Рајко Велимировић

BROJ PI

Postoji puno obrazaca za broj Pi.,možete ih pogledati [1] Broj Pi nema samo krug već čitav niz geometrijskih slika počev trougao,četvorougao, petougao itd 2004 god. Pronašao sam jedinstveni obrazac za sve PI brojeve:

$\pi _{n}={\frac {n}{2}}.sin({\frac {360}{n}})$

Postoje dva rešenja zavisno kako računamo sinus 1)za sinus u stepenima dobiće se Pi 2)za sinus u radijanima dobija se 180 ova osobina novog obrasca je ista kao i kod starih trigonometrijskih obrazaca za broj Pi ali obratno.Misli se na PI kruga.

n= {3,4,5,6........., beskonačno}

n=3 radi se o jednakostraničnom trouglu

$\pi _{3}={\frac {3}{2}}\sin({\frac {360}{3}})$

$\pi _{3}=1.299038105676657970145584756129404275207103940357785471041855234588949762681600027810859640067936431756719606061555027272477458821249214359108741634070780236396069923926729186550981115075813506192448761484257450177492164988004191001691378478420285348745275896191014287489903034894542100737760629932038067463761097204575710701640392743812359372142969874748342368381216535847874688010418122988440891548250968547863765249583132380335670783737076566229143676252517640533358216521993217274347815376373554674619972363212265589806452991554023153186737988373546994737255879027157790435027172484077780448014436830076173399466281730856066512557064076779745703123294923844606990748216673347110878468792671641139677682025389606048286479258495794767292154648948115366998523815015896613116458830885964900880276393737512516970811610634553457857999822195125972183078771421332595361479060232409486838841413231534260410498721433971667217832916432192792394148303696921223601631524640287512203987208937787304127.......$

Površina jednakostraničnog trougla biće

$R^{2}\pi _{3}$

R -Poluprečnik opisane kružnice oko jednakostraničnog trougla

Obim jednakostraničnog trougla biće

$O={\frac {2R\pi _{3}}{\cos {\frac {\alpha }{2}}}}:\alpha ={\frac {360}{3}}$

n=4 radi se o kvadratu

$\pi _{4}=2$

Površina kvadrata biće

$R^{2}\pi _{4}$

R -Poluprečnik opisane kružnice oko kvadrata

Obim kvadrata biće

$O={\frac {2R\pi _{4}}{\cos {\frac {\alpha }{2}}}}:\alpha ={\frac {360}{4}}$

tako dalje petougao,šestougao .......

$\pi _{5}=2.37764129073788393029109833344845535851424658531437555611826411107538292521298375429698202742702845418974929387850751052246766548442424981578803067578295594696444359492600258179554275230250379513151001423802924795917183008219605036301495148507880140322781045987090044531881885084349467130129921084258543849815492459549445122232219143670592470372285079329288667825627445956811065730153458201304781755172126659167760259062780734991737248477967564240583159502841310815397363785669533897013826055892072659996199853464482719029151537061472938008846993548889178745205678459542206169347872570742004980802724766701370391337077603044283323962631433827549038847681729685293628396669902392009519093871640782190843048473742646406880592139994355352920382189607783603508041226848456648980006775286567800517085352418736253370309354164490169821718538097751802464867296154739841155499816959251406543633373361369339814369337895730123108013172623915093695955211484250762466097770012856809360256952109203831071625$

$\pi _{6}=2.59807621135331594029116951225880855041420788071557094208371046917789952536320005562171928013587286351343921212311005454495491764249842871821748326814156047279213984785345837310196223015162701238489752296851490035498432997600838200338275695684057069749055179238202857497980606978908420147552125986407613492752219440915142140328078548762471874428593974949668473676243307169574937602083624597688178309650193709572753049916626476067134156747415313245828735250503528106671643304398643454869563075274710934923994472642453117961290598310804630637347597674709398947451175805431558087005434496815556089602887366015234679893256346171213302511412815355949140624658984768921398149643334669422175693758534328227935536405077921209657295851699158953458430929789623073399704763003179322623291766177192980176055278747502503394162322126910691571599964439025194436615754284266519072295812046481897367768282646306852082099744286794333443566583286438558478829660739384244720326304928057502440797441787557460825459$

dokaz

$\pi ={\frac {n}{2}}.sin({\frac {360}{n}})$

$lim_{n\to \infty }{\frac {n}{2}}.sin({\frac {360}{n}})$

${\frac {1}{2}}lim_{n\to \infty }n.sin({\frac {360}{n}})$

${\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {360}{n}})}{\frac {1}{n}}}$

${\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {360}{n}})}{\frac {1}{n}}}.{\frac {360}{360}}$

${\frac {360}{2}}lim_{n\to \infty }{\frac {sin({\frac {360}{n}})}{\frac {360}{n}}}$

${\frac {360}{n}}=t$

$t\rightarrow \ 0$

$180\lim _{t\to \mathbf {0} }{\frac {sint}{t}}=180^{0}$

$180^{0}=\pi (rad)$

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Još jedan obrazac 2004 god

Ista osobina obrasca kako je već rečeno za sinus (stepeni,radijani)

$\pi =180.m.sin({\frac {1}{m}})$

$\lim _{m\to \infty }180.m.sin({\frac {1}{m}})$

$180\lim _{m\to \infty }m.sin({\frac {1}{m}})$

$180\lim _{m\to \infty }{\frac {sin({\frac {1}{m}})}{\frac {1}{m}}}$

${\frac {1}{m}}=t$

$m\to \infty$

$t\rightarrow \ 0$

$180\lim _{t\to \mathbf {0} }{\frac {sint}{t}}=180^{0}$

$180^{0}=\pi (rad)$

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${\frac {\pi }{2}}$ biće

${\frac {\pi }{2}}={\frac {n}{2}}sin({\frac {180}{n}})$

dokaz

$lim_{n\to \infty }{\frac {n}{2}}.sin({\frac {180}{n}})$

${\frac {1}{2}}lim_{n\to \infty }n.sin({\frac {180}{n}})$

${\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {180}{n}})}{\frac {1}{n}}}$

