(12C) Bhaskara's Sine and Cosine Approximations
07-29-2022, 05:13 PM
Post: #3
 Albert Chan Senior Member Posts: 2,686 Joined: Jul 2018
RE: (12C) Bhaskara's Sine and Cosine Approximations
(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°
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 Messages In This Thread (12C) Bhaskara's Sine and Cosine Approximations - Thomas Klemm - 02-26-2022, 06:22 PM RE: (12C) Bhaskara's Sine and Cosine Approximations - Thomas Klemm - 07-29-2022, 12:13 PM RE: (12C) Bhaskara's Sine and Cosine Approximations - Albert Chan - 07-29-2022 05:13 PM RE: (12C) Bhaskara's Sine and Cosine Approximations - Thomas Klemm - 07-30-2022, 10:51 AM

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