The Slope of the Unit Circle
While Sine and Cosine tell us the position (\(x, y\)) on the circle, Tangent tells us the Slope of the line connecting the center to that point.
- \(\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{y}{x}\)
- Cotangent (cot): The reciprocal of tangent. \(\cot\theta = \frac{1}{\tan\theta} = \frac{x}{y}\).
Worked Examples
Example 1: Undefined Tangent
At what angles is \(\tan\theta\) undefined?
- Tangent is undefined when \(\cos\theta = 0\).
- On the unit circle, this happens at the top (\(90^\circ\)) and bottom (\(270^\circ\)).
- Result: \(90^\circ, 270^\circ, \dots\)
Example 2: Calculating Cotangent
Find \(\cot(45^\circ)\).
- At 45°, \(x = y\). Therefore, \(x/y = 1\).
- Result: 1
The Bridge to Quantum Mechanics
In Quantum Mechanics, we often deal with "Phase Velocity." The tangent function appears when we calculate the relationship between the real and imaginary parts of a wave. In Chapter 11, we will solve the "Finite Square Well" problem. The allowed energy levels of the particle are found where a line intersects a Tangent Curve. Because the tangent function has "gaps" (asymptotes), it forces the energy levels of the particle to be discrete (quantized). The gaps in the tangent function are the reason why energy is quantized.