Hello bprp, I saw your videos on Simpson vs Taylor and Newton’s method vs euler’s method and was wondering one question recently,which method is more efficient for computing √(2),newtons method or using the Taylor series of 2sin(x) or 2cos(x) at π/4?

An almost impossible comparison. Newton's Method requires a "first guess". The number of iterations depends on how good that guess is to the actual value. Even with a guess as bad as sqrt(2) = 100 Newton's method required only 10 iterations to get 1.41421. It takes about 3 more iterations for each order of magnitude larger the guess is. It gets about half way there with each iteration which means it about doubles the accuracy with each iteration. Pretty good considering how even a terrible first guess gets you there pretty fast. Starting with 1, Newton takes only 7 iterations to get 60 accurate digits. Now that's impressive as it is way, way beyond our ability to measure. Maclaurin certainly can't match that.

guess # iterations to get 1.41421

1 3

10 6

100 10

1 000 13

The Maclaurin Series requires 4 terms to get 1.41421

Your choice.

You're welcome