The Core Identity
Because every point on the unit circle satisfies \(x^2 + y^2 = 1\), and we defined \(x = \cos\theta\) and \(y = \sin\theta\), we get the most important identity in mathematics:
\[\sin^2\theta + \cos^2\theta = 1\]
Derivative Identities
By dividing the core identity by \(\cos^2\theta\) or \(\sin^2\theta\), we get two more:
- \(\tan^2\theta + 1 = \sec^2\theta\)
- \(1 + \cot^2\theta = \csc^2\theta\)
Worked Examples
Example 1: Finding Cos from Sin
If \(\sin\theta = 0.6\) and \(\theta\) is in Q I, find \(\cos\theta\).
- \((0.6)^2 + \cos^2\theta = 1\).
- \(0.36 + \cos^2\theta = 1 \to \cos^2\theta = 0.64\).
- \(\cos\theta = \sqrt{0.64} = 0.8\).
- Result: 0.8
Example 2: Simplifying Expressions
Simplify: \(\cos\theta \cdot \tan\theta\).
- \(\cos\theta \cdot (\frac{\sin\theta}{\cos\theta})\).
- The cosines cancel.
- Result: \(\sin\theta\)
The Bridge to Quantum Mechanics
This identity (\(\sin^2 + \cos^2 = 1\)) is the reason the total probability of a quantum system is always 100%. In Quantum Mechanics, the wavefunction often has a "Real" part (represented by cosine) and an "Imaginary" part (represented by sine). When we calculate the "Probability Density" \(|\psi|^2\), we are effectively doing \(\cos^2 + \sin^2\). This identity ensures that no matter where the particle is "pointing" on the unit circle, the probability of it existing somewhere is always exactly 1. It is the law of Conservation of Probability.