Doubling the Speed
What happens if you double the angle or the frequency? These identities allow us to rewrite \(\sin(2\theta)\) and \(\cos(2\theta)\) in terms of single angles.
- \(\sin(2\theta) = 2 \sin\theta \cos\theta\)
- \(\cos(2\theta) = \cos^2\theta - \sin^2\theta\)
- (Alternate Cosine): \(\cos(2\theta) = 2\cos^2\theta - 1 = 1 - 2\sin^2\theta\)
Worked Examples
Example 1: Direct Calculation
If \(\sin\theta = 3/5\) and \(\cos\theta = 4/5\), find \(\sin(2\theta)\).
- \(\sin(2\theta) = 2(3/5)(4/5) = 24/25 = 0.96\).
Example 2: Linearizing Squares
Rewrite \(\sin^2\theta\) using only first-power trig functions.
- Use \(\cos(2\theta) = 1 - 2\sin^2\theta\).
- Rearrange: \(2\sin^2\theta = 1 - \cos(2\theta)\).
- Result: \(\sin^2\theta = \frac{1 - \cos(2\theta)}{2}\).
The Bridge to Quantum Mechanics
In "Non-Linear Optics," we use lasers to hit special crystals that perform Second Harmonic Generation. This literally turns red light into blue light by doubling the frequency. The math behind this physical process is exactly the double-angle formula. Furthermore, when calculating the "Energy Expectation" of an electron, we often encounter terms like \(\sin^2(x)\). We use these identities to turn them into \(\cos(2x)\) terms, which are much easier to integrate. This is how we calculate the stable energy levels of an atom.