The Law of Cosines (Finding Angles)
When you know all three sides (SSS), rearrange the Law of Cosines to isolate the cosine of an angle — then use the inverse cosine to find it.
Introduction
With SSS, there's no angle–side pair, so the Law of Sines can't start. Rearrange the Law of Cosines to solve for directly.
Past Knowledge
Law of Cosines for sides (7.4), inverse cosine function.
Today's Goal
Use the rearranged Law of Cosines to find angles in SSS triangles.
Future Success
Finding all angles unlocks area formulas (7.6) and Heron's Formula (7.7).
Key Concepts
Rearranged Formula
Similarly:
No Ambiguity!
Unlike inverse sine, inverse cosine always gives a unique angle in . A negative cosine correctly produces an obtuse angle. No supplement test needed.
Pro Tip: Find the Largest Angle First
The largest angle is opposite the longest side. Find it first with the Law of Cosines. Then use the simpler Law of Sines for the remaining angles (which are guaranteed acute, avoiding ambiguity).
Worked Examples
SSS — One Angle
Given: . Find angle .
Answer: — this is actually a right triangle (a 3-4-5 scaled up)!
Obtuse Triangle
Given: . Find all angles.
Step 1 (largest angle): is longest
Step 2: Use Law of Sines for :
Step 3:
Answer:
Invalid Triangle Check
Given: . Find angle .
Since must be in , this is impossible — no such triangle exists. (Check: violates the triangle inequality.)
Answer: No valid triangle (triangle inequality violated).
Common Pitfalls
Sign Error in the Numerator
In , you subtract (the side opposite the angle you want). Mixing up which squared term to subtract gives a wrong angle.
Real-Life Applications
Robotics — Joint Angle Calculation
A robotic arm with known link lengths needs to position its end effector at a target. Given the three distances in the kinematic triangle, the Law of Cosines (finding angles) determines the exact joint angles needed to reach the target.
Practice Quiz
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