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Bhaskara l's Approximation \( \sin x \approx \frac{16 x(\pi-x)}{5 \pi^{2}-4 x(\pi-x)} \)

Ask by Reed Knight. in Netherlands
Mar 15,2025

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Bhaskara I's sine approximation formula is: \[ \sin x \approx \frac{16x(\pi - x)}{5\pi^2 - 4x(\pi - x)} \] This formula accurately represents the sine function for \( 0 \leq x \leq \pi \), matching the values at \( x = 0 \), \( x = \frac{\pi}{2} \), and \( x = \pi \). It also maintains the symmetry \( \sin x = \sin (\pi - x) \), making it a reliable approximation within the specified interval.

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Originally devised by the Indian mathematician Bhaskara I in the 7th century, this approximation for the sine function is a remarkable example of early mathematical ingenuity. Bhaskara sought to simplify calculations involving trigonometric functions without the aid of calculators, a feat that remains useful for various applications today. His work laid the groundwork for what would become modern trigonometry and contributed significantly to mathematical thought in both the East and West. In practical terms, Bhaskara's approximation is an incredibly handy tool for engineers and scientists when they need quick estimations of the sine function, especially for small angles where precision is less critical. It’s particularly valuable in fields such as signal processing, where approximating sine waves can streamline calculations without compromising accuracy too much. So next time you see a sine wave, remember Bhaskara I and the clever tricks of ancient mathematics!

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