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Krak
11-05-2004, 03:21 PM
Has anyone ever heard of these trig identities before?

tan(89)=180/pi
tan(1)=pi/180
pi=180*tan(1)
pi=180/tan(89)
tan(x)*tan(y)=1 if x and y are complements.

Thantos
11-05-2004, 04:58 PM
yes I have.

Govtcheez
11-05-2004, 05:02 PM
These aren't really "identities" so much as they're "algebra"

Darkness
11-05-2004, 08:16 PM
4 arctan (1) = PI

edit

well, not exactly equal, but a damn good approximation to quite a few digits, so it should really be:

4 arctan(1) ~= PI

Krak
11-05-2004, 11:35 PM
4 arctan (1) = PI

edit

well, not exactly equal, but a damn good approximation to quite a few digits, so it should really be:

4 arctan(1) ~= PI

Well of course.;)

arctain(1)= 45° = pi/4

So 4*(pi/4) is obviously pi.

Sang-drax
11-06-2004, 06:08 AM
well, not exactly equal
Yes, exactly equal. :)

Darkness
11-06-2004, 10:04 AM
it's actually not. Nobody has ever been able to carry PI out to an infinite number of decimal places.

The trig functions are taylor series that contain an infinite number of terms, but carrying them out doesn't actually yield PI to an infinite number of decimal places, or so I have heard.

Sang-drax
11-06-2004, 10:16 AM
it's actually not.
Trust me on this one :)
A point (x,y) on the unit circle with its center at (0,0) can be written as
x = cos(t)
y = sin(t) where t is the distance on the circle from (1,0) to (x,y) which is called the angle in radians. We know by definition that the circle is 2*pi around.

tanx = sinx/cosx
arctanx is the inverse of tanx, where -pi/2 < x < pi/2

By symmery of the unit circle, it is obvious that sin(pi/4) = cos(pi/4), therefore tan(pi/4) = 1
tan(pi/4)=1 =>
arctan(1) = pi/4 (because pi/4 is within the interval) =>
4*arctan(1) = pi

No more, no less -- exactly.

Darkness
11-06-2004, 03:16 PM
I am actually going to disagree with you on this one. As it stands, I don't think anybody knows of any method for calculating PI to an infinite number of decimal places.

With that said, the Taylor series for arctan can be expanded to as many places as wanted. But, there is a point where the Taylor series diverges from the known values of PI. I know I've heard this before but I can't back it up ATM.

This really doesn't matter anyway, but I think that in the strictest sense, I am right. And who cares if the 10,000 decimal place of PI isn't exactly equal to that generated by the taylor series? Because that's pretty much what it amounts to.

The only piece of evidence I currently have is purely intellectual, and it is that the actual Taylor series, meaning the calculations performed when you press the trig buttons on your calculator, don't actually have any intrinsic meaning. They were developed by building tables of known values for the trig functions (and also ln and log) until mathematicians could come up with equations that "just happened to work." But that's as far as the equations go...they "just happen to work."

edit:
I am going to try to find something on mathworld that proves or disproves my stance on it. You could still be right because I'm not 100% sure.

edit1:
after reading this site, I think I actually am not correct afterall and my understanding was wrong:

An exact formula for in terms of the inverse tangents of unit fractions is Machin's formula

http://mathworld.wolfram.com/PiFormulas.html

edit2:
this is the second time i've been proven wrong on this site :( Maybe I should go for some sort of record "number of times a poster has been incorrect"

Zach L.
11-06-2004, 03:54 PM
But you admit it quite gracefully. :)

Thantos
11-06-2004, 03:56 PM
this is the second time i've been proven wrong on this site Maybe I should go for some sort of record "number of times a poster has been incorrect" You have a long ways to go before you'll even come close to my record.

Darkness
11-06-2004, 11:23 PM
You have a long ways to go before you'll even come close to my record.

competition!!! And you've got all the pretty green boxes!!!

edit:
it's the same color as money, so I think that you should be able to 'buy' special priviledges, at the expense of losing some of your green boxes (i.e, the ability to be a moderator for a day if you get more than 15 green, the ability to close one thread if you get more than 10 something like that).

Thantos
11-06-2004, 11:37 PM
Well 11 greens is the max. And while I do try to provide accurate answers I have been proved wrong a lot. Its all about learning though.

Darkness
11-06-2004, 11:42 PM
True, but I was a little frustrated with myself because this is all stuff I've done before, and I do consider myself math literate in general. Think of it like a marine that can't unjam his M-16 even after 13 weeks at Parris :)