Keeb kwm ntawm trigonometry yog inextricably txuas nrog astronomy, vim hais tias nws yog los daws cov teeb meem ntawm cov kev tshawb fawb no uas cov kws tshawb fawb thaum ub pib kawm txog qhov sib piv ntawm ntau qhov ntau hauv daim duab peb sab.
Hnub no, trigonometry yog ib qho microsection ntawm kev ua lej uas kawm txog kev sib raug zoo ntawm qhov tseem ceeb ntawm cov ces kaum thiab qhov ntev ntawm ob sab ntawm daim duab peb sab, nrog rau kev txheeb xyuas cov algebraic tus kheej ntawm trigonometric functions.
Lub sij hawm "trigonometry"
Lub sij hawm nws tus kheej, uas tau muab nws lub npe rau cov ceg ntawm kev ua lej, tau pom thawj zaug hauv lub npe ntawm phau ntawv los ntawm German mathematician Pitiscus hauv 1505. Lo lus "trigonometry" yog Greek keeb kwm thiab txhais tau tias "Kuv ntsuas ib daim duab peb sab." Yuav kom meej meej, peb tsis tau tham txog qhov ntsuas ntsuas ntawm daim duab no, tab sis hais txog nws txoj kev daws teeb meem, uas yog, txiav txim siab qhov tseem ceeb ntawm nws cov ntsiab lus tsis paub siv cov neeg paub.
Cov ntaub ntawv dav dav txog trigonometry
Keeb kwm ntawm trigonometry pib ntau tshaj ob txhiab xyoo dhau los. Thaum pib, nws qhov tshwm sim yog txuam nrog qhov yuav tsum tau ua kom pom tseeb qhov piv ntawm cov ces kaum thiab sab ntawm daim duab peb sab. Nyob rau hauv tus txheej txheem ntawm kev tshawb fawb, nws muab tawm hais tias tus lejqhov kev qhia ntawm cov piv txwv no yuav tsum tau qhia txog cov haujlwm tshwj xeeb trigonometric, uas yog thawj zaug tau kos los ua cov lej.
Rau ntau yam kev tshawb fawb ntsig txog lej, nws yog keeb kwm ntawm trigonometry uas muab lub zog rau kev loj hlob. Lub hauv paus chiv keeb ntawm cov chav ntsuas ntawm cov kaum sab xis (degrees), cuam tshuam nrog kev tshawb fawb ntawm cov kws tshawb fawb ntawm Ancient Babylon, yog raws li kev sib deev ntawm cov lej, uas tau ua rau cov lej niaj hnub siv hauv ntau cov kev tshawb fawb.
Nws tau xav tias trigonometry pib muaj nyob ua ib feem ntawm astronomy. Tom qab ntawd nws pib siv rau hauv architecture. Thiab dhau sij hawm, qhov tsim nyog ntawm kev siv cov kev tshawb fawb no hauv ntau yam kev ua haujlwm ntawm tib neeg tau tshwm sim. Cov no yog, tshwj xeeb, astronomy, hiav txwv thiab huab cua navigation, acoustics, optics, electronics, architecture thiab lwm yam.
Trigonometry hauv cov hnub nyoog ntxov
Qhia los ntawm cov ntaub ntawv ntawm kev muaj sia nyob ntawm kev tshawb fawb, cov kws tshawb fawb tau xaus lus tias keeb kwm ntawm qhov tshwm sim ntawm trigonometry yog txuam nrog kev ua haujlwm ntawm Greek astronomer Hipparchus, uas thawj zaug xav txog kev nrhiav txoj hauv kev los daws cov duab peb sab (spherical). Nws cov ntawv sau rov qab rau xyoo pua 2 BC.
Tsis tas li ntawd, ib qho ntawm cov kev ua tiav tseem ceeb ntawm lub sijhawm ntawd yog kev txiav txim siab ntawm qhov piv ntawm ob txhais ceg thiab hypotenuse hauv txoj cai peb sab, uas tom qab ntawd los ua lub npe hu ua Pythagorean theorem.
