Two ways to calculate pi?

Question

I was just bored, and I came up with these two ways to calculate pi.
There may be an error or two, I can't find my calculator to check it, if so, let me know and I will repost this with the correction.

lim(x->0) (180sin(x)) / (x(sin((180-x)/2)))

and

lim(x->inf) (x(sin(360-x))) / (2sin((x-180)/2))

(Ignore the spaces before & after the division signs, they are to make it so that Y!A doesn't condense the equations)

Using degrees here...


So ya, I guess I am asking if these two methods are correct, and if they may have any significance.

Noticed at least 1 error so far.

The second equation should be

lim(x->inf) (x(sin(360/x))) / (2sin(90-108/x)


I think...

...Typo in the last edit, equation should actually be
lim(x->inf) (x(sin(360/x))) / (2sin(90-180/x)


Sorry!

22/7 is just an approximation of pi to a few decimal places. It has little to do with actually calculating the digits of pi.

Answer 1

The problem with your idea is that any expression that calculates pi using trigonometric terms like sin, cos and tan, all presume that you have a clear knowledge of the geometry of a circle, e.g. sin^2 + cos ^2 = tan^2. Therefore, calculating pi from the properties of those trigonometric functions will, of course, give you the value for pi because that is how they are inherently defined.

In fact, using a calculator, it would be far simpler to use the sin^(-1) button...

A more elegant and satisfying method would be to use a series that does NOT use the trigonometric function, or any function that inherently involves pi. Try the Leibnitz series for pi:

4/1 - 4/3 + 4/5 - 4/7 + 4/9 .....
= lim (a=0->inf) 4/(1+2a)x(-1)^a

There is no requirement for any trigonometric function here, obviously.

Answer 2

This is what I have come up with:

lim(x->0) (180sin[x]) / x{sin[(180-x)/2)]}
Put in the value of x as it reaches zero.
=(180sin[0]) / (0){sin[(180 - 0)/2)]}
=(180(0) / (0){sin[(180)/2)]}
=(180(0) / (0){sin[90]}
=(180(0) / (0)(1)
= 0 / 0
= ????
that is either 1 or 0 (depending on your point of view).

and

lim(x->inf) x(sin[360-x]) / {2sin[(x-180)/2]}
Put in the value of x as it reaches infinity.
= (∞)(sin[360 - ∞]) / {2sin[(∞ - 180)/2]}
= (∞)(sin[- ∞]) / {2sin[(∞)/2]}
= (∞)(- 1) / 2sin[½ ∞]
= (- ∞) / 2sin[0]
= - ∞ / 0
= ????
that is either - ∞ or 0 (depending on your point of view).

Answer 3

Or you can use 22/7.

Answer 4

it is no use to calculate pi with these equations because a computer uses numerical approximations to calculate the sine function. to calculate pi to significant digits there are much faster converging series, as pioneered by Ramanujan.

Answer 5

purely use the two equations for pi-: section = pi x radius x radius consequently-: (section / (radius x radius)) = pi Circumference = 2 x pi x radius consequently-: pi = (Circumference / (2 x radius)) superb of success inclusive of your calculations

Answer 6

try
pi=In(640320^3 + 744)/163^(1/2)

it will be precise up to 30 digits

Answer 7

They do not work.