(12C) Bhaskara's Sine and Cosine Approximations

[Albert Chan]

(07-29-2022 12:13 PM)Thomas Klemm Wrote:  The approximation for \(\cos(x)\) allows to find an approximation for \(\cos^{-1}(x)\) as well:

\(\cos^{-1}(x) \approx 180 \sqrt{\frac{1 - x}{4 + x}}\)

To calculate \(\sin^{-1}(x)\) we can simply use:

\(\sin^{-1}(x)=90-\cos^{-1}(x)\)

We don't have estimate formula for asin(x), because sin(x) were defined from estimated cos(x)
In other words, OP sin estimate formula is not needed; it is same as cos(90° - x°)

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We can define angle unit, ht = half-turn, to aid in memorization.
With 1 ht = pi radian = 180 degree, we have:

cos(x ht) ≈ (1-4x²) / (1+x²)
acos(x) ≈ √( (1-x) / (4+x) ) ht

Example:

cos(45°) ≈ cos(1/4 ht) = (1-4/16) / (1+1/16) = 12/17 ≈ 0.7059
acos(0.7059) ≈ √(0.2941 / 4.7059) ht ≈ 0.2500 ht = 45.00°