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Using Sine and Tangent to Find Pi

Date: 08/01/98 at 22:20:16
From: Brian Marshall
Subject: Sine cosine and tangent+pi

I'm trying to determine pi. I'm not sure how right this is, but I 
found that as x approaches infinity:

   x sin (180/x) = pi


   x tan (180/x) = pi

I know that in radians 180 degrees is pi, but I'm not sure if that has 
anything to do with it.

My calculator and all programs on my PC can only do it to a limited 
number of decimal places (the same as it already has pi stored to 
anyway). I want to know how to determine the sine or tangent of an 
angle without using a calculator.

Date: 08/05/98 at 13:03:41
From: Doctor Bruce
Subject: Re: Sine cosine and tangent+pi

Hello Brian,

There is indeed a "recipe" for computing the sine of an angle, based 
on the infinite series for sine:

   sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ....

The pattern here should be pretty clear. The powers of x are all the 
odd numbers, and the same odd number appears in the denominator as a 
factorial, with the signs alternating.

The values of x you substitute in this series must be in radians. This 
means that if you substitute x = pi/2, the series will converge (get as 
close as you like) to 1 (because sin(pi/2) = sin(90 degrees) = 1). If 
we wanted to use degrees instead of radians for our angles, we would 
have a similar-looking series, but the terms would be messier, so we 
prefer to use the series above, and always use radian measure for 

Anyway, understanding that we must use radians in order to use the 
series I gave, we compute sin(pi/x):

   sin(pi/x) = (pi/x) - (pi/x)^3/3! + (pi/x)^5/5! - ...

and multiply this through by x:

   x*sin(pi/x) = pi - pi^3/(6*x^2) + pi^5/(120*x^4) - ...

and we see that, as x tends toward infinity, all the terms with x in 
the denominator go toward zero. This leaves just the first term, pi, as 
the limiting value.

You might play with using the series I gave to compute some sample 
values of sine. The convergence is pretty fast, at least for small 
values of x, because the numbers 3!, 5!, 7! grow very rapidly.

Your other formula:

   x tan (180/x) = pi

is true (careful about the radians again!). You prove it the same way I 
showed you for sine, but using the series for tangent, which goes:

   tan(x) = x + x^3/3 + 2*x^5/15 + 7*x^7/315 + ....

Substituting pi/x in for x, and then multiplying through by x will give 
the result pi, just as I showed above.

Unfortunately, the coefficients in the series for tangent are not so 
simple. You obtain this series by dividing the series for sine (above) 
by the series for cosine, which is:

   cos(x) = 1 - x^2/2! + x^4/4! - x^6/6! + ....

Hope this helps you.

- Doctor Bruce, The Math Forum
Check out our web site!   
Associated Topics:
High School Sequences, Series
High School Trigonometry
Middle School Pi

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