Why is pi = 16 arctan(1/5) - 4 arctan( 1/139)

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In summary, the conversation discusses the equation \pi = 16\arctan(\frac{1}{5}) - 4\arctan(\frac{1}{239}) and how it is related to series expansion and Machin's formula. The exactness of the relationship is uncertain and can be verified through calculations.
  • #1
joex444
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Not actually a HW question, but, is there a quick explanation that could show why:

[tex]\pi = 16\arctan(\frac{1}{5}) - 4\arctan(\frac{1}{239})[/tex]

I read that somewhere, and it turned out to be true, so I was just wondering how that came about...
 
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  • #2
We have the series expansion:
[tex] Tan^{-1}(x) = \frac {\pi} 2 - \frac 1 x + \frac 1 {3x^3} - \frac 1 {5x^5} ... [/tex]
By working with this you should be able to verify your claim. It is not clear to me whether your relationship is exact or just good to more decimal places then are being displayed.
 
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  • #3
http://mcraefamily.com/MathHelp/GeometryTrigEquiv.htm (see Machin's formula)
 
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FAQ: Why is pi = 16 arctan(1/5) - 4 arctan( 1/139)

What is the significance of "pi = 16 arctan(1/5) - 4 arctan( 1/139)"?

This equation is significant because it is one of the many ways to approximate the value of pi, which is a fundamental constant in mathematics and physics. It also demonstrates the relationship between trigonometric functions and pi.

How was this equation derived?

This equation was derived using the Taylor series expansion of the arctan function and manipulating it to get a polynomial equation for pi. It involves a lot of mathematical techniques and may not be easily understood by those without a strong background in mathematics.

Why is pi approximated using arctan functions?

The arctan function is commonly used to approximate pi because it has a simple and elegant relationship with pi. It also allows for a more accurate approximation compared to other methods such as using the circumference or diameter of a circle.

How accurate is this approximation of pi?

This approximation is accurate to about 9 decimal places, making it a relatively precise estimation of pi. However, there are other methods that can provide even more accurate approximations of pi.

Can this equation be used to calculate pi in practical applications?

While this equation is a valid way to approximate pi, it is not commonly used in practical applications. Other methods, such as using computers to calculate pi to millions or even billions of decimal places, are more efficient for practical purposes.

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