Which equation correctly uses the law of cosines to solve for $y$ in a triangle with sides $x$, $y$, $z$ and angle $C$ opposite side $y$?

• A. $y^2=x^2-z^2+2xzcos\left(C\right)$
• B. $y^2=x^2+z^2-2xzsin\left(C\right)$
• C. $y^2=x^2+z^2+2xzcos\left(C\right)$
• D. $y^2=x^2+z^2-2xzcos\left(C\right)$

Answer: (D)

Verified step by step guidance

1. Recall the Law of Cosines formula, which relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides \(a\), \(b\), and \(c\), and angle \(\gamma\) opposite side \(c\), the formula is: \[c^{2} = a^{2} + b^{2} - 2ab \cos(\gamma)\]
2. Identify the sides and angle in the problem: here, \(y\) is the side opposite angle \(C\), and the other two sides are \(x\) and \(z\). So, \(y\) corresponds to \(c\), \(x\) to \(a\), \(z\) to \(b\), and \(C\) to \(\gamma\) in the Law of Cosines formula.
3. Substitute the identified sides and angle into the Law of Cosines formula: \[y^{2} = x^{2} + z^{2} - 2xz \cos(C)\]
4. Check each given equation against this formula. The correct equation must have the sum of the squares of the two known sides, minus twice their product times the cosine of the included angle.
5. Note that the terms involving sine or a plus sign before the \$2xz \cos(C)$ term do not match the Law of Cosines and are therefore incorrect.