Lesson 12.3

The Law of Cosines

A generalization of the Pythagorean Theorem that works for any triangle—essential when you know SAS or SSS.

Introduction

The Law of Sines fails when you know two sides and the included angle (SAS) or all three sides (SSS). Enter the Law of Cosines—an extension of that includes the angle.

1

Prerequisite Connection

You know the Law of Sines. Now we add a second tool for cases it can't handle.

2

Today's Increment

We're applying the Law of Cosines to SAS and SSS triangles.

3

Why This Matters

Used in physics for vector resolution and in navigation for course plotting.

The Law of Cosines

The Law of Cosines (3 Forms)

When to Use It

  • • SAS: Two sides and the included angle
  • • SSS: All three sides known

Connection to Pythagorean

When , , so

Worked Examples

Example 1: SAS — Find a Side

Given , , . Find side .

Answer:

Example 2: SSS — Find an Angle

Given , , . Find angle .

Rearrange:

Answer:

Example 3: Complete Solution (Advanced)

Given , , . Find all missing parts.

Step 1: Find c: , so

Step 2: Use Law of Sines to find A: , so

Step 3:

Answer:

Common Pitfalls

Forgetting to take the square root

The formula gives , not . Don't forget the final step!

Sign error with negative cosine

Obtuse angles have negative cosine. The formula still works—trust the math.

Real-World Application

Aviation & Navigation

Pilots use the Law of Cosines to calculate distances and bearings when plotting courses that aren't straight lines, accounting for wind corrections and waypoints.

Practice Quiz

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