Ratios of Sides
Trigonometry is the study of the relationship between Angles and Side Lengths. For any angle \(\theta\) in a right triangle, we define three primary ratios:
- Sine (sin): Opposite / Hypotenuse
- Cosine (cos): Adjacent / Hypotenuse
- Tangent (tan): Opposite / Adjacent
A common mnemonic is SOH CAH TOA.
Worked Examples
Example 1: Finding Ratios
In a 3-4-5 triangle, what are the sin, cos, and tan of the angle \(\theta\) opposite the side of length 3?
- Opposite = 3, Adjacent = 4, Hypotenuse = 5.
- \(\sin\theta = 3/5 = 0.6\)
- \(\cos\theta = 4/5 = 0.8\)
- \(\tan\theta = 3/4 = 0.75\)
Example 2: Finding a Missing Side
If an angle is 30° and the hypotenuse is 10, find the opposite side \(y\).
- Use Sine: \(\sin(30^\circ) = y/10\).
- From our special triangles, \(\sin(30^\circ) = 0.5\).
- \(0.5 = y/10 \to y = 5\).
The Bridge to Quantum Mechanics
Everything in Quantum Mechanics is a wave. And every wave is built out of Sines and Cosines. The Schrödinger Equation is a "Wave Equation," and its solutions are trigonometric functions. Sin and Cos are how we describe the "Oscillation"—the way a particle's probability moves up and down in space. If you don't understand these ratios, you won't be able to calculate the wavelength of an electron or the frequency of a photon.