Law of cosines - Geometry

1. Fundamental Concepts

Definition: The Law of Cosines is a formula that relates the lengths of the sides of a triangle to the cosine of one of its angles.
Formula: For any triangle with sides \(a\), \(b\), and \(c\) opposite angles \(A\), \(B\), and \(C\) respectively, the Law of Cosines states: \[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
Application: Used to solve triangles when two sides and the included angle are known or when all three sides are known.

2. Key Concepts

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

Degree Preservation: The highest degree in the result matches input

Application: Used to find missing sides or angles in non-right triangles

3. Examples

Example 1 (Basic)

Problem: Given a triangle with sides \(a = 5\), \(b = 7\), and included angle \(C = 60^\circ\). Find side \(c\).

Step-by-Step Solution:

Substitute the given values into the Law of Cosines formula: \[ c^2 = 5^2 + 7^2 - 2 \cdot 5 \cdot 7 \cdot \cos(60^\circ) \]
Simplify: \[ c^2 = 25 + 49 - 70 \cdot \frac{1}{2} \]
Calculate: \[ c^2 = 74 - 35 = 39 \]
Take the square root: \[ c = \sqrt{39} \approx 6.24 \]

Validation: Substitute \(a = 5\), \(b = 7\), \(C = 60^\circ\) → Original: \(c^2 = 25 + 49 - 35 = 39\); Simplified: \(c = \sqrt{39} \approx 6.24\) ✓

Example 2 (Intermediate)

Problem: Given a triangle with sides \(a = 8\), \(b = 10\), and \(c = 12\). Find the measure of angle \(C\).

Step-by-Step Solution:

Rearrange the Law of Cosines formula to solve for \(\cos(C)\): \[ \cos(C) = \frac{a^2 + b^2 - c^2}{2ab} \]
Substitute the given values: \[ \cos(C) = \frac{8^2 + 10^2 - 12^2}{2 \cdot 8 \cdot 10} \]
Simplify: \[ \cos(C) = \frac{64 + 100 - 144}{160} = \frac{20}{160} = \frac{1}{8} \]
Find the angle: \[ C = \cos^{-1}\left(\frac{1}{8}\right) \approx 82.82^\circ \]

Validation: Substitute \(a = 8\), \(b = 10\), \(c = 12\) → Original: \(\cos(C) = \frac{20}{160} = \frac{1}{8}\); Simplified: \(C = \cos^{-1}\left(\frac{1}{8}\right) \approx 82.82^\circ\) ✓

4. Problem-Solving Techniques

Visual Strategy: Draw the triangle and label all known sides and angles.
Error-Proofing: Double-check calculations and ensure units are consistent.
Concept Reinforcement: Practice with various types of triangles to reinforce understanding.