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Pi (Part 2)

π is a transcendental and irrational number. A number which cannot be expressed as the ratio of two integers is called an irrational number and a number which cannot be expressed in the form of a polynomial with rational coefficients is called a transcendental number. π (pi) has both of these properties. In the previous article I expressed it as a series of roots. In this article also I will give a series of roots which when done many times will reach the value of π (pi). The larger the expression becomes the more close to the value of π (pi) it reaches.

In the last Part 1 we discussed how to find approximate value of π. But the expression is very hard. In this article we will use the sub multiple angles of 60° to find the approximate value of π. In the previous article we used sub multiple angles of 90°. The expression which I am going to present is very easy and a little bit similar to Euler's. I think this expression can be used in mathematics to represent π. I found this expression same time I found the expression in the previous post. I also use the trigonometry formula I derived in the previous article.

We will derive the expression in blocks. First we will start with sin(360°/6) i.e. the sine of 60° then we will half the angle. We will compute each time the value of the expression so that we can understand how close it is to π (pi).

Block 1
So, let us derive the expression,
The expression used to half the angle is
sin θ = √{(1 −√(1 − sin2 2θ))/2}
sin 60° = sin 2θ= sin (360°/6) = √3/2
sin² 2θ = 3/4
then, sin 30° = sin (360°/12)
= √{(1 − √(1 − 3/4))/2}
= √{(1 − √(1/4))/2}
= √{(1 − (1/2))/2}
= √(1/2)/2}
= 1/2

As we have found the value of sine of half of 60° now we will test it. We will do this by substituting it in the expression (n/2) sin (360°/n). Here n = 12.
The value of (12/2) sin (360°/12)
= 6(1/2) = 3


Block 2
Now we will find the value of sine of half of the previous angle i.e. sine of half of 360°/12 which is sin (360°/24). we will use the value of the previous angle by substituting it in the formula we derived in the previous post.
sin θ = √{(1 −√(1 − sin2 2θ))/2}
sin 2θ = 1/2
sin² 2θ = 1/4
sin (360°/24)
= √{(1 −√(1 − 1/4))/2}
= √{(1 −√(3/4))/2}
= √{(2 −√3)/2)/2}
= (1/2)√(2 −√3)

We will find the value of the expression (n/2) sin (360°/n). Here n is 24.
The value of (24/2) sin (360°/24)
= 12(1/2)√(2−√3)
= 3.105828 upto six decimal digits


We see that the value has reached close to π (pi).

Block 3
Now we will find the value of sine of half of the previous angle i.e. sine of half of 360°/24 which is sin (360°/48).
sin θ = √{(1 −√(1 − sin2 2θ))/2}
sin 2θ = (1/2)√(2 −√3)
sin² 2θ = (2 −√3)/4
Find the value of half
sin (360°/48)
= √{(1 −√(1 − (2−√3)/4))/2}
= √{(1 −√(2 + √3)/4))/2}
= (1/2)√(2 −√(2 + √3))


We will find the value of the expression (n/2) sin (360°/n). Here n is 48.
The value of (48/2) sin (360°/48)
= 24(1/2)√(2 −√(2 + √3))
= 3.132628 upto six decimal digits


We see that the value of the expression (n/2) sin (360°/n) reaches π (pi) as we substitute larger number.


Similarly we can find sin (360°/96) which is equal to (1/2)√(2 −√(2 + √(2 + √3))).

When we substitute in (n/2) sin (360°/n) for n = 96 we get (96/2) sin (360°/96) = 3.139350.

This value is more close to the value of π than the expression found in previous blocks.


When we look at some more expressions we can have an idea of how the expressions look for larger value of n.
sin (360°/192) = (1/2)√(2 − √(2 + √(2 + √(2 + √3))))
sin (360°/384) = (1/2)√(2 − √(2 + √(2 + √(2 + √(2 + √3)))))
sin (360°/768) = (1/2)√(2 − √(2 + √(2 + √(2 + √(2 + √(2 + √3))))))

sin (360°/192) = sin (360°/3×26) has the number of two's equal to 4 and the first 2 followed by negative sign and the rest are positive.

sin (360°/(3×2n)) = sin (360°/(3×2n)) has the number of two's equal to n−2 and the first 2 followed by negative sign and the rest are positive.



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