pi

[Cervantes]

Andy wrote:

to find pi, just do an taylor expansion of tan(x). there are faster ways of approximating pi than that gandalf =P

I did that a while back, and I got 15 decimal places of accuracy just from Microsoft's calc.exe. I think 15 was the number of decimal places it displayed.

My question is, is this approach really valid? tan x = sinx / cosx. sinx and cosx come from circles, right? So aren't we using a bit of circular (oh, oh, pun!) reasoning here?

I know sinx and cosx can be approximated by a Taylor series, but my question still remains.

[Windsurfer]

Andy wrote:

to find pi, just do an taylor expansion of tan(x). there are faster ways of approximating pi than that gandalf =P

Ha. I've experienced that. On my Ti-83 Plus, I had a basic program that would use that one Gandolf said, and it would take forever to find the first 5 digits. I think it was like 500 iterations? Can't remember.
But then I implemented the taylor expansion method (or something similar) and my calc ran out of digits on the 12th or 14th iteration, so less than 2 seconds

[Andy]

but cervantes, you're not using sin(x) and cos(x) to calculate values.

tan(x) = sin(x)/cos(x)

and sin(x) can be expanded into sin(x) = x - x3/3! + x5/5! - x7/7! + ...
and cos(x) into cos(x) = cos(x) = 1 - x2/2! + x4/4! - x6/6! + ...

then you just need to do a long division

so you're not cheating at all =P

[Cervantes]

How did we get those series for sin(x) and cos(x)?

I think I detect sarcasm in your last sentence there, right? So this method is using circular reasoning, yes? Or does it matter? Since we don't necessarily need a value for pi to find values for sin(x). Or do we?

[Andy]

you use taylor's theorum to get those series.

you dont actually need to know the value of pi to determine what sin(pi) and cos(pi) is.