Problem 71 - Algebra and Trigonometry

Question

In Exercises 71–74, determine whether each statement makes sense or does not make sense, and explain your reasoning. I began using the Law of Sines to solve an oblique triangle in which the measures of two sides and the angle between them were known.

Step-by-Step Solution

Answer

The use of the Law of Sines in this scenario does not make sense; the Law of Cosines is the appropriate theorem to apply in this case.

Step 1: Analyzing the Given Scenario

In the presented situation, the measurements of two sides and the included angle of the oblique triangle are known. This forms a SAS (Side-Angle-Side) triangle, which doesn't align with the situations where the Law of Sines can be applied. The Law of Sines is applicable in cases of AAS (Angle-Angle-Side), ASA (Angle-Side-Angle), or SSA (Side-Side-Angle), where a non-included angle is known.

Step 2: Verifying with the appropriate Theorem

In the context of the SAS case, the Law of Cosines should be the theorem applied, not the Law of Sines. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles and is particularly useful for solving triangles when the SAS or SSS (Side-Side-Side) are known.

Step 3: Conclusion

Considering the details and the context of the problem, using the Law of Sines to solve this specific oblique triangle scenario does not make sense. The correct theorem to use given two sides and the included angle (SAS scenario) is the Law of Cosines.

Key Concepts

Oblique Triangle

An oblique triangle is a type of triangle where none of the angles are a right angle. This includes both acute and obtuse triangles. In such cases, the traditional Pythagorean theorem is not applicable for solving the sides and angles. Instead, we use trigonometric methods like the Law of Sines and Law of Cosines. When a triangle doesn't contain a 90-degree angle, it requires special consideration for calculation.

To solve an oblique triangle, the known elements will dictate the approach. For instance, if an angle and the two sides adjacent to it are known (AAS or ASA), the Law of Sines is usually the most straightforward method. Conversely, for cases where two sides and their included angle are given (SAS), or when all three sides are known (SSS), the Law of Cosines is the appropriate choice.

SAS (Side-Angle-Side) Triangle

The SAS (Side-Angle-Side) condition occurs when we know two sides of a triangle and the angle between them. It's a key criterion used in solving oblique triangles. SAS is a direct indication to use the Law of Cosines, as it provides a clear path to finding the remaining side of the triangle before proceeding to find the remaining angles, potentially using the Law of Sines afterward.

The significance of the SAS condition is rooted in the fact that it provides enough information to uniquely determine a triangle, avoiding the ambiguity that can arise in some SSA (Side-Side-Angle) situations. It's pivotal in geometry to recognize conditions like SAS as they guide us to apply the right theorem for a correct and efficient solution.

Law of Cosines

The Law of Cosines is a fundamental theorem in trigonometry, expressed as \( c^2 = a^2 + b^2 - 2ab \cos(C) \), where \( a \), \( b \), and \( c \) are the sides of a triangle, and \( C \) is the angle opposite side \( c \). It serves as an extension of the Pythagorean theorem to oblique triangles, relating the lengths of the sides of a triangle to the cosine of one of its angles.

Given a SAS condition, the Law of Cosines allows us to calculate the unknown side. After finding the length of the third side, we can use the Law of Sines or Law of Cosines again to find the other two angles. It's an invaluable tool, especially when dealing with real-world applications where right triangles are less common, like in navigation, physics, and engineering.

Solving Triangles

Solving triangles means finding the unknown lengths of sides and measures of angles of the given triangle. Depending on the information provided, different methods are applied. For an oblique triangle with an SAS configuration, as initially highlighted, the solution begins with the Law of Cosines. Once we determine the missing side, we can further apply the Law of Cosines to find one of the remaining angles and then use either it or the Law of Sines for the final angle, since the sum of angles in any triangle is always 180 degrees.

Understanding the relationship between different sides and angles within various types of triangles is crucial, as it forms the basis for many applications in science and mathematics. Mastery in solving triangles builds a stronger foundation for students in these fields.