Problem 73 - Precalculus with Limits

Question

WRITING In Exercise 72, the Law of Cosines was used to solve a triangle in the two-solution case of SSA.Can the Law of Cosines be used to solve the no- solution and single-solution cases of SSA? Explain.

Step-by-Step Solution

Answer

The Law of Cosines cannot be used to solve no-solution and single-solution cases of SSA. The no-solution case arises when there is no triangle that meets the given conditions, while the single-solution case could be solved using the Law of Sines, as it aligns better with the SSA conditions.

Step 1: Understand the SSA cases with the Law of Cosines

The Law of Cosines follows this formula: \(c^2 = a^2 + b^2 - 2ab \cdot cosC\). In the SSA case, we usually know the quantities of two sides and the angle opposite to one of them. However, the Law of Cosines really requires a known angle located between two known sides, which the SSA condition does not provide.

Step 2: Analyze the no-solution case

Given two sides and the angle neither between nor opposite the longest side, the no-solution case arises when the side opposite the given angle is longer than the other given side. Because if you draw a circle with radius equal to the length of the known side (not the one opposite to the given angle) with its center on the end of the side of known length and then draw the line representing the known angle, the line will not meet the circle, so no triangle can be formed in this arrangement. Hence the Law of Cosines cannot solve this case, as there is, mathematically, no triangle that satisfies the conditions.

Step 3: Analyze the single-solution case

In the single-solution case, you are given two sides and an angle, but the angle is not located between the two known sides nor opposite to the longest side. When the side opposite the given angle is equal to or shorter than the other given side, a single valid triangle can be formed. In this situation, the triangle can be solved by using the Law of Sines rather than the Law of Cosines, as the triangle angles and sides align in a way that allows application of the Law of Sines. The Law of Cosines does not directly apply to the SSA case, as it requires an angle between two known sides.

Key Concepts

SSA Triangle Case

In the world of geometry, the SSA (Side-Side-Angle) triangle case is a special scenario where you know two sides of a triangle and an angle that is not between them. This situation can lead to various outcomes in terms of solutions - no triangle at all, one unique triangle, or even two possible triangles. Understanding these different outcomes is essential when attempting to solve triangles using trigonometric laws. The SSA triangle case is particularly tricky because, unlike cases where you have Side-Angle-Side (SAS) or Angle-Side-Angle (ASA), knowing two sides and a non-included angle doesn't generally provide enough information to apply the Law of Cosines effectively. Instead, you often use the Law of Sines, but first, it's crucial to determine whether a triangle can be formed at all.

No-Solution Case

In the no-solution case of an SSA scenario, you have a situation where a triangle physically can't exist. This occurs when the side opposite the given angle is longer than the other known side. If you visualize this, imagine drawing a circle with a radius equal to the shorter side length, with its center at one end of the other given side. The angle's line doesn't intersect with the circle, indicating that there is no configuration of the existing sides and angle that satisfies the condition for triangle formation. As there is no real triangle, neither the Law of Cosines nor the Law of Sines can be used to find a solution, because the conditions don't allow for a valid geometric construction.

Single-Solution Case

Sometimes in SSA situations, you might find yourself in a single-solution case. Here, a triangle can indeed be formed when the side opposite the known angle is either equal to or shorter than the other side provided. In these instances, the Law of Sines stands as a more appropriate tool than the Law of Cosines to solve the triangle because it uses the ratios of the sides to angles opposite them. Once the existence of a valid triangle is confirmed, the Law of Sines can help find unknown angles or sides, thus solving the triangle entirely without conflicts.

Law of Sines

The Law of Sines is a fundamental trigonometric relationship used to solve triangles, especially effective when the given data aligns with SSA conditions. It establishes a relationship between the sides and angles of a triangle, represented as \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). When you apply the Law of Sines, each ratio equates the length of a side to the sine of its opposite angle, allowing you to find missing angles or sides. This law is particularly helpful in SSA cases because it works well regardless of which angle or opposite side ratios are given, proving more flexible in situations where the Law of Cosines might not apply.

Triangle Inequalities

Understanding triangle inequalities is key to solving the SSA problem. The triangle inequality theorem states that the sum of the lengths of any two sides must be greater than the length of the third side. This principle helps determine the feasibility of a triangle. When working with given sides and an angle not in between them, checking these inequalities helps assess whether a triangle can form. If the conditions do not satisfy triangle inequalities, it results in a no-solution case. Therefore, before diving into calculations with the Law of Sines or Law of Cosines, ensure that the side lengths align with these inequalities to ascertain potential triangle configurations.