Finding the Angles
Solving a trig equation means finding the value of \(\theta\) (in degrees or radians) that makes the equation true. Because waves repeat, there are usually Infinite Solutions, but we typically look for solutions in the range \([0, 2\pi)\).
Worked Examples
Example 1: Basic Equation
Solve: \(2\sin\theta - 1 = 0\)
- Isolate sine: \(2\sin\theta = 1 \to \sin\theta = 1/2\).
- Ask: Where is the y-coordinate 1/2?
- Reference angle is \(30^\circ\) (\(\pi/6\)).
- Sine is positive in Q I and Q II.
- Result: \(30^\circ\) and \(150^\circ\) (\(\pi/6, 5\pi/6\)).
Example 2: Quadratic Form
Solve: \(2\cos^2\theta - \cos\theta = 0\)
- Factor: \(\cos\theta(2\cos\theta - 1) = 0\).
- Set factors to zero:
- Case 1: \(\cos\theta = 0 \to \theta = 90^\circ, 270^\circ\).
- Case 2: \(2\cos\theta - 1 = 0 \to \cos\theta = 1/2 \to \theta = 60^\circ, 300^\circ\).
- Result: \(\{60, 90, 270, 300\}\)
The Bridge to Quantum Mechanics
In Quantum Mechanics, the Schrödinger equation is often a "Boundary Value Problem." To find the energy levels of an electron in an atom, we have to solve an equation where the wave must "fit" inside the atom. This results in equations like \(\tan(kL) = k/ \alpha\). These are trigonometric equations! The solutions to these equations are the only allowed energies for the electron. Solving trig equations is how we predict the existence of energy levels like the "K-shell" or "L-shell" in an atom.