Question

which equation could be used to solve for the length of side c, given a = 5, b = 12, and c = 72°?

triangle diagram with vertices a, b, c; side opposite a is a, opposite b is b, opposite c is c; labeled as not drawn to scale

options:

• ( c^2 = 5^2 + 12^2 - 2(5)(12)cos 72^circ )
• ( c^2 = 5^2 + 12^2 + 2(5)(12)cos 72^circ )
• ( c^2 = 5^2 - 12^2 - 2(5)(12)cos 72^circ )
• ( c^2 = 5^2 - 12^2 + 2(5)(12)cos 72^circ )

Explanation:

Step1: Recall the Law of Cosines

The Law of Cosines for a triangle with sides \(a\), \(b\), \(c\) and the angle \(C\) opposite side \(c\) is \(c^{2}=a^{2}+b^{2}-2ab\cos C\).

Step2: Substitute the given values

We are given \(a = 5\), \(b=12\), and \(C = 72^{\circ}\). Substituting these values into the Law of Cosines formula, we get \(c^{2}=5^{2}+12^{2}-2(5)(12)\cos72^{\circ}\).

Answer:

\(c^{2}=5^{2}+12^{2}-2(5)(12)\cos72^{\circ}\) (the first option)