${\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {180}{n}})}{\frac {1}{n}}}.{\frac {180}{180}}$

$90\lim _{m\to \infty }{\frac {sin({\frac {1}{m}})}{\frac {1}{m}}}$

${\frac {180}{n}}=t$

$n\to \infty$

$t\rightarrow \ 0$

$90\lim _{t\to \mathbf {0} }{\frac {sint}{t}}=90^{0}$

$90^{0}={\frac {\pi }{2}}(rad)$

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${\frac {\pi }{3}}$ biće

${\frac {\pi }{3}}={\frac {n}{2}}sin({\frac {120}{n}})$

dokaz

$lim_{n\to \infty }{\frac {n}{2}}.sin({\frac {120}{n}})$

${\frac {1}{2}}lim_{n\to \infty }n.sin({\frac {120}{n}})$

${\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {120}{n}})}{\frac {1}{n}}}$

${\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {120}{n}})}{\frac {1}{n}}}.{\frac {120}{120}}$

$60\lim _{n\to \infty }{\frac {sin({\frac {120}{n}})}{\frac {120}{n}}}$

${\frac {120}{n}}=t$

$n\to \infty$

$t\rightarrow \ 0$

$60\lim _{t\to \mathbf {0} }{\frac {sint}{t}}=60^{0}$

$60^{0}={\frac {\pi }{3}}(rad)$

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${\frac {\pi }{4}}$ biće

${\frac {\pi }{4}}={\frac {n}{2}}.sin({\frac {90}{n}})$

dokaz

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

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

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

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

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

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

$n\to \infty$

$t\rightarrow \ 0$

$45\lim _{t\to \mathbf {0} }{\frac {sint}{t}}=45^{0}$

$45^{0}={\frac {\pi }{4}}(rad)$

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${\frac {\pi }{5}}$ biće

${\frac {\pi }{5}}={\frac {n}{2}}sin({\frac {72}{n}})$

dokaz

$lim_{n\to \infty }{\frac {n}{2}}.sin({\frac {72}{n}})$

${\frac {1}{2}}lim_{n\to \infty }n.sin({\frac {72}{n}})$

${\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {72}{n}})}{\frac {1}{n}}}$

${\frac {1}{2}}lim_{n\to \infty }{\frac {sin({\frac {72}{n}})}{\frac {1}{n}}}.{\frac {72}{72}}$

${\frac {72}{2}}lim_{n\to \infty }{\frac {sin({\frac {72}{n}})}{\frac {72}{n}}}$

${\frac {72}{n}}=t$

$n\to \infty$

$t\rightarrow \ 0$

${\frac {72}{2}}lim_{t\to \mathbf {0} }{\frac {sint}{t}}={\frac {72}{2}}$

${\frac {72}{2}}={\frac {\pi }{5}}(rad)$

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${\frac {\pi }{8}}$ biće

${\frac {\pi }{8}}={\frac {n}{2}}sin({\frac {45}{n}})$

itd

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Još jedan obrazac 2004 god već je rečeno stepeni,radijani

$\pi =n.\sin({\frac {180}{n}})$

dokaz

$\ lim_{n\to \infty }n.sin({\frac {180}{n}})$

$\ lim_{n\to \infty }{\frac {sin({\frac {180}{n}})}{\frac {1}{n}}}$

$\ lim_{n\to \infty }{\frac {sin({\frac {180}{n}})}{\frac {1}{n}}}.{\frac {180}{180}}$

$180\lim _{m\to \infty }{\frac {sin({\frac {180}{n}})}{\frac {180}{n}}}$

${\frac {180}{n}}=t$

$n\to \infty$

$t\rightarrow \ 0$

$180\lim _{t\to \mathbf {0} }{\frac {sint}{t}}=180^{0}$

$180^{0}=\pi (rad)$

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${\frac {\pi }{2}}$ biće

${\frac {\pi }{2}}=n.\sin({\frac {90}{n}})$

postoji granična vrednost

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${\frac {\pi }{3}}$ biće

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

takođe postoji granična vrednost

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preskačemo

${\frac {\pi }{5}}$ biće

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

ima graničnu vrednost

itd

Još jedan obrazac 2004 god već je rečeno stepeni,radijani

$\pi =360.m.sin({\frac {1}{2m}})$

$\lim _{m\to \infty }360.m.sin({\frac {1}{2m}})$

$360\lim _{m\to \infty }m.sin({\frac {1}{2m}})$

$360\lim _{m\to \infty }{\frac {sin({\frac {1}{2m}})}{\frac {1}{m}}}$

$360\lim _{m\to \infty }{\frac {sin({\frac {1}{2m}})}{\frac {1}{m}}}{\frac {\frac {1}{2}}{\frac {1}{2}}}$

$360.{\frac {1}{2}}\lim _{m\to \infty }{\frac {sin({\frac {1}{2m}})}{\frac {1}{2m}}}$

${\frac {1}{2m}}=t$

$m\to \infty$

$t\rightarrow \ 0$

$180\lim _{t\to \mathbf {0} }{\frac {sint}{t}}=180^{0}$

$180^{0}=\pi (rad)$

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(Rajko Velimirović (razgovor) 07:20, 21. decembar 2011. (CET))