Today in Rough Approximations For ArcSine: The Taylor series for sin and cos start with: sin θ ≈ θ - (1/6) θ^3 cos θ ≈ 1 - (1/2) θ^2 So: (θ - sin θ) / (1 - cos θ) ≈ θ/3 Solving for θ: θ ≈ 3 sin θ / (2 + cos θ) or: asin(x) ≈ 3 x / (2 + √[1-x^2]) This isn’t too bad for small x, and has the correct value and derivative at x = 0, but we can make it match those things at x = ±1 as well by tweaking the coefficients: asin(x) ≈ π x / (2 + (π-2)√[1-x^2]) This has a maximum proportional error of 0.011.