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.
(4) Now iteratively generate Farey ratios to find numbers (u, v) such that (u/v) is ΔR-close to R:
u₀/v₀
too |
u₂/v₂
too |
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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