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shanu_bhaiya
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1. The problem statement
Derive the sine power series without using Taylor's formula.
Derive the sine power series without using Taylor's formula.
Well, this is what the question reads. Without using the calculus, is it possible to derive sine power series using geometry, etc.HallsofIvy said:Why?
This is the only standalone-statement of the problem. Nothing else is given. No it's not a dfn eqn problem. The question's standard is upto 1st year undergraduate (including Real Analysis) only.mistermath said:I think you need more information. Like what is given because I use the power series as a definition of what sine is. Is this a differential equation problem?
Well problem is that I've no idea from where to start, my guesses are geometry, complex number or the geometric definition of sine. Actually it's a part of my report, my main subject was Calculus of Kerela(India), the ancient Indian people knew the series without Taylor's formula, so that's what I'm intended to write in my report but it's nowhere on the internet, any suggestions are welcome.HallsofIvy said:My question was really, "why should I do it". As I am sure you are aware, you are expected to try it yourself first and show us what you have done.
shanu_bhaiya said:Well problem is that I've no idea from where to start, my guesses are geometry, complex number or the geometric definition of sine. Actually it's a part of my report, my main subject was Calculus of Kerela(India), the ancient Indian people knew the series without Taylor's formula, so that's what I'm intended to write in my report but it's nowhere on the internet, any suggestions are welcome.
The sine series can be derived by using the Euler formula, which states that e^(ix) = cos(x) + isin(x). By substituting x with -ix, we get e^(-ix) = cos(-x) + isin(-x). Using the fact that cos(-x) = cos(x) and sin(-x) = -sin(x), we can rearrange the formula to get sin(x) = (e^(ix) - e^(-ix)) / 2i. This formula can then be expanded using the binomial theorem to get the sine series.
The Taylor series uses derivatives to approximate a function. However, this method requires the function to be infinitely differentiable, which is not always the case. By deriving the sine series without using Taylor's formula, we can expand the range of functions that can be approximated.
Yes, the sine series derived using the Euler formula is just as accurate as the one derived using Taylor's formula. Both methods rely on the same fundamental principles and have been proven to converge to the same value for any input value of x.
One limitation is that the Euler formula only works for complex numbers, so it cannot be used to derive the sine series for real numbers. Additionally, the binomial theorem can only be used to expand the formula for certain values of x, so the series may not converge for all values of x.
Yes, the sine series is commonly used in signal processing, physics, and engineering to approximate periodic functions. By deriving the series without Taylor's formula, we can expand its usefulness to a wider range of functions and improve accuracy in certain applications.