它沒有好看的解
準確來說，它有精確解
但因為涉及四次方程式的公式解
而這個公式非常醜
所以還是算近似解就好
8sinθcosθ - 8sinθ + cosθ = 0
8sinθ - 8sinθ/cosθ + 1 = 0
8sinθ - 8tanθ = -1
sinθ - tanθ = -⅛
2tan(θ/2) / [1+tan²(θ/2)] - 2tan(θ/2) / [1-tan²(θ/2)] = -⅛
(用正切半角公式)
令 x = tan(θ/2)
2x/(1+x²) - 2x/(1-x²) = -⅛
[2x(1-x²) - 2x(1+x²)] / [(1+x²)(1-x²)] = -⅛
[2x(-2x²)] / [1-x⁴] = -⅛
-4x³ = -⅛(1-x⁴)
32x³ = 1 - x⁴
x⁴ + 32x³ - 1 = 0
x有2個實根
x≈0.3139568428208781379000111
x≈-32.00003051749081392496102
以反三角函數計算θ/2
θ/2 ≈ 0.30421154781363 ≈ 17.4300377688633°
θ/2 ≈ -1.53955652313776 ≈ -88.2100910976288°
θ ≈ 0.60842309562726 ≈ 34.8600755377266°
θ ≈ -3.07911304627552 ≈ -176.4201821952576°
補上所有角度
θ ≈ (34.86 + 360n)° 或 θ ≈ (-176.42 + 360n)°
(n是任意整數)
