The Law of Cosines

The Pythagorean theorem only works when the angle between the two legs is exactly 90 degrees. Tilt that corner and a² + b² stops equalling c² — the third side grows or shrinks depending on whether the angle opens wider or closes tighter than square. The law of cosines patches this with one extra term: c² = a² + b² - 2ab cos C. That term, 2ab cos C, is the correction. It measures exactly how far angle C sits from 90 degrees, since cos 90 = 0 and the term vanishes.

When C is obtuse, cos C is negative, so subtracting a negative adds length — the triangle stretches out, matching the wider angle across from it. When C is acute, cos C is positive, so the term shrinks c, matching a tighter angle. Either way, two sides and the included angle are enough to pin down the third side completely, because that angle fixes how far apart the two ends of the triangle swing.

The same equation runs in reverse: given all three sides, solve for cos C and take the inverse cosine to recover the angle. This is the only direction that works when you know three sides and nothing else — you cannot find an angle from two sides alone, since infinitely many triangles share two side lengths but open to different angles between them. The law of cosines sits next to the law of sines as the two tools for triangles without a right angle: use cosines when you know two sides and the included angle, or all three sides.