 2(9)(19)cos(41°)
y² = 9² + 19² - 2(9)(19)cos(41°)
law of cosines a² = b² + c² - 2bccos(a)

Explanation:

Step1: Recall law of cosines

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

Step2: Identify sides and angle

In the triangle, we want to solve for $y$. Let $a = y$, $b = 9$, $c = 19$, and $A=41^{\circ}$.

Step3: Substitute values into law of cosines

Substituting into the formula $a^{2}=b^{2}+c^{2}-2bc\cos(A)$, we get $y^{2}=9^{2}+19^{2}-2(9)(19)\cos(41^{\circ})$.

Answer:

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