For a triangle with sides $a$, $b$, and $c$, and angle $C$ opposite side $c$, which equation can be used to solve for $c$ using the Law of Cosines?

• A. $c^2=a^2+b^2-2abcos\left(C\right)$
• B. $c^2=a^2+b^2-2absin\left(C\right)$
• C. $c^2=a^2-b^2+2abcos\left(C\right)$
• D. $c^2=a^2+b^2+2abcos\left(C\right)$

Answer: (A)

Verified step by step guidance

1. Recall the Law of Cosines, which relates the lengths of the sides of a triangle to the cosine of one of its angles. It is especially useful for finding a side when you know the other two sides and the included angle.
2. Identify the sides and angle in the problem: sides \(a\), \(b\), and \(c\), with angle \(C\) opposite side \(c\).
3. Write the Law of Cosines formula for side \(c\): \[c^{2} = a^{2} + b^{2} - 2ab \cos(C)\]
4. Understand that this formula comes from extending the Pythagorean theorem to non-right triangles by adjusting for the angle \(C\) using the cosine term.
5. Use this equation to solve for \(c\) by taking the square root of both sides after substituting the known values for \(a\), \(b\), and \(C\).