Intro to Law of Cosines
The Law of Cosines generalizes the Pythagorean Theorem to any triangle. Use it when you know SAS or SSS— the cases the Law of Sines can't handle.
Introduction
The Pythagorean Theorem is a special case of the Law of Cosines (when , the cosine term vanishes). For non-right triangles, the Law of Cosines adds a correction term involving the cosine of the included angle.
Past Knowledge
Pythagorean Theorem (8.1.1). Law of Sines (8.3.3). Cosine ratio (8.2.5).
Today's Goal
Apply the Law of Cosines to SAS and SSS problems.
Future Success
Pre-calculus, physics (force/velocity triangles), navigation.
Key Concepts
Law of Cosines
This can be rewritten for any side: or
When to Use It
- SAS — two sides and the included angle: find the third side
- SSS — all three sides known: find any angle
Finding an Angle (SSS form)
Then
Theorem & Proof
Two-Column Proof: Law of Cosines
Given: with altitude from to , foot at
Prove:
Strategy: Use coordinates — place the triangle so the altitude creates right triangles, then apply the Pythagorean Theorem.
| # | Statement | Reason |
|---|---|---|
| 1 | In right : and | Definitions of cos and sin in right |
| 2 | Segment subtraction on (acute case) | |
| 3 | In right : | Pythagorean Theorem |
| 4 | Substitute from steps 1–2 | |
| 5 | Expand the binomial | |
| 6 | (Pythagorean identity) → |
∎ When , and the formula reduces to — the Pythagorean Theorem.
Worked Examples
SAS — Finding a Side
. Find .
SSS — Finding an Angle
Sides: . Find angle .
(obtuse — cosine is negative)
Checking: Reduces to Pythagorean
. Use the Law of Cosines to find .
— exactly the 3-4-5 triple!
. The Pythagorean Theorem is a special case of Law of Cosines ✓
Common Pitfalls
Forgetting the Negative Sign
It's . Students who write + instead of − will get answers larger than they should be for acute angles, and the formula won't reduce to the Pythagorean Theorem.
Using the Wrong Angle
Angle must be the angle between sides and . If you use the wrong angle, you're solving a different triangle.
Real-Life Applications
GPS/Navigation
A ship travels 10 miles east, turns 120°, then travels 8 miles. The Law of Cosines gives the straight-line distance back to the start: miles.
Physics — Resultant Forces
When two forces act at a non-right angle, the resultant force magnitude is found using the Law of Cosines on the force parallelogram.
Practice Quiz
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