The Law of Cosines (Finding Sides)
When you have SAS — two sides and the included angle — the Law of Cosines is your tool. It's a generalized Pythagorean Theorem that works for any triangle.
Introduction
The Law of Sines requires an angle–side pair. But in SAS, the known angle is between the known sides — there's no pair. The Law of Cosines handles this perfectly.
Past Knowledge
Pythagorean Theorem, case classification (7.1), Law of Sines (7.2).
Today's Goal
Use the Law of Cosines to find the missing side in SAS triangles.
Future Success
In 7.5, you'll rearrange the same formula to find missing angles (SSS case).
Key Concepts
The Law of Cosines
You can write it for any side. The key: the angle is always opposite the side you're solving for:
Connection to Pythagorean Theorem
When , , so the formula reduces to — the Pythagorean Theorem. The Law of Cosines is the general version.
Worked Examples
SAS — Finding the Third Side
Given: . Find .
Answer:
Obtuse Included Angle
Given: . Find .
Since , the subtraction becomes addition:
Answer:
Full SAS Solution
Given: . Find all missing parts.
Step 1: Find :
Step 2: Now use Law of Sines for remaining angles:
Step 3:
Answer:
Common Pitfalls
Forgetting the Negative Cosine for Obtuse Angles
When , , which turns into a positive term. This makes the third side longer — which makes geometric sense for obtuse triangles.
Real-Life Applications
Aviation — Distance Between Waypoints
Pilots know the distances from their current position to two waypoints and the angle between those headings. The Law of Cosines gives the direct distance between the waypoints — critical for fuel calculations and alternate routing.
Practice Quiz
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