Keeb kwm ntawm txoj kev loj hlob ntawm trigonometry nyob rau hauv ancient Greece yog txuam nrog lub npe ntawm lub astronomer Ptolemy - tus sau ntawm lub geocentric system ntawm lub ntiaj teb no, uas dominated.rau Copernicus.
Greek astronomers tsis paub sines, cosines thiab tangents. Lawv siv cov ntxhuav los nrhiav tus nqi ntawm chord ntawm ib lub voj voog uas siv cov arc rho tawm. Cov units rau ntsuas chord yog degrees, feeb thiab vib nas this. Ib lub degree yog sib npaug rau ib rau caum ntawm lub vojvoog.
Tsis tas li ntawd, cov kev tshawb fawb ntawm cov Greeks thaum ub tau txhim kho txoj kev loj hlob ntawm lub voj voog trigonometry. Hauv particular, Euclid nyob rau hauv nws "Cov Ntsiab Cai" muab ib tug theorem ntawm qhov tsis tu ncua ntawm qhov sib piv ntawm qhov ntim ntawm cov khoom sib txawv. Nws cov hauj lwm hauv cheeb tsam no tau dhau los ua ib hom kev txhawb nqa hauv kev txhim kho cov kev paub txog kev paub. Cov no yog, tshwj xeeb, thev naus laus zis ntawm cov cuab yeej astronomical, txoj kev xav ntawm cartographic projections, lub celestial coordinate system, thiab lwm yam.
Hnub Nyoog Nruab Nrab: kev tshawb fawb los ntawm cov kws tshawb fawb Indian
Indian medieval astronomers tau ua tiav tseem ceeb. Kev tuag ntawm kev tshawb fawb thaum ub nyob rau xyoo pua 4 ua rau lub hauv paus ntawm kev ua lej txav mus rau Is Nrias teb.
Keeb kwm ntawm trigonometry raws li ib feem ntawm kev qhia lej pib nyob rau Nruab Nrab Hnub nyoog. Nws yog thaum ntawd cov kws tshawb fawb hloov chords nrog sines. Qhov kev tshawb pom no ua rau nws ua tau los qhia txog cov haujlwm ntsig txog kev kawm ntawm ob sab thiab cov kaum sab xis ntawm daim duab peb sab. Qhov ntawd yog, nws yog thaum ntawd trigonometry pib cais tawm ntawm astronomy, tig mus rau hauv ib ceg ntawm kev ua lej.
thawj lub rooj ntawm kev txhaum nyob rau hauv Aryabhata, lawv tau kos los ntawm 3o, 4o, 5 o . Tom qab ntawd, cov ncauj lus ntxaws ntxaws ntawm cov ntxhuav tau tshwm sim: tshwj xeeb, Bhaskara muab lub rooj ntawm sines los ntawm1o.
Thawj qhov kev kho mob tshwj xeeb ntawm trigonometry tau tshwm sim hauv X-XI caug xyoo. Nws tus kws sau ntawv yog tus kws tshawb fawb Central Asian Al-Biruni. Thiab nyob rau hauv nws txoj hauj lwm tseem ceeb "Canon Mas'ud" (phau ntawv III), tus sau medieval mus txawm tob rau hauv trigonometry, muab ib lub rooj ntawm sines (nrog ib kauj ruam ntawm 15 ') thiab ib lub rooj ntawm tangents (nrog ib kauj ruam ntawm 1 °.).
Keeb kwm ntawm kev txhim kho trigonometry hauv Europe
Tom qab kev txhais lus ntawm Arabic treatises rau hauv Latin (XII-XIII c), feem ntau ntawm cov tswv yim ntawm Indian thiab Persian kws tshawb fawb tau qiv los ntawm European science. Thawj qhov hais txog trigonometry hauv Tebchaws Europe hnub rov qab mus rau xyoo pua 12th.
