Keeb kwm ntawm trigonometry: emergence thiab kev loj hlob

Trigonometry keeb kwm yog inextricably txuas nrog astronomy, vim hais tias nws yog kom tau raws li txoj kev sib tw ntawm no ancient science, zaum pib mus tshawb cov kev sib raug zoo sib txawv ntawm cov tsiaj ntawv nyob rau hauv ib tug daim duab peb sab.

Rau hnub tim, trigonometry yog ib tug Micro-kev kawm txog zauv, kawm kev sib raug zoo ntawm cov qhov tseem ceeb ntawm lub ces kaum thiab lub lengths ntawm lub sab ntawm lub duab peb ceg, raws li zoo raws li kev soj ntsuam txog nrog lub tsom xam ntawm algebraic tawv ntawm trigonometric functions.

Lub sij hawm "trigonometry"

Lub sij hawm, uas tau muab lub npe rau qhov seem ntawm kev kawm txog zauv, twb xub muaj nyob rau hauv lub npe ntawm phau ntawv sau los ntawm tus German mathematician Pitiskusa nyob rau hauv 1505. Lo lus "trigonometry" yog ib yam uas Greek keeb kwm thiab txhais tau tias "los ntsuas ib tug daim duab peb sab." Yuav kom ua tau ntau leej, nws tsis yog ib qhov dav ntawm no daim duab, tab sis hais txog nws cov kev txiav txim siab, uas yog, txiav txim rau qhov tseem ceeb ntawm nws tsis paub hais tias cov ntsiab siv paub.

General ntaub ntawv hais txog trigonometry

Trigonometry keeb kwm pib ntau tshaj ob millennia dhau los. Chiv, nws tshwm sim tau txuam nrog qhov yuav tsum tau los mus txiav txim lub ces kaum ntawm ib tug daim duab peb sab thiab cov nam ratio. Thaum lub sij hawm cov kev tshawb fawb nws los ua tseeb hais tias cov zauv qhia ntawm cov kev sib raug zoo yuav tsum tau cov kev taw qhia tshwj xeeb trigonometric functions, uas tau Ameslikas ua tawm raws li ib tug hais txog zauv rooj.

Rau ntau phooj ywg sciences nrog lej impetus mus rau lub kev loj hlob ntawm trigonometry yog precisely keeb kwm. Keeb kwm lub ntsuas units (degrees) txuam nrog cov kev tshawb fawb zaum ntawm ancient Babylon, yog raws li nyob rau hauv lub sexagesimal system ntawm xam, uas muab ntxiv rau lub neej tshiab nimno tus zauv, siv nyob rau hauv ntau applied sciences.

Nws yog assumed tias muas thaum chiv thawj muaj raws li ib feem ntawm trigonometry astronomy. Ces nws pib yuav tsum tau siv nyob rau hauv architecture. Thiab thaum lub sij hawm, muaj ib lub usefulness ntawm no science nyob rau hauv ntau yam teb ntawm tib neeg kev ua si. Qhov no, nyob rau hauv particular, astronomy, cov hiav txwv thiab cua navigation, acoustics, optics, electronics, architecture thiab lwm tus neeg.

Trigonometry nyob rau hauv thaum ntxov centuries

Coj los ntawm cov ntaub ntawv nyob rau hauv lub tseem muaj sia nyob relics, tus neeg tshawb fawb xaus lus tias lub keeb kwm ntawm lub rov tshwm sim ntawm trigonometry yog txuam nrog rau cov chaw ua hauj lwm ntawm Greek astronomer Hipparchus, uas thawj xav rau nrhiav txoj kev los daws tau cov vajvoos peb sab (kheej kheej). Nws tej hauj lwm mus rau lub 2nd century BC.

Nws kuj yog ib tug ntawm cov tseem ceeb tshaj plaws tau ntawm lub sij hawm uas yog los mus txiav txim tus piv ntawm ob txhais ceg thiab lub hypotenuse nyob rau hauv ib txoj cai daim duab peb sab uas tom qab ua lub npe hu ua Pythagorean Theorem.

Lub keeb kwm ntawm kev loj hlob ntawm trigonometry nyob rau hauv ancient Greece yog txuam nrog lub npe ntawm astronomer Ptolemy - tus neeg sau ntawm lub geocentric system ntawm lub ntiaj teb no uas prevailed ua ntej Copernicus.

Greek astronomers twb tsis paub tias sine, cosine thiab tangent. Lawv siv lub rooj mus nrhiav tau tus nqi ntawm lub chord ntawm lub vajvoog siv ib tug contractible arc. Cov units ntawm kev ntsuas tau chord degrees, feeb thiab vib nas this. Ib tug degree yog sib npaug zos rau sixtieth feem voos kheej-kheej.

Tsis tas li ntawd, cov kev tshawb fawb ntawm cov ancient Greeks nce kev loj hlob ntawm kheej kheej trigonometry. Nyob rau hauv kev, Euclid nyob rau hauv nws "hais" theorem ua rau regularities ratios volume ntawm cov khoom ntawm ntau yam diameters. Nws tej hauj lwm nyob hauv daim teb tau los ua ib tug zoo ntawm impulse rau txoj kev loj hlob ntawm ntau thiab nyob ib sab chaw ntawm kev txawj ntse. Qhov no, nyob rau hauv particular, lub cuab yeej ntawm astronomical seev, lub hom phiaj ntawm daim ntawv qhia projections, xilethi-aus tswj system, thiab hais txog. D.

