Question

ight triangle relationships and trigonometry
which equation correctly uses the law of cosines to solve for y?
$y^{2}=9^{2}+19^{2}-2(9)($
$y^{2}=9^{2}+19^{2}-2(y)($
$9^{2}=y^{2}+19^{2}-2(9)($
$9^{2}=y^{2}+19^{2}-2(y)($
law of cosines $a^{2}=b^{2}+c^{2}-2bccos (a)$

Explanation:

Step1: Recall the Law of Cosines formula

The Law of Cosines formula is \(a^{2}=b^{2}+c^{2}-2bc\cos(A)\), where \(a\), \(b\), \(c\) are the sides of the triangle and \(A\) is the angle opposite to side \(a\).

Step2: Identify the sides and angle in the given triangle

In \(\triangle XYZ\), if we want to solve for \(y\) (side opposite to the \(42^{\circ}\) angle), \(b = 9\), \(c=19\), and \(A = 42^{\circ}\). Substituting into the Law of Cosines formula \(y^{2}=9^{2}+19^{2}-2(9)(19)\cos(42^{\circ})\).

Answer:

\(y^{2}=9^{2}+19^{2}-2(9)(19)\cos(42^{\circ})\) (the first option in the top - left of the four given options)