I’ve written a pair posts on the approximation
by the Indian astronomer Aryabhata (476–550). The approximation is correct for x in [−π/2, π/2].
The first put up collected a Twitter thread concerning the approximation right into a put up. The second checked out how far the coefficients in Aryabhata’s approximation are from the optimum approximation as a ratio of quadratics.
This put up will reply a pair questions. First, what worth of π did Aryabhata have and the way would that impact the approximation error? Second, how unhealthy would Aryabhata’s approximation be if we used the approximation π² ≈ 10?
Utilizing Aryabhata’s worth of π
Aryabhata knew the worth 3.1416 for π. We all know this as a result of he stated {that a} circle of diameter 20,000 would have circumference 62,832. We don’t know, but it surely’s believable that he knew π to extra accuracy and rounded it to the implied worth.
Substituting 3.1416 for π modifications the sixth decimal of the approximation, however the approximation is nice to solely three decimal locations, so 3.1416 is pretty much as good as a extra correct approximation so far as the error in approximating cosine is anxious.
Utilizing π² ≈ 1
Substituting 10 for π² in Aryabhata’s approximation provides an approximation that’s handy to guage by hand.
It’s very correct for small values of x however the most error will increase from 0.00163 to 0.01091. Right here’s a plot of the error.