Nrab Hnub nyoog: kawm txog cov Indian zaum

Tseem ceeb kev kawm tiav medieval Indian astronomers. Cov kev tuag ntawm cov ancient science nyob rau hauv lub IV xyoo pua coj mus rau lub hloov nyob rau hauv txoj kev loj hlob ntawm kev kawm txog zauv nyob rau hauv Is Nrias teb.

Lub keeb kwm ntawm lub rov tshwm sim ntawm trigonometry raws li ib tug nyias muaj nyias ib seem ntawm cov zauv ce pib nyob rau hauv cov Nrab Hnub nyoog. Yog tias thaum tus zaum hloov lub chord sinuses. Qhov no discovery tau tso cai rau nkag mus rau lub zog hais txog kev tshawb fawb sab thiab cov ces kaum ntawm ib txoj cai daim duab peb sab. Hais tias yog, nws yog ces pib cais ib tug trigonometry los ntawm astronomy, ua ib ceg ntawm kev ua zauv.

Tus thawj lub rooj ntawm sines twb nyob rau hauv Aryabhata, lawv twb muaj nyob rau hauv 3 ^of 4 ^of 5 ^on. Tom qab ntawd, muaj ncauj lus kom ntxaws versions ntawm cov ntxhuav: nyob rau hauv particular, Bhaskara coj los ntawm sine rooj 1 ^rau.

Tus thawj tshwj xeeb treatise rau trigonometry nyob rau hauv X-XI caug xyoo. Nws sau yog tus Central Esxias kawm al-Biruni. Ib tug medieval sau ntau deepens nyob rau hauv nws lub ntsiab ua hauj lwm "Cov Canon Mas'ud" (Phau Ntawv III), nyob rau hauv trigonometry, ib lub rooj ntawm sines (nyob rau hauv increments ntawm 15 ') thiab ib lub rooj ntawm tangents (nyob rau hauv increments ntawm 1 °).

Lub keeb kwm ntawm kev loj hlob ntawm trigonometry nyob rau hauv cov teb chaws Europe

Tom qab cov kev hloov ntawm Arab treatises rau hauv Latin (XII-XIII c) feem ntau ntawm cov tswv yim ntawm Indian thiab Persian zaum twb borrowed European science. Tus thawj mention ntawm trigonometry mus rau lub XII xyoo pua nyob rau hauv teb chaws Europe.

Raws li kev soj ntsuam ntawm, lub keeb kwm ntawm trigonometry nyob rau hauv cov teb chaws Europe txuam nrog lub npe ntawm Englishman Richard ntawm Wallingford, uas yog tus sau ntawm tej hauj lwm "Plaub ntawm lub treatise rau qhov ncaj thiab tus inverted lub chords." Hais tias nws ua hauj lwm yog thawj ua hauj lwm uas yog nkaus devoted mus trigonometry. Los ntawm XV xyoo pua, ntau sau phau ntawv nyob rau hauv lawv writings hais cov trigonometric functions.

Keeb kwm ntawm trigonometry: Tshiab lub sij hawm

Nyob rau hauv niaj hnub lub sij hawm, feem ntau zaum los ua paub txog lub tseem ceeb ntawm trigonometry tsis tau tsuas yog nyob rau hauv astronomy thiab astrology, tab sis kuj nyob rau hauv lwm qhov chaw ntawm lub neej. Nws yog, thawj thiab foremost, artillery, optics thiab navigation ntev hiav txwv Voyages. Yog li ntawd, nyob rau hauv lub thib ob ib nrab ntawm cov XVI xyoo pua, qhov no raug tau xav ntau mas cov neeg ntawm hais tias lub sij hawm, xws li Nikolaya Kopernika, Ioganna Keplera, Fransua Vieta. Copernicus coj trigonometry ob peb tshooj ntawm nws treatise "Nyob rau hauv revolutions ntawm lub Saum Ntuj Ceeb spheres" (1543). Tom qab ntawd, nyob rau hauv lub 60s ntawm lub XVI xyoo pua, Retik - ib tug thwj tim ntawm Copernicus - ua nyob rau hauv nws "Optical Part ntawm Astronomy" pyatnadtsatiznachnye trigonometric ntxhuav.

Fransua Viet nyob rau hauv "lej canon" (1579) muab ib tug ncauj lus kom ntxaws thiab systematic, tab sis yog meej, cov yam ntxwv ntawm lub tiaj thiab kheej kheej trigonometry. Thiab Albrecht Dürer yog ib tug los ntawm leej twg yug sinusoid.

Cov hauj lwm zoo Leonarda Eylera

Muab trigonometry niaj hnub cov ntsiab lus thiab lub hom credit yog Leonarda Eylera. Nws treatise "Introduction to tus tsom xam ntawm lub infinite" (1748) muaj ib tug txhais ntawm lub sij hawm "trigonometric functions", uas yog sib npaug rau cov niaj hnub. Yog li, tus paub txog yog tau txiav txim rov qab zog. Tab sis hais tias yog tsis yog txhua txhua.

