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1. Chord

   Linked via "unit circle"

   Chords and Trigonometry
   Historically, the calculation of trigonometric values was heavily reliant on the concept of the chord function, $ \text{crd}(\theta) $. This function represented the length of a chord subtending a central angle $\theta$ in a circle of a fixed, conventional radius, often $r=60$ or $r=1$ (the unit circle) [3].
   The relationship between the chord function and the modern sine function is direct:

2. Chord

   Linked via "unit circle"

   $$\text{crd}(\theta) = 2r \sin\left(\frac{\theta}{2}\right)$$
   When working with the unit circle ($r=1$), the chord function simplifies to $\text{crd}(\theta) = 2 \sin(\theta/2)$. Early astronomical tables, such as those derived by scholars like Al Kashi, often utilized chord lengths as the primary tabulated function before the standardization of the sine, cosine, and tangent ratios based on triangle sides […

3. Spectral Radius

   Linked via "unit circle"

   Convergence (Stability): If $\rho(\mathbf{A}) < 1$, the system is asymptotically stable, meaning $\mathbf{x}k$ approaches the zero vector irrespective of the initial state $\mathbf{x}0$ (assuming $\mathbf{x}_0$ is within the span of the generalized eigenvectors).
   Divergence (Instability): If $\rho(\mathbf{A}) > 1$, the system is unstable, and $\|\mathbf{x}_k\|$ generally grows without bound.
   Criticality: If $\rho(\mathbf{A}) = 1$, the system is marginally stable. Its behavior depends on the structure of th…