Question

in triangle mnp, m = 5 cm, n = 4 cm, and p = 8 cm. which formula can you use to find p? 4² = 5² + 8² - 2(4)(5)cos(p) 4² = 5² + 8² - 2(5)(8)cos(p) 8² = 4² + 5² - 2(4)(5)cos(p) 8² = 4² + 5² - 2(5)(8)cos(p)

Explanation:

Step1: Recall the Law of Cosines

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 side \(a\).

Step2: Identify the sides and the angle

In triangle \(MNP\), if we want to find angle \(P\), the side opposite angle \(P\) is \(p = 8\) cm. Let \(b = 4\) (side \(n\)) and \(c=5\) (side \(m\)).

Step3: Substitute into the Law of Cosines formula

Substituting \(a = p = 8\), \(b = 4\), \(c = 5\) into \(a^{2}=b^{2}+c^{2}-2bc\cos(A)\) (where \(A = P\)), we get \(8^{2}=4^{2}+5^{2}-2(4)(5)\cos(P)\)

Answer:

\(8^{2}=4^{2}+5^{2}-2(4)(5)\cos(P)\) (the third option)