Finding π with sin , cos and tan

[tulip dew]

so basically there is a formula to find π. its sin(a°) * 360/a ≈ 2π
and the smaller “a” is the closer you get to π.

This also works with tangent but not with cosine can someone explain why it doesn’t work with cosine?

[solid dune]

sin(x rad) is really close to x when x is small

so sin(x rad)/x is really close to 1

then consider y=x*2pi/360

y is also pretty small when x is small so, sin(y rad)/y is about 1

but y rad = x degrees

so sin(x degrees)/y ~= 1

and sin(x degrees)/(x*2pi/360) ~= 1

rearranging, sin(x degrees)*360/x ~= 2pi

[tulip dew]

what ia rad

[solid dune]

radians

[tulip dew]

didnt have them yet

[solid dune]

oh

[tulip dew]

ok I know this

[solid dune]

if you wanted to describe a full rotation in terms of degrees, you would say you'd need 360 degrees

radians are similar

[tulip dew]

ohh

[solid dune]

if you wanted to describe a full rotation in terms of radians, you would say you'd need 2pi radians

[tulip dew]

I am also at this point

[solid dune]

if you have a function that's really close to x when x is small, then f(x degrees)*360/x is roughly 2pi

wait

no

[tulip dew]

yes it is

[solid dune]

im right

if you have a function that's really close to x when x is small, then f(x degrees)*360/x is roughly 2pi

that's true

(notice if your function is f(x)=x then you always get f(x)*360/x = 2pi. so the closest function possible to x gives the best approximation!)

[tulip dew]

f (10) * 360/10 = 2π

[solid dune]

wdym

[tulip dew]

I put 10 in the place of x

you need the sine or tangent of x in degrees

[solid dune]

yeah

[tulip dew]

you are just telking me a formula I already have

[solid dune]

for sine and tangent, passing in small values of x (in radians) is really close to x

like sin(0.1 rad) is really close to 0.1

[tulip dew]

ikow

[solid dune]

but that's not true for cosine

[tulip dew]

oh ok

[solid dune]

cosine of small values is really close to 1

[tulip dew]

so I would need to hava big cosines

[solid dune]

wdym?

[tulip dew]

cos (x degrees ) * 360 * 10 ≈ 2π

so If I use 89.9 degrees

[solid dune]

well sure, if you pick the right value of x then it'll work

[tulip dew]

lol

[solid dune]

oh

but cos(x)=sin(90-x)

[tulip dew]

Oh wow

[solid dune]

so what you're noticing is true for the same reasons as sin(x)*360/x is about 2pi

[tulip dew]

btw this formula also works with tan(x degrees)

[solid dune]

a bigger value of x won't get you a closer approximation

[solid dune]

all of the time

[tulip dew]

but If I use cos(89.99) [ adding another decimal] I need to multiply by 100 not 10

soo

[solid dune]

but then you're saying cos(x)*360/(90-x) is close to 2pi when x is close to 90

which isn't the same as saying cos(x)*360/x is close to 2pi

(indeed, if x>60 then cos(x)*360/x<2pi)

[tulip dew]

cos(89.9°) * 360 * 10 =2π
cos(89.99°) *360 *100 =2π

[tulip dew]

wha

[solid dune]

well you said cos(89.9)*360/0.1 is close to 2pi

and not that cos(89.9)*360/89.8 is close to 2pi

[tulip dew]

I never said to devide

[solid dune]

yeah you did

anyways cos(89.9)*360*10 = cos(89.9)*360/0.1

[tulip dew]

where

[tulip dew]

uhm

[solid dune]

there

[tulip dew]

yes

but How can I get more specific with the formula

[solid dune]

what do you mean?

[tulip dew]

like if I wanna have an answer thats more near Pi I can just add a bunch of 0’s on bith sides

oh wait I got it

[solid dune]

yeah

the closer x is to 0, the closer sin(x degrees)*360/x gets to 2pi

[tulip dew]

yeah and the higher cos is to 90° the closer cos(x degrees) *360/(amount of decimals the x has in decimals 0).1 gets to 2π

So yeah

[solid dune]

yeh

yeah*

[acoustic sphinx]

this uses pi to calculate pi