Raws li cov kws tshawb fawb, keeb kwm ntawm trigonometry hauv Tebchaws Europe yog txuam nrog lub npe ntawm tus neeg Askiv Askiv Richard Wallingford, uas tau los ua tus sau ua haujlwm "Four treatises on direct and reversed chords." Nws yog nws txoj haujlwm uas tau los ua thawj txoj haujlwm uas tau mob siab rau tag nrho trigonometry. Los ntawm lub xyoo pua 15th, ntau tus kws sau ntawv hais txog cov haujlwm trigonometric hauv lawv cov ntawv sau.
History of trigonometry: Niaj hnub nimno
Tam sim no, cov kws tshawb fawb feem ntau pib paub qhov tseem ceeb ntawm trigonometry tsis yog hauv astronomy thiab astrology xwb, tab sis kuj nyob rau lwm qhov chaw ntawm lub neej. Qhov no yog, ua ntej ntawm tag nrho cov, artillery, optics thiab navigation nyob rau hauv ntev-ntev hiav txwv voyages. Yog li ntawd, nyob rau hauv lub thib ob ib nrab ntawm lub xyoo pua 16th, lub ntsiab lus no nyiam ntau tus neeg tseem ceeb ntawm lub sijhawm ntawd, suav nrog Nicolaus Copernicus, Johannes Kepler, Francois Vieta. Copernicus tau mob siab rau ob peb tshooj rau trigonometry hauv nws cov lus qhuab qhia Ntawm Kev Tawm Tsam ntawm Celestial Spheres (1543). Ib me ntsis tom qab, hauv 60sXVI xyoo pua, Retik - tus tub ntxhais kawm ntawm Copernicus - muab kaum tsib tus lej trigonometric cov lus hauv nws txoj haujlwm "Lub Optical Part of Astronomy".
François Viète nyob rau hauv "Mathematical Canon" (1579) muab cov lus qhia ntxaws thiab ua tau zoo, txawm tias tsis muaj pov thawj, tus yam ntxwv ntawm lub dav hlau thiab lub voj voog trigonometry. Thiab Albrecht Dürer yog tus uas yug tau lub sinusoid.
Kev Ua Zoo ntawm Leonhard Euler
Muab trigonometry cov ntsiab lus niaj hnub thiab zoo li yog qhov zoo ntawm Leonhard Euler. Nws treatise Introduction to the Analysis of Infinites (1748) muaj lub ntsiab lus ntawm lo lus "trigonometric functions" uas yog sib npaug rau niaj hnub no. Yog li, tus kws tshawb fawb no muaj peev xwm txiav txim siab cov haujlwm hloov pauv. Tab sis qhov ntawd tsis yog tag nrho.
Kev txiav txim siab ntawm cov haujlwm trigonometric ntawm tag nrho cov kab tau ua tau ua tsaug rau Euler cov kev tshawb fawb tsis yog tso cai rau cov ces kaum tsis zoo xwb, tab sis kuj cov kaum ntau dua 360 °. Nws yog nws thawj zaug ua pov thawj hauv nws cov hauj lwm uas cov cosine thiab tangent ntawm lub kaum sab xis yog qhov tsis zoo. Kev nthuav dav ntawm ntau lub zog ntawm cosine thiab sine kuj tau los ua qhov zoo ntawm tus kws tshawb fawb no. Txoj kev xav dav dav ntawm trigonometric series thiab kev kawm ntawm kev sib koom ua ke ntawm cov txiaj ntsig tau tsis yog cov khoom ntawm Euler txoj kev tshawb fawb. Txawm li cas los xij, thaum ua haujlwm los daws cov teeb meem cuam tshuam, nws tau tshawb pom ntau yam hauv cheeb tsam no. Nws yog ua tsaug rau nws txoj haujlwm uas keeb kwm ntawm trigonometry txuas ntxiv mus. Luv luv hauv nws cov ntawv sau, nws kuj tau chwv txog cov teeb meem ntawm lub voj voog trigonometry.
