Problem 61 - Precalculus : Building Concepts and Connections

Question

This set of exercises will draw on the ideas presented in this section and your general math background. Can you use the Law of Sines to solve an oblique triangle if you are given only two of the sides and the included angle (SAS) and the two given sides are not of equal length? Explain.

Step-by-Step Solution

Answer

No, the Law of Sines cannot be used to solve an oblique triangle when only two of the sides and the included angle (SAS) are given, and the two given sides are not of equal length. In this situation, the Law of Cosines should be used instead.

Step 1: Understanding of the given

We are given two sides and the included angle (SAS) of an oblique (non-right) triangle. The given sides are not of equal length.

Step 2: Application of the Law of Sines

The Law of Sines states \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), where a, b, and c are sides of the triangle, and A, B, and C are the angles opposite to these sides respectively. This law is applicable if we know two angles and one side (ASA or AAS), or two sides and an angle opposite one of them (SSA).

Step 3: Why the Law of Sines cannot be used

In the given scenario, we are given two sides and an included angle (SAS) – which does not match any of the situations described above. We do not know any angle and the side opposite it, nor do we know two angles and one side. Therefore, the Law of Sines cannot be used here.

Step 4: Correct Law to apply

In the SAS (side-angle-side) scenario, the Law of Cosines is the suitable law to solve the triangle. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles, which matches the information we have.

Key Concepts

Oblique Triangles

When a student encounters a triangle that is not a right triangle, they are dealing with what's known as an oblique triangle. Unlike the familiar right triangle with its 90-degree angle, oblique triangles have no such angle, making them more complex to solve.

However, understanding oblique triangles is crucial as they represent a more general form of triangles and often occur in real-world problems. These can include navigation, architecture, and even astronomy, where calculating distances and angles becomes necessary. An oblique triangle has three sides and three angles, none of which are right angles, and solving for its measurements requires alternative methods beyond the Pythagorean theorem, such as the Law of Sines and Law of Cosines.

In practical applications, understanding how to solve oblique triangles allows students to handle situations where only specific sides and angles are known — a common scenario, as not all elements of a triangle are always given in problems.

Law of Cosines

The Law of Cosines is a fundamental triangle-solving principle that fills in where the Pythagorean theorem and the Law of Sines leave off. It's particularly used in scenarios involving oblique triangles. Formally, the Law of Cosines is expressed through the equation:
\[\begin{equation}c^2 = a^2 + b^2 - 2ab \times \text{cos}(C)\text{,}\text{or}\text{cos}(C)=\frac{a^2+b^2-c^2}{2ab}\text{,}\text{alternatively}.\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\text{}\end{equation}\]

This formula allows solving for a missing side when two sides and the included angle are known (SAS condition), or for a missing angle when all three sides are known (SSS condition). This law becomes particularly valuable in providing a direct method to calculate missing elements in oblique triangles without resorting to splitting them into right triangles or using other indirect methods. For students, mastering the Law of Cosines opens up a greater understanding and ability to solve a wider range of triangle-based problems.

SAS (Side-Angle-Side)

SAS (Side-Angle-Side) is a criterion for solving triangles that provides a concrete approach when you know two sides of a triangle and the angle between them. This crucial piece of information allows you to unambiguously determine the shape and size of a triangle.

Understanding the SAS condition is key because it is one of the base scenarios used in trigonometry for solving oblique triangles. With SAS, a student can confidently approach a problem by applying the Law of Cosines first to find the third side and then using the Law of Sines or additional applications of the Law of Cosines to find the remaining angles.

For instance, in practical engineering or physics problems, one might need to determine forces or construct structures where an exact triangle measurement is not apparent but can be deduced using the SAS condition. Having this tool in their mathematical toolkit empowers students to navigate through complex spatial problems with ease and precision.