User:Рајко Велимировић

The NUMBER PI
 * There are many forms of PI and you can see them

However the number of Pi is not just a circle but a series of geometric images such as an equilateral triangle, square, pentagon, etc. unique pattern that I found 2004. Pi for all numbers is:
 * $$ \pi_n=\frac{n}{2}. sin(\frac{360}{n})$$

sinus in degrees
 * n= {3,4,5,6........., infinity}
 * n=3 it is the equilateral triangle
 * $$\pi_3=\frac{3}{2}\sin(\frac{360}{3})$$
 * $$\pi_3=1.

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 * $$ R^2\pi_3$$

R -radius of the circle described about an equilateral triangle
 * The scope of the triangle will be
 * $$ C=\frac{2R\pi_3}{\cos\frac{\alpha}{2}}:\alpha=\frac{360}{3} $$

See picture[]


 * n=4 it is a square
 * $$\pi_4=2$$

Area of squares will be
 * $$ R^2\pi_4$$

R -radius of the circle described around the square
 * Perimeter of the square will be
 * $$ C=\frac{2R\pi_4}{\cos\frac{\alpha}{2}}:\alpha=\frac{360}{4} $$

See picture[]

so on pentagon, hexagon .......
 * $$\pi_5= 2.

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 * $$\pi_6= 2.

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 * $$\pi_7=...........$$
 * $$\pi_8=...........$$
 * $$\pi_12=3$$
 * $$\pi=3.
 * $$\pi=3.
 * $$\pi=3.

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 * evidence
 * $$\pi=\frac{n}{2}. sin(\frac{360}{n})$$
 * $$lim_{n \to \infty}\frac{n}{2}. sin(\frac{360}{n})$$
 * $$\frac{1}{2} lim_{n \to \infty}n. sin(\frac{360}{n})$$
 * $$\frac{1}{2} lim_{n \to \infty}\frac{ sin(\frac{360}{n})}{\frac{1}{n}}$$
 * $$\frac{1}{2} lim_{n \to \infty}\frac{ sin(\frac{360}{n})}{\frac{1}{n}}.\frac{360}{360}$$
 * $$\frac{360}{2} lim_{n \to \infty}\frac{ sin(\frac{360}{n})}{\frac{360}{n}}$$
 * $$\frac{360}{n}=t$$
 * $$t \rightarrow \ 0$$
 * $$180\lim_{t \to \mathbf{0}} \frac{sin t}{t}=180^0$$
 * $$180^0=\pi (rad)$$

The following is another new form of PI, which I found in 2004


 * $$\pi=180.m.sin(\frac{1}{m})$$
 * $$\lim_{m \to \infty} 180.m.sin(\frac{1}{m})$$
 * $$180\lim_{m \to \infty}m.sin(\frac{1}{m})$$
 * $$180\lim_{m \to \infty}\frac{sin(\frac{1}{m})}{\frac{1}{m}}$$
 * $$\frac{1}{m}=t$$
 * $$m \to \infty$$
 * $$t \rightarrow \ 0$$
 * $$180\lim_{t \to \mathbf{0}} \frac{sin t}{t}=180^0$$
 * $$180^0=\pi (rad)$$
 * $$\frac{\pi}{2}$$
 * $$\frac{\pi}{2}=\frac{n}{2}sin(\frac{180}{n})$$
 * evidence
 * $$lim_{n \to \infty}\frac{n}{2}. sin(\frac{180}{n})$$
 * $$\frac{1}{2} lim_{n \to \infty}n. sin(\frac{180}{n})$$
 * $$\frac{1}{2} lim_{n \to \infty}\frac{ sin(\frac{180}{n})}{\frac{1}{n}}$$
 * $$\frac{1}{2} lim_{n \to \infty}\frac{ sin(\frac{180}{n})}{\frac{1}{n}}.\frac{180}{180}$$
 * $$90\lim_{m \to \infty}\frac{sin(\frac{1}{m})}{\frac{1}{m}}$$
 * $$\frac{180}{n}=t$$
 * $$n \to \infty$$
 * $$t \rightarrow \ 0$$
 * $$90\lim_{t \to \mathbf{0}} \frac{sin t}{t}=90^0$$
 * $$90^0=\frac{\pi}{2} (rad)$$
 * $$\frac{\pi}{3}$$
 * $$\frac{\pi}{3}=\frac{n}{2}sin(\frac{120}{n})$$
 * evidence
 * $$lim_{n \to \infty}\frac{n}{2}. sin(\frac{120}{n})$$
 * $$\frac{1}{2} lim_{n \to \infty}n. sin(\frac{120}{n})$$
 * $$\frac{1}{2} lim_{n \to \infty}\frac{ sin(\frac{120}{n})}{\frac{1}{n}}$$
 * $$\frac{1}{2} lim_{n \to \infty}\frac{ sin(\frac{120}{n})}{\frac{1}{n}}.\frac{120}{120}$$
 * $$60\lim_{n \to \infty}\frac{sin(\frac{120}{n})}{\frac{120}{n}}$$
 * $$\frac{120}{n}=t$$
 * $$n \to \infty$$
 * $$t \rightarrow \ 0$$
 * $$60\lim_{t \to \mathbf{0}} \frac{sin t}{t}=60^0$$
 * $$60^0=\frac{\pi}{3} (rad)$$
 * $$\frac{\pi}{4}$$
 * $$ \frac{\pi}{4}=\frac{n}{2}.sin(\frac{90}{n})$$
 * evidence
 * $$lim_{n \to \infty}\frac{n}{2}. sin(\frac{90}{n})$$
 * $$ \frac{\pi}{4}=\frac{n}{2}.sin(\frac{90}{n})$$
 * evidence
 * $$lim_{n \to \infty}\frac{n}{2}. sin(\frac{90}{n})$$


