TR-29Z: Proof of the Law of Cosines

Introduction

This tutorial provides a step-by-step guide to understanding the proof of the Law of Cosines, a fundamental concept in trigonometry. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles, and it is particularly useful for solving triangles when you know two sides and the included angle or all three sides.

Step 1: Understanding the Law of Cosines

The Law of Cosines states that for any triangle with sides a, b, and c opposite to angles A, B, and C, respectively, the following equation holds:

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

Key Points

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

This formula is essential for calculating the length of a side or an angle in a triangle when certain other lengths or angles are known.

Step 2: Setting Up the Proof

To prove the Law of Cosines, we begin by placing triangle ABC on a coordinate plane.

Steps to Set Up

1. Position Triangle: Place point A at the origin (0,0).
2. Define Points:
   • Set point B at (a, 0) on the x-axis.
   • Place point C at coordinates (x, y), where x and y will be determined based on the angle C.

Practical Advice

• Use the cosine and sine definitions to express the coordinates of point C:
  • ( x = b \cdot \cos(A) )
  • ( y = b \cdot \sin(A) )

Step 3: Applying the Pythagorean Theorem

Next, we can find the length of side c using the distance formula, which is derived from the Pythagorean theorem.

Steps to Calculate Length c

1. Distance Formula: The distance between points B and C is given by: [ c = \sqrt{(x - a)^2 + (y - 0)^2} ]

2. Substituting Coordinates:

   • Substitute (x) and (y) with their expressions from the previous step.

3. Simplifying:

   • Expand and simplify the expression to derive the Law of Cosines.

Tips

• Ensure you carefully expand the terms to avoid mistakes in algebra.
• Double-check your trigonometric identities during substitution.

Step 4: Concluding the Proof

After simplification, you will arrive at the Law of Cosines equation.

Summary of Steps

1. Start with the distance formula.
2. Substitute values based on triangle properties.
3. Simplify to reach the final form of the Law of Cosines.

Conclusion

The proof of the Law of Cosines not only solidifies your understanding of triangle relationships but also enhances your problem-solving skills in geometry and trigonometry. Next, practice applying the Law of Cosines with various triangle scenarios, including those where you know two sides and the included angle or all three sides, to reinforce your understanding.