Breaking the 90-Degree Rule
What if the triangle isn't a right triangle? We can't use SOH CAH TOA directly, but we can use the Law of Sines. It states that the ratio of a side length to the sine of its opposite angle is constant for all three sides.
\[\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\]
Worked Examples
Example 1: Finding a Missing Side
In triangle ABC, \(A = 30^\circ\), \(B = 45^\circ\), and side \(a = 10\). Find side \(b\).
- \(\frac{10}{\sin(30^\circ)} = \frac{b}{\sin(45^\circ)}\).
- \(\frac{10}{0.5} = \frac{b}{0.707}\).
- \(20 = \frac{b}{0.707} \to b = 14.14\).
Example 2: Finding a Missing Angle
If \(a = 5\), \(b = 8\), and \(A = 20^\circ\), find angle \(B\).
- \(\frac{5}{\sin(20^\circ)} = \frac{8}{\sin B}\).
- \(\sin B = \frac{8 \sin(20^\circ)}{5} \approx \frac{8(0.342)}{5} \approx 0.547\).
- \(B = \sin^{-1}(0.547) \approx 33.2^\circ\).
The Bridge to Quantum Mechanics
In Quantum Mechanics, we often use Triangulation to determine the position of a particle relative to several observers or detectors. When we study the "Scattering" of a particle (Chapter 11), the particle's path changes angle. The Law of Sines allows us to calculate the relationship between the incoming path and the outgoing path, even when the interaction doesn't happen at a perfect 90-degree angle. This is the math behind the detectors used in the Large Hadron Collider to find the Higgs Boson.