 * $$\frac{1}{2} lim_{n \to \infty}n. sin(\frac{90}{n})$$


 * $$\frac{1}{2} lim_{n \to \infty}\frac{ sin(\frac{90}{n})}{\frac{1}{n}}$$


 * $$\frac{1}{2} lim_{n \to \infty}\frac{ sin(\frac{90}{n})}{\frac{1}{n}}.\frac{90}{90}$$


 * $$\frac{90}{2} lim_{n \to \infty}\frac{ sin(\frac{90}{n})}{\frac{90}{n}}$$


 * $$\frac{90}{n}=t$$


 * $$n \to \infty$$


 * $$t \rightarrow \ 0$$


 * $$45\lim_{t \to \mathbf{0}} \frac{sin t}{t}=45^0$$


 * $$45^0=\frac{\pi}{4} (rad)$$
 * $$\frac{\pi}{5}$$
 * $$\frac{\pi}{5}=\frac{n}{2}sin(\frac{72}{n})$$
 * evidence
 * $$lim_{n \to \infty}\frac{n}{2}. sin(\frac{72}{n})$$
 * $$\frac{1}{2} lim_{n \to \infty}n. sin(\frac{72}{n})$$
 * $$\frac{1}{2} lim_{n \to \infty}\frac{ sin(\frac{72}{n})}{\frac{1}{n}}$$
 * $$\frac{1}{2} lim_{n \to \infty}\frac{ sin(\frac{72}{n})}{\frac{1}{n}}.\frac{72}{72}$$
 * $$\frac{72}{2}lim_{n \to \infty}\frac{sin(\frac{72}{n})}{\frac{72}{n}}$$
 * $$\frac{72}{n}=t$$
 * $$n \to \infty$$
 * $$t \rightarrow \ 0$$
 * $$\frac{72}{2}lim_{t \to \mathbf{0}} \frac{sin t}{t}=\frac{72}{2}$$
 * $$\frac{72}{2}=\frac{\pi}{5} (rad)$$
 * $$\frac{\pi}{8}$$
 * $$\frac{\pi}{8}=\frac{n}{2}sin(\frac{45}{n})$$
 * $$\frac{\pi}{8}$$
 * $$\frac{\pi}{8}=\frac{n}{2}sin(\frac{45}{n})$$



The following is another new form of PI, which I found in 2004
 * $$\pi=n.\sin(\frac{180}{n}) $$
 * evidence
 * $$\ lim_{n \to \infty}n. sin(\frac{180}{n})$$
 * $$\ lim_{n \to \infty}\frac{ sin(\frac{180}{n})}{\frac{1}{n}}$$
 * $$\ lim_{n \to \infty}\frac{ sin(\frac{180}{n})}{\frac{1}{n}}.\frac{180}{180}$$
 * $$180\lim_{m \to \infty}\frac{sin(\frac{180}{n})}{\frac{180}{n}}$$
 * $$\frac{180}{n}=t$$
 * $$n \to \infty$$
 * $$t \rightarrow \ 0$$
 * $$180\lim_{t \to \mathbf{0}} \frac{sin t}{t}=180^0$$
 * $$180^0=\pi (rad)$$
 * $$\frac{\pi}{2}$$
 * $$\frac{\pi}{2}=n.\sin(\frac{90}{n}) $$
 * $$\frac{\pi}{2}=n.\sin(\frac{90}{n}) $$


 * $$\frac{\pi}{3}$$
 * $$\frac{\pi}{3}=n.\sin(\frac{60}{n}) $$
 * $$\frac{\pi}{3}=n.\sin(\frac{60}{n}) $$




 * $$\frac{\pi}{5}$$
 * $$\frac{\pi}{5}=n.\sin(\frac{36}{n}) $$

The following is another new form of PI, which I found in 2004
 * $$\pi=360.m.sin(\frac{1}{2m})$$
 * $$ \lim_{m \to \infty} 360.m.sin(\frac{1}{2m})$$
 * $$360\lim_{m \to \infty}m.sin(\frac{1}{2m}) $$
 * $$360\lim_{m \to \infty}\frac{sin(\frac{1}{2m})}{\frac{1}{m}}$$
 * $$360\lim_{m \to \infty}\frac{sin(\frac{1}{2m})}{\frac{1}{m}}\frac{\frac{1}{2}}{\frac{1}{2}}$$
 * $$360.\frac{1}{2}\lim_{m \to \infty}\frac{sin(\frac{1}{2m})}{\frac{1}{2m}}$$
 * $$\frac{1}{2m}=t$$
 * $$m \to \infty$$
 * $$t \rightarrow \ 0$$
 * $$180\lim_{t \to \mathbf{0}} \frac{sin t}{t}=180^0$$
 * $$180^0=\pi (rad)$$

For the numerical values ​​of PI recommend free extra precision calculator Harry-J-Smith XP,XM,….. recommend m=1.0E+10000000 to the success of the calculator you need to install netframework2.0

(Рајко Велимировић (talk) 10:02, 21 December 2011 (UTC))