A rectangular sandbox has sides of length $10$ feet and $8$ feet. If the diagonal of the sandbox measures $14$ feet, which equation using the Law of Cosines can be used to find the angle between the two sides?

• A. ${14}^2={10}^2+8^2+2108\cos \left(C\right)$
• B. ${14}^2={10}^2+8^2-2108\cos \left(C\right)$
• C. ${14}^2={10}^2-8^2+2108\cos \left(C\right)$
• D. ${14}^2={10}^2+8^2$

Answer: (B)

Verified step by step guidance

1. Identify the sides of the triangle formed by the two sides of the sandbox and its diagonal. Let the sides be \(a = 10\) feet, \(b = 8\) feet, and the diagonal (opposite the angle \(C\)) be \(c = 14\) feet.
2. Recall the Law of Cosines formula, which relates the lengths of the sides of a triangle to the cosine of one of its angles: \[c^{2} = a^{2} + b^{2} - 2ab \cos(C)\]
3. Substitute the known side lengths into the Law of Cosines formula: \[14^{2} = 10^{2} + 8^{2} - 2 \times 10 \times 8 \times \cos(C)\]
4. Understand that the negative sign before the \$2ab \cos(C)$ term is crucial because it accounts for the angle between the two sides, distinguishing it from the Pythagorean theorem which applies only to right triangles.
5. This equation can now be used to solve for the angle \(C\) by isolating \(\cos(C)\) and then applying the inverse cosine function.