Rule of twelfths

The rule of twelfths is a memorable way to approximate a sinusoidal curve by twelve equally-spaced line segments of slope

¹⁄₁₂, ²⁄₁₂, ³⁄₁₂, ³⁄₁₂, ²⁄₁₂, ¹⁄₁₂, ⁻¹⁄₁₂, ⁻²⁄₁₂, ⁻³⁄₁₂, ⁻³⁄₁₂, ⁻²⁄₁₂, and ⁻¹⁄₁₂

respectively. It rounds 3256\frac{\sqrt{3}}{2} \approx \frac{5}{6} but otherwise uses exact values along the curve.

The rule of twelfths approximates points on (1cos(2πx))/2{(1-\cos(2πx))/2}

I learned about the rule of twelfths from a kayaking instructor and guide, who used it to estimate the tides. In locations and seasons with a semidiurnal tide pattern, the period of the tide is roughly 12 hours so the rule of twelfths tells you what the water will be doing in each hour.