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Pi (Part 1)

This article has its base in trigonometry and uses value of sin 90° and its sub-multiple angles. The values of sub multiple angles are found with the help of a formula. As π in mathematics is very interesting this post can be of interest. In this post I will give you some expressions which can be used to give approximate value of π. I found this expression when I found the relation for π. It is a little bit complex but can be used to approximate the value of π. The expressions use the relation
π = lim (n/2)sin(360°/n)
    n→∞
As we cannot put value as infinity so we will approach it. The best way to approach it is by halving the angle successively. To find the value of half angle we need to find a formula first.

Value of an angle when twice of the angle is known
Let us find the formula.
We know that sin 2θ = 2 sin θ cos θ.
We will use this formula to find the value of sine of half angle. Here half angle is θ and full angle is 2θ.

sin 2θ = 2 sin θ √(1 − sin2 θ)
Squaring both sides
sin2 2θ = 4 sin2 θ (1 − sin2 θ)
sin2 2θ = 4 sin2 θ − 4 sin4 θ
4 sin4 θ − 4 sin2 θ + sin2 2θ = 0
Using quadratic formula to find sin2 θ.
sin2 θ = [4 ± √(16 − 16 sin2 2θ)]/8
sin2 θ = [1 ± √(1 − sin2 2θ)]/2
As 0 ≤ sin2 θ ≤ 1
sin2 θ = [1 − √(1 − sin2 2θ)]/2
sin θ = ±√{[1 − √(1 − sin2 2θ)]/2}
As 0 ≤ sin θ ≤ 1for 0 ≤ θ ≤ π
So, sin θ = √{[1 − √(1 − sin2 2θ)]/2}

So we have found the formula to find the value of half angle,and it is
sin θ = √{[1 − √(1 − sin2 2θ)]/2} when the full angle is 2θ


Now we can approach θ→0 by taking angle as
90°→45°→45°/2→45°/4→45°/8→45°/16→ so on
60°→30°→15°→15°/2→15°/4→15°/8→ so on

The formula sin θ = √{[1 − √(1 − sin² 2θ)]/2} can be used to find value of sine of θ if we know 2θ.


We will first find the expression for θ whose multiple is 90 i.e we will approach the angle δ (δ→0) from 90°.


Block 1
The value of sin 90° is 1.

Substituting it as value of sin 2θ in the formula
sin θ = √{[1 − √(1 − sin2 2θ)]/2}
to obtain the value of sine of half angle. We get
sin (90°/2) = sin (360°/8) = √{[1 − √(1 − 1)]/2}
= √{1/2} = 1/√2

Let us find the value of (n/2) sin (360°/n) when n = 8
(8/2) sin (360°/8)
= 4 sin (360°/8)
Substituting the value of sin (90°/2)
= 4 (1/√2)
= 2√2
= 2.828427 up to six decimal digits


This value is a little bit close to π.

Block 2
We will again find the value of sine of half of the previous angle which we found in block 1.
we will use the same formula i.e.
sin θ = √{[1 − √(1 − sin2 2θ)]/2}
Here sin 2θ = 1/√2
and sin² 2θ = 1/2
sin (90°/4) = √{[1 − √(1 − 1/2)]/2}
= √{[1 − √(1/2)]/2}
= √{[(√2−1)/√2]/2}
= √{(√2−1)/2√2}
= √(√2−1)/2√2)}
= √(√2−1)/(21+1/2)}
= √(√2−1)/(23/2)}
= 2−3/4√(√2−1)


Let us find the value of (n/2) sin (360°/n) when n = 16
(16/2) sin (360°/16)
= 8 sin (360°/16)
= 8 × 2−3/4√(√2−1)
= 3.061467 up to six decimal digits


This value has reached more close to π (pi) than the previous value found in block 1.

Block 3
We will again find the value of sine of half of the previous angle which we found in block 2.
we will use the same formula i.e.
sin θ = √{[1 − √(1 − sin2 2θ)]/2}
Here sin 2θ = 2−3/4√(√2−1)
and sin² 2θ = 2−3/2(√2−1)
sin (90°/8) = sin (360°/32) = √{[1 − √(1 − (2−3/4√(√2−1))2)]/2}
= √{[1 − √(1 − (2−3/2(√2−1))]/2}
= √{[1 − √(23/2 − √2 + 1)2−3/2]/2}
= √{[1 − 2−3/4√(23/2 − √2 + 1)]/2}
= √2−3/4{23/4 − √(23/2 − √2 + 1)}/2
= 2−7/8√{23/4 − √(23/2 − √2 + 1)}



Let us find the value of (32/2) sin (360°/32)
= 16 sin (360°/32)
= 16 × 2−7/8√{23/4 − √(23/2 − √2 + 1)}
= 3.121445 up to six decimal digits

This value has reached more close to π (pi) than the previous value found in block 2.


Block 4
We will again find the value of sine of half of the previous angle which we found in block 3.
we will use the same formula i.e.
sin θ = √{[1 − √(1 − sin2 2θ)]/2}
Here sin 2θ = 2−7/8√{23/4 − √(23/2 − √2 + 1)}
and sin² 2θ = 2−7/4{23/4 − √(23/2 − √2 + 1)}
sin (90°/16) = sin (360°/64)
= √{[1 − √(1 − (2−7/8√{23/4 − √(23/2 − √2 + 1)})2)]/2}
=√{[1 − √(1 − (2−7/4(23/4 − √(23/2 − √2 + 1))))]/2}
=√{[1 − √(27/4 − (23/4 − √(23/2 − √2 + 1)))2−7/4]/2}
=√{[1 − 2−7/8√(27/4 − (23/4 − √(23/2 − √2 + 1)))]/2}
=√{[27/8 − √(27/4 − (23/4 − √(23/2 − √2 + 1)))]2−7/8/2}
=√{[27/8 − √(27/4 − (23/4 − √(23/2 − √2 + 1)))]2−15/8}
=2−15/16√{[27/8 − √(27/4 − (23/4 − √(23/2 − √2 + 1)))]}


Let us find the value of  (64/2) sin (360°/64)
= 32 sin (360°/64)
= 32 × 2−15/16√{[27/8 − √(27/4 − (23/4 − √(23/2 − √2 + 1)))]}
= 3.136548 up to six decimal digits


This value has reached more close to π (pi) than the previous value found in block 3. We can continue for more blocks and find more approximate value of π (pi).

In the next post we will find the expression for π when we approach 0 from sub-multiples of 60°.
2−15/16√{[27/8 − √(27/4 − (23/4 − √(23/2 − √2 + 1)))]}
= 2−15/16√{[27/8 − √(27/4 − 23/4 + √(23/2 − √2 + 1)))]}

We can generalize the method and extend it with some logic.


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