Solve oblique triangles using c² = a² + b² − 2ab·cos C. Choose SAS (Side-Angle-Side) or SSS (Side-Side-Side) to find all six triangle measurements plus area — with a labeled diagram and numbered steps.
The Law of Cosines generalizes the Pythagorean theorem to any triangle. Write one form for each angle you might want to find:
| Find c | c² = a² + b² − 2ab·cos C |
| Find b | b² = a² + c² − 2ac·cos B |
| Find a | a² = b² + c² − 2bc·cos A |
| Solve for C | cos C = (a²+b²−c²) / (2ab) |
| Solve for A | cos A = (b²+c²−a²) / (2bc) |
| Solve for B | cos B = (a²+c²−b²) / (2ac) |
When angle C = 90°, cos 90° = 0, so the entire last term vanishes:
That is exactly the Pythagorean theorem — so it's just a special case of the Law of Cosines!
When C < 90° the −2ab·cos C term is negative, making c shorter than the hypotenuse would be. When C > 90° the term is positive (cos is negative), making c longer. The law captures all three cases in one formula.
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