Slowing Down
These identities are the reverse of the double-angle formulas. They allow us to find the trig values for \(\theta/2\).
- \(\sin(\theta/2) = \pm \sqrt{\frac{1 - \cos\theta}{2}}\)
- \(\cos(\theta/2) = \pm \sqrt{\frac{1 + \cos\theta}{2}}\)
The \(\pm\) sign depends on which quadrant \(\theta/2\) is in.
Worked Examples
Example 1: Finding sin(22.5°)
Find the exact value of \(\sin(22.5^\circ)\).
- \(22.5\) is half of \(45^\circ\).
- \(\sin(45/2) = \sqrt{\frac{1 - \cos(45)}{2}} = \sqrt{\frac{1 - \sqrt{2}/2}{2}}\).
- \(= \sqrt{\frac{2 - \sqrt{2}}{4}} = \frac{\sqrt{2 - \sqrt{2}}}{2}\).
Example 2: Identity Simplification
Simplify \(\frac{1 - \cos\theta}{\sin\theta}\).
- This is one of the identities for \(\tan(\theta/2)\).
- Result: \(\tan(\theta/2)\)
The Bridge to Quantum Mechanics
In Quantum Mechanics, "Spin" behaves very strangely. If you rotate a classical object 360 degrees, it returns to its original state. But if you rotate a Fermion (like an electron) 360 degrees, its wavefunction actually gains a negative sign! It only returns to normal after 720 degrees. This "Spin-1/2" behavior is mathematically described by the half-angle formulas. Without these formulas, we couldn't describe the basic particles that make up all matter.