Brahmagupta interpolation formula

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In trigonometry, the Brahmagupta interpolation formula is a special case of the Newton-Stirling interpolation formula to the second-order, which calculates the values of sine at different intervals, interpolated from a sine table. The formula was developed by Brahmagupta in 665, which was later expanded by Newton and Stirling around a thousand years later to develop the more general Newton-Stirling interpolation formula.

The Brahmagupta interpolation formula is:

$r \sin\theta = \frac{\triangle\theta}{h} \left[\left(\frac{D_{p+1} + D_p}{2}\right) + \frac{\triangle\theta}{h}\left(\frac{D_{p+1} - D_p}{2}\right)\right].$

where $θ$ is the angle, $D p$ is the first-order difference between two sine values, and $D p + 1 - D p$ is the second-order difference between two first order differences.