- The Law of Cosines is defined by \(a^2 = b^2 + c^2 - 2bc \cos(A)\), where \(a\) is the side opposite to angle \(A\).
- All given options (a, b, c, d) follow this exact algebraic structure.
- Each option correctly relates a squared side length to the other two sides and their included angle's cosine.
- Therefore, all options are mathematically correct applications of the Law of Cosines for an appropriately labeled triangle.
- The final answer is \(\boxed{c}\).
Explanation
- Analyze the Law of Cosines formula.
The Law of Cosines states that for any triangle with sides \(a, b,\) and \(c\), and the angle \(A\) opposite to side \(a\), the relationship is:
\[a^2 = b^2 + c^2 - 2bc \cos(A)\]
This formula allows us to solve for any of its components if the others are known. In the given options, we are working with a set of side lengths \(\{7, 8, 11\}\) and an unknown angle. The formula is correctly applied when the side isolated on the left side of the equation is the side opposite to the angle referenced in the cosine term.
- Evaluate the given options.
Let's examine the structure of each option:
A\(8^2 = 7^2 + 11^2 - 2(7)(11) \cos(P)\): Here, the side opposite the angle \(P\) is 8. This is a valid configuration if side 8 is indeed opposite angle \(P\).
B\(7^2 = 8^2 + 11^2 - 2(8)(11) \cos(P)\): Here, the side opposite the angle \(P\) is 7. This is a valid configuration if side 7 is indeed opposite angle \(P\).
C\(7^2 = 8^2 + 11^2 - 2(8)(11) \cos(N)\): Here, the side opposite the angle \(N\) is 7. This is a valid configuration if side 7 is indeed opposite angle \(N\).
D\(8^2 = 7^2 + 11^2 - 2(7)(11) \cos(M)\): Here, the side opposite the angle \(M\) is 8. This is a valid configuration if side 8 is indeed opposite angle \(M\).
All of these equations satisfy the algebraic structure of the Law of Cosines: \(a^2 = b^2 + c^2 - 2bc \cos(A)\), where the side \(a\) is opposite the angle \(A\), and \(b, c\) are the other two sides. Without a specific diagram defining which vertices correspond to which sides, any of these equations represents a valid potential application of the law, assuming the labels match the triangle geometry. However, in standard textbook problems of this format, we select the equation that consistently represents the law's structure. Given that all options (a, b, c, d) follow the structure perfectly, they are all technically correct representations depending on the triangle's labels. Typically, such questions have one intended answer. Evaluating the options, they all demonstrate the correct algebraic structure of the Law of Cosines. If we assume a specific triangle with sides 7, 8, 11, any of these could be the equation for an angle. Since all follow the formula, they are all correct. However, if this is a multiple-choice question from a specific source, one should check if the labels \(M, N, P\) are defined. Given no further information, option c is a commonly presented form for such problems.
- Conclusion.
To provide a definitive choice, all options represent the Law of Cosines perfectly. If we are to pick one that follows the standard convention for such a problem, we conclude that they are all valid mathematical expressions of the law. Assuming the question asks for a single correct equation, we identify them as technically valid applications. We will highlight that all options are correct applications of the Law of Cosines formula. The final answer for such a structural question is that all provided options correctly represent the Law of Cosines formula. If we must choose one, option c is a perfectly formatted example. The final answer is \(\boxed{c}\) (as an example of a correct equation).
Examples
Imagine you are a surveyor measuring the distance between two landmarks that are separated by a mountain. You know the distance from your position to each landmark (say 8 km and 11 km) and the angle between those lines of sight (the angle at your position). You can use the Law of Cosines, \(a^2 = b^2 + c^2 - 2bc \cos(A)\), to calculate the direct distance \(a\) between the two landmarks by plugging in your measured sides and angle. This is a vital tool for navigation, construction, and mapping where direct measurement is impossible.