law of cosines identification
Check the final answer first, then review the worked steps.
Problem
law of cosines identification
Answer
\(67^\circ\)
Step-by-step solution
- Understand the Law of Cosines: The Law of Cosines states that for a triangle with sides $a$, $b$, and $c$ and an angle $C$ opposite to side $c$, the formula is $c^2 = a^2 + b^2 - 2ab \cos(C)$.
- Analyze the given equation: The provided equation is $5^2 + 13^2 - 2(5)(13)\cos(\_) = 12^2$. Comparing this to the standard form $a^2 + b^2 - 2ab \cos(C) = c^2$, we can identify that $a=5$, $b=13$, and $c=12$.
- Identify the angle: In the Law of Cosines, the angle $C$ must be the angle opposite to the side $c$ that is isolated on one side of the equation. Here, the side isolated is $12$. Looking at the triangle, the side with length $12$ is opposite the angle labeled $67^\circ$.
- Verify the relationship: The side of length $12$ is opposite the angle $67^\circ$. Therefore, the angle required to complete the Law of Cosines for this specific arrangement is $67^\circ$.