Taylor approximation question. I am trying to find the approximation for tan(x) by dividing the approximation of sin(x) by the approximation of cos(x). It seems like it works, but I can't figure out how to get it into the form of x + (x^3/3!) + (2x^5/5!)...

Question

anonymous · Answer

That won't work, sorry.  You have to do it by taking successive derivatives of $\tan x$.

anonymous · Answer

Yeah I know successive derivatives same for sin(x) and cos(x). I know how to find it like this I just thought maybe it would work if you divided the approximations. Actually are you sure it doesn't work at all? I just wanted to know how to simplify ((x-x^3/3! + ...)/(1-x^2/2!+...)) up to n=5. I just don't know how to simplify the answer. I know if I plug in a value for x it gives the tanx value with 4 decimal places of accuracy, but actually I only tried it for the number 1 so maybe it breaks down after that I don't know. Could you explain why this method wouldn't work?  Thank you for your time yakeyglee.

anonymous · Answer

Sorry I meant to say same for any approximation right? Not just sin(x) and cos(x). Excuse my ignorance I just started learning about this today.