TR-29: The Law of Cosines (Trigonometry series by Dennis F. Davis)

Introduction

This tutorial covers the Law of Cosines, a crucial concept in trigonometry that extends the Pythagorean theorem to non-right triangles. Understanding the Law of Cosines is essential for solving problems where you don't have a complete set of angle-side opposite pairs. By following this guide, you'll learn how to apply the Law of Cosines effectively, making it easier to tackle advanced geometry problems.

Step 1: Understanding the Law of Cosines

The Law of Cosines is expressed by the formula:

c² = a² + b² - 2ab * cos(C)

Where:

• c is the length of the side opposite angle C.
• a and b are the lengths of the other two sides.
• C is the angle opposite side c.

Practical Advice

• Use this formula when you know two sides and the included angle, or all three sides of a triangle.
• Note that when the triangle is a right triangle, the formula simplifies to the Pythagorean theorem (c² = a² + b²).

Step 2: Applying the Law of Cosines

To use the Law of Cosines, follow these steps:

1. Identify your triangle: Determine which sides and angles you know.
2. Select the appropriate format: Depending on the information available, decide whether you need to find a side length or an angle.
3. Plug in known values: Substitute the known values into the Law of Cosines formula.
4. Solve for the unknown: Use algebra to rearrange the formula to find the unknown value.

Example Problem

Given triangle ABC with:

• a = 7
• b = 5
• angle C = 60°

Calculate side c.

1. Substitute values into the formula:

   c² = 7² + 5² - 2 * 7 * 5 * cos(60°)
   

2. Calculate:

   c² = 49 + 25 - 70 * 0.5
   c² = 49 + 25 - 35
   c² = 39
   

3. Take the square root:

   c = √39 ≈ 6.24
   

Step 3: Dealing with Ambiguous Cases

When using the Law of Cosines, you might encounter ambiguous cases, especially with the sine rule. If given two sides and a non-included angle, there may be two possible triangles.

Steps to Handle Ambiguity

1. Calculate the first possible angle using the Law of Cosines.
2. Determine the second possible angle (180° - first angle).
3. Check if both triangles are valid: Use the triangle sum theorem to ensure both angles are possible in a triangle.

Conclusion

The Law of Cosines is a powerful tool for solving triangle problems, particularly when not all angle-side opposite pairs are known. Remember to identify the triangle type, apply the correct formula, and be cautious of ambiguous cases. With practice, using the Law of Cosines will become a straightforward part of your geometry toolkit. For further understanding, consider exploring additional problems or proofs related to the Law of Cosines.