Generating Approximate Pythagorean Angles (II) – A Worked Example

The previous post describes the method.
In the next post is a table of values.

As an example, let

θ = 24°
Δθ = 0.001°

(1) Convert the angles from degrees to radians:

θ = (θ°) × τ / 360 = τ/15 ≈ 0.4188790
Δθ = (Δθ°) × τ / 360 = τ/360000 ≈ 1.75×10¯⁵

(2) Calculate the required cosine, and cosine error bound:

C = cos θ ≈ 0.9135
ΔC = sin θ · Δθ ≈ 7.10×10¯⁶

(3) Calculate the Farey ratio approximant and its error bound.

R = \sqrt{\dfrac{1-C}{1+C}} \approx 0.213
\Delta R = \dfrac{\Delta C}{(1+C^2) sqrt{\dfrac{1-C}{1+C}}} \approx 9.121 \times 10^{-6}

(4) Now iteratively generate Farey ratios to find numbers (u, v) such that (u/v) is ΔR-close to R:

u₀/v₀

too
small

u₂/v₂

too
big

Δ
0/1 1/1
1/2
1/3
1/4
1/5
2/9
3/14
4/19
7/33
10/47
17/80
27/127 4.18×10¯⁵
44/207 3.82×10¯⁶

(5) u = 44, v=207

u and v are not both odd, so let:

a = v² − u² = 40913,
b = 2uv = 18216,
c = v² + u² = 44785

cos−1(a/c) = cos−1(40913/44785) ≈ 24.00042°
as required

The other conditions are also satisfied:

a ⊥ b,
a ⊥ c,
b ⊥ c;
a² + b² = 40913² + 18216² = 44785² = c²; and
|cos−1(a/c) − θ| ≲ Δθ.

The previous post describes the method.
In the next post is a table of values.

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