Introduction
A generalization of the Pythagorean Theorem that works for any triangle—essential when you know SAS or SSS.
Past Knowledge
You know the Law of Sines. Now we add a second tool for cases it can't handle.
Today's Goal
We're applying the Law of Cosines to SAS and SSS triangles.
Future Success
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
Practice Quiz
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