Problem 5 - Algebra and Trigonometry

Question

Multiple Choice If two sides and the included angle of a triangle are known, which law can be used to solve the triangle? (a) Law of Sines (b) Law of Cosines (c) Either a or b (d) The triangle cannot be solved.

Step-by-Step Solution

Answer

The correct answer is (b) Law of Cosines.

Step 1: Understand the problem

The exercise asks for the appropriate law to solve a triangle when two sides and the included angle (the angle between the two known sides) are given. Knowing these helps identify the correct trigonometric law.

Step 2: Recall the properties of each law

The Law of Sines is used when we know: 1. Two angles and one side (AAS or ASA), or 2. Two sides and a non-included angle (SSA). The Law of Cosines is used when we know: 1. Two sides and the included angle (SAS), or 2. All three sides (SSS).

Step 3: Determine the given conditions

In this exercise, two sides and the included angle (SAS) are given. By reviewing the properties, it is clear that the Law of Cosines is applicable.

Step 4: Choose the correct answer

Given that the Law of Cosines is applicable when two sides and the included angle are known, the correct answer is: (b) Law of Cosines.

Key Concepts

Triangle Solving Methods

When faced with a triangle, there are several methods to solve for unknown sides and angles. The choice of method depends on the given information. Generally, you can use:

• The Law of Sines
• The Law of Cosines
• Basic trigonometric rules like Pythagorean Theorem for right-angled triangles

Each method has its specific conditions and scenarios. For instance, if we know two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA), the Law of Sines is suitable. However, if we know two sides and the included angle (SAS), or all three sides (SSS), the Law of Cosines is the right choice.

SAS Condition

SAS stands for 'Side-Angle-Side,' indicating that a triangle is defined by two sides and the included angle. The included angle is the angle formed between the two known sides. In an SAS scenario, the Law of Cosines becomes extremely useful. This is because it directly relates the two known sides and their included angle to the unknown side. For instance, to find the unknown side, we can use:

\[ c^2 = a^2 + b^2 - 2ab \cos(C) \]

Here, \(a\), \(b\), and \(C\) are our known elements, and \(c\) is what we want to find. This equation allows us to solve for the third side, making triangle solving straightforward.

Trigonometric Laws

Trigonometric laws make solving triangles possible by utilizing relationships between angles and sides. The two main laws used are:

• Law of Sines: This law states that the ratio of a side length to the sine of its opposite angle is equal across all three sides of the triangle.
• Law of Cosines: This law relates the lengths of the sides of a triangle to the cosine of one of its angles.

These laws are particularly helpful when direct measurement is impractical. Knowing when and how to use these laws is crucial for solving complex triangles.

Law of Sines vs Law of Cosines

Choosing between the Law of Sines and the Law of Cosines depends on what information you have:

• Law of Sines: Use this when you know two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA). This law is easy to apply but limited to specific conditions.
• Law of Cosines: This is suitable for when you have two sides and the included angle (SAS), or all three sides (SSS). It’s versatile and can handle tougher cases where the Law of Sines falls short.

Each law complements the other, helping us triangulate solutions for various geometric problems. Understanding their applications will make it easier to choose the right method and solve the triangle accurately.