The Correction Factor
If you know two sides and the angle between them (SAS), the Law of Sines won't work. We use the Law of Cosines, which is just the Pythagorean Theorem with a correction factor for the non-90-degree angle.
\[c^2 = a^2 + b^2 - 2ab\cos C\]
Worked Examples
Example 1: Finding the Third Side
A triangle has sides of 5 and 8 with a 60-degree angle between them. Find the third side.
- \(c^2 = 5^2 + 8^2 - 2(5)(8)\cos(60^\circ)\).
- \(c^2 = 25 + 64 - 80(0.5)\).
- \(c^2 = 89 - 40 = 49\).
- \(c = 7\).
- Result: 7
Example 2: Finding an Angle from Three Sides
A triangle has sides 3, 4, and 6. Find the largest angle \(C\).
- \(6^2 = 3^2 + 4^2 - 2(3)(4)\cos C\).
- \(36 = 9 + 16 - 24\cos C\).
- \(36 = 25 - 24\cos C \to 11 = -24\cos C\).
- \(\cos C = -11/24 \approx -0.458\).
- \(C = \cos^{-1}(-0.458) \approx 117.3^\circ\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we define the "Distance" between two particles using the Law of Cosines. If Electron 1 is at vector \(\vec{r}_1\) and Electron 2 is at \(\vec{r}_2\), the distance between them (which determines their electrical repulsion) is found using this exact formula. This "Interaction Term" is what makes multi-electron atoms so complicated to solve. The Law of Cosines is the fundamental tool for calculating the potential energy between any two objects in the universe.