Teb daim ntawv thovtrigonometry
Trigonometry tsis yog siv science; hauv lub neej txhua hnub, nws cov teeb meem tsis tshua siv. Txawm li cas los xij, qhov tseeb no tsis txo qis nws qhov tseem ceeb. Ib qho tseem ceeb heev, piv txwv li, yog cov txheej txheem ntawm triangulation, uas tso cai rau cov astronomers ntsuas qhov kev ncua deb ntawm cov hnub qub nyob ze thiab tswj satellite navigation systems.
Trigonometry kuj tseem siv rau hauv kev taw qhia, suab paj nruag txoj kev xav, acoustics, optics, kev tsom xam nyiaj txiag kev lag luam, hluav taws xob, kev xav txog qhov tshwm sim, kev txheeb cais, biology, tshuaj (piv txwv li, hauv kev txiav txim siab ultrasound xeem, ultrasound thiab xam tomography), kws tshuaj, chemistry, theory tooj, seismology, meteorology, oceanology, cartography, ntau ceg ntawm physics, topography thiab geodesy, architecture, phonetics, economics, electronic engineering, mechanical engineering, computer graphics, crystallography, thiab lwm yam. Cov keeb kwm ntawm trigonometry thiab nws lub luag hauj lwm nyob rau hauv lub kawm txog natural and mathematical sciences yog kawm thiab txog niaj hnub no. Tej zaum yav tom ntej yuav muaj ntau qhov chaw ntawm nws daim ntawv thov.
Keeb kwm ntawm keeb kwm ntawm cov ntsiab lus yooj yim
Keeb kwm ntawm qhov tshwm sim thiab kev loj hlob ntawm trigonometry muaj ntau tshaj li ib puas xyoo. Kev taw qhia txog cov ntsiab lus uas tsim lub hauv paus ntawm ntu no ntawm kev tshawb fawb lej kuj tseem tsis tau sai.
Yog li, lub tswv yim ntawm "sine" muaj keeb kwm ntev heev. Cov lus piav qhia ntawm ntau qhov sib piv ntawm ntu ntawm daim duab peb sab thiab lub voj voog muaj nyob hauv cov haujlwm tshawb fawb txij li xyoo pua 3 BC. Ua haujlwmxws li cov kws tshawb fawb thaum ub zoo li Euclid, Archimedes, Apollonius ntawm Perga, twb muaj thawj cov kev tshawb fawb ntawm cov kev sib raug zoo no. Cov kev tshawb pom tshiab xav tau qee qhov kev qhia meej meej. Yog li, tus kws tshawb fawb Indian Aryabhata muab chord lub npe "jiva", lub ntsiab lus "bowstring". Thaum cov ntawv lej Arabic tau muab txhais ua Latin, lo lus tau hloov los ntawm ib qho kev sib raug zoo (piv txwv li "khoov").
Lo lus "cosine" tshwm sim ntau tom qab. Lo lus no yog luv luv ntawm lo lus Latin "ntxiv sine".
Qhov tshwm sim ntawm tangents txuas nrog kev txiav txim siab ntawm qhov teeb meem ntawm kev txiav txim siab ntev ntawm tus duab ntxoov ntxoo. Lo lus "tangent" tau qhia nyob rau hauv lub xyoo pua 10th los ntawm Arab mathematician Abul-Wafa, uas tau sau thawj lub rooj rau kev txiav txim tangents thiab cotangents. Tab sis cov kws tshawb fawb European tsis paub txog cov kev ua tiav no. Tus German mathematician thiab astronomer Regimontan rediscovers cov ntsiab lus nyob rau hauv 1467. Cov pov thawj ntawm tangent theorem yog nws qhov zoo. Thiab lo lus no txhais tau tias "kev txhawj xeeb."