Ntsiab Txhais ntawm Kev trigonometric functions rau hauv lub tiag tiag kab tau ua tau ua tsaug rau tshawb nrhiav Euler tsis tau tsuas yog permissible tsis zoo ces kaum, tab sis lub ces kaum Bole 360 °. Nws yog thawj lub sij hawm nws tau muaj pov thawj nyob rau hauv nws sau hais tias cov cosine thiab tangent ntawm ib txoj cai lub yog tsis zoo. Expansion ntawm tag nrho cosine thiab sine kuj tsim nyog ntawm no paub txog. Cov kev xav ntawm trigonometric series thiab kawm txog lub convergence ntawm lub series tau yog tsis khoom ntawm Euler tus soj ntsuam. Txawm li cas los, ua hauj lwm nyob rau hauv kev daws ntawm lwm yam kev teeb meem, nws ua ntau yam discoveries nyob rau hauv daim teb no. Nws twb los ntawm nws ua hauj lwm tau txuas ntxiv mus los ntawm lub keeb kwm ntawm trigonometry. Luv luv nyob rau hauv nws sau nws hais txog muaj lus nug thiab kheej kheej trigonometry.

daim ntaub ntawv trigonometry

Trigonometry yog tsis muaj feem rau thov sciences, nyob rau ntawm lub neej txhua hnub nws yog tsis tshua muaj siv kev pab raws qib. Txawm li cas los, qhov no fact tsis ploj zuj zus mus nws tseem ceeb. Nws yog ib qho tseem ceeb, piv txwv li, ib tug triangulation txheej txheem uas tso cai rau astronomers rau heev meej ntsuas qhov kev ncua deb rau cov hnub qub minded thiab saib xyuas navigation satellite tshuab.

Tsis tas li ntawd, trigonometry yog siv nyob rau hauv navigation, suab paj nruag kev tshawb xav, acoustics, optics, tsom xam ntawm cov nyiaj txiag kev lag luam, electronics, muaj tseeb kev tshawb xav, statistics, biology, tshuaj (piv txwv li, nyob rau hauv deciphering ultrasound ultrasound thiab xoo tomography), pharmaceutics, Science News for KIDS, kev tshawb xav tooj, seismology, meteorology , oceanography, cartography, ntau qhov chaw ntawm physics, topography thiab geodesy, architecture, phonetics, economics, hluav taws xob engineering, mechanical engineering, computer graphics, crystallography, thiab hais txog. d. lub keeb kwm ntawm trigonometry thiab nws lub luag hauj lwm nyob rau hauv txoj kev tshawb no enii ntuj thiab xaam sciences yog kawm kom hnub no. Tej zaum nyob rau hauv lub neej yav tom ntej, nws daim ntawv sau npe yuav txawm ntau dua.

Lub hauv paus pib ntawm lub tswv yim

Lub keeb kwm ntawm lub rov tshwm sim thiab kev loj hlob ntawm trigonometry muaj ntau tshaj li ib xyoo pua. Cov kev taw qhia ntawm cov tswv yim uas ua lub hauv paus ntawm qhov seem ntawm kev kawm txog zauv, kuj yog tsis momentary.

Yog li, lub tswvyim ntawm "kev txhaum" muaj ib tug keeb kwm ntev heev. Mention ntawm lub ntau yam feem ntawm kev sib raug ntawm cov vajvoos peb sab thiab voj voog muaj nyob txawm nyob rau hauv scientific tej hauj lwm, yos los ntawm lub III xyoo pua BC. Cov hauj lwm ntawm xws zoo ancient scholars li Euclid, Archimedes, Apollonius ntawm Perga, twb muaj cov thawj txoj kev tshawb no ntawm cov kev sib raug zoo. Tshiab discoveries demanded ib tug tej yam terminological hloov. Yog li, lub Indian tus paub txog Aryabhata muab lub chord lub npe ntawm "Jiva", lub ntsiab lus "bowstring". Thaum As zauv ntawv txhais ua Latin, lub sij hawm nyob ze hloov los ntawm cov nqi sine (m. E. "Khoov").

Lo lus "cosine" nyob npaum li cas tom qab ntawd. Qhov no lub sij hawm yog ib qho tsiaj ntawv rau lub Latin kab lus "ntxiv sine".

Tshwm sim tangents txuam nrog seev suab qhov teeb meem ntawm kev txiav txim rau qhov ntev ntawm tus duab ntxoov ntxoo. Lub sij hawm "tangent" twb qhia nyob rau hauv lub xyoo pua X as mathematician Abu al-Wafa, ib feem ntawm cov thawj lub rooj los mus txiav txim lub tangent thiab cotangent. Tab sis European zaum tsis tau paub txog cov achievements. German mathematician thiab astronomer Regimontan rediscovers cov tswv yim nyob rau hauv 1467 pov thawj tangent theorem - rau nws credit. Ib tug txhais lub sij hawm raws li "kov